F Barranco
email: barranco@us.es

R A Broglia
email: broglia@mi.infn.it

E Vigezzi
email: vigezzi@mi.infn.it

Quantum size effects in the inner crust of neutron stars

It is generally accepted that the inner crust of neutron stars is formed by a Coulomb lattice of nuclei immersed in a gas of quasi-free neutrons. We discuss the implications of the inhomogeneity of the crust and of nuclear shell effects on the linear response and on the superfluid properties of the system, as well as on the structure of vortices.

Introduction

The first study of the structure of the inner crust of neutron stars based on detailed quantum calculations was the paper published by Negele and Vautherin in 1973 [1]. Their work was truly ahead of their times, as is demonstrated by the fact that it has received about two thirds of its about 400 citations after the year 1995, and only about 10% in the decade following its publication [2]. This reflects the theoretical effort deployed in recent times towards a detailed understanding of the complex physics of the crust. It also indicates that the basic structure of the crust as described by Negele and Vautherin -a Coulomb lattice formed by spherical nuclei immersed in a sea of quasi-free neutrons, with a lattice step that decreases going towards the core of the star -is still valid today, at least for nucleon densities up to about 0.03 fm -3 . The precise determination of the density range in which spherical nuclei are energetically favoured, as compared to other phases with different spatial structures, is an open and important issue. Also the isotopic composition of the crust depends on the details of the calculation and in particular on pairing correlations [3]. For a recent, general review of the physics of the inner crust, we refer to [4].

In the conclusions of their study, Negele and Vautherin remarked that 'the degree to which the nuclei in the free neutron regime resemble ordinary nuclei' is striking, and that 'this similarity is also manifested in the behaviour of the single-particle energies'. In the present contribution, various issues associated with the persistence of the shell structure in the low-density part of the inner crust will be discussed. Of notice that shell effects have mostly been considered in connection with the calculation of ground state energies. Here, we shall focus on the role of inhomogeneities and quantal finite-size effects concerning the linear response and the superfluidity of the crust. These effects turn out to be particularly important in the microscopic description of vortices, which, arguably, are the most direct manifestation of superfluidity of the crust in a rotating pulsar, as in atomic Fermi gases [5].

Shell effects and linear response

Going from the surface towards the center of a neutron star, the separation energy of neutrons from nuclei diminishes, until at densities larger than about 4 ×10 11 g cm -3 , neutrons start to drip out of nuclei and to occupy states in the continuum part of the spectrum. This defines the beginning of the inner crust, which extends over a density range of about three orders of magnitude, until the core of the star is reached for densities of the order of 1.4 ×10 14 g cm -3 ( or 0.08 fm -3 ), and neutron star matter becomes homogeneous.

Many studies have addressed the structure of the inner crust, using the liquid drop approximation, semiclassical methods, Hartree-Fock quantum calculations or molecular dynamics simulations. A Coulomb lattice of spherical nuclei appears generally to be the favoured phase up to about 0.03 fm -3 . Most calculations in this phase have been based on the Wigner-Seitz (WS) approximation, in which one deals with a single elementary cell of the Coulomb lattice, approximating it with a sphere. One then minimizes the energy of neutron, proton and electron matter in β-equilibrium within the WS cell, whose radius is considered as a variational parameter, determining the isotopic composition of the cell. According to the few studies which have taken the band structure of the Coulomb lattice into account [6,7], the WS approximation turns out to be rather reliable, but this is a topic that should be further investigated, in particular when the radius of the WS cell becomes smaller than about 20 fm and the results of calculations show a marked dependence on the specific boundary conditions [8].

In the following we shall present results obtained in a WS cell of radius R W S = 33.1 fm, containing 1314 neutrons and with Z = 50 protons at the center. This cell is representative of nucleon density n ≈ 1.5 × 10 13 g/cm 3 ( n ≈ 0.01 fm -3 ).

In Fig. 1 we show the neutron and proton mean fields, as well as the neutron and proton densities resulting from a self-consistent, spherical HF calculation performed with the effective interaction SLy4, imposing that the wavefunctions vanish at the edge of the cell . The neutron and proton Fermi energies lie at E F = 5.5 MeV and -35.1 MeV respectively. Notice that the neutron continuum spectrum starts at E ≈ -3.5 MeV, and that the bound neutron single-particle levels are occupied by about 130 neutrons.

In the HF calculation, we deal with discrete states in the spherical WS cell, and the potential goes to zero at the edge of the cell. It proves of interest to extrapolate the HF potential of the WS cell from the wide central region where it is about constant up to infinity. The energy dependence of the phase shifts calculated in the extrapolated neutron potential is shown in Fig. 2. One finds several resonant single-particle states in the continuum, corresponding to values of the orbital angular momentum l up to 10. The centroid and the width of the resonances are given in Table 1. The centroid is taken to be the equal to the energy at which the phase shift δ lj (E) reaches the value π/2 or 3π/2 with positive slope, while the width is obtained from the full width at half maximum of the function dδ lj (E)/dE.

As displayed in Table 1, some of the states have a rather narrow width, of the order of a few MeV. The spacing between resonances in 1364 Sn corresponds to a typical oscillator frequency ω ≈ 8 MeV, a value which is not very different from that typical of a heavy nucleus of mass number A ≈ 200 (that is, with a similar number of bound nucleons), according to the standard estimate ω ≈ 41A -1/3 MeV. These resonances can play an important role in different contexts. In particular, they determine the linear response of the nucleus immersed in the neutron sea.

In Fig. 3 the particle-hole transitions which are associated with the largest values of the strength of the quadrupole and octupole operators dU/drY 20 and dU/drY 30 are displayed; U denotes the local neutron HF field associated with the zero-range SLy4 force (similar results would be obtained using the neutron density instead of U ). In all cases, they involve unbound states whose quantum numbers {l, j} correspond to those reported in Table 1, and whose energies lie close to those of the corresponding centroids. Easy to recognize are low-lying octupole transitions, having an energy approximately given by ∆E ≈ ω, and high-lying quadrupole and octupole transitions having ∆E ≈ 2 ω and ∆E ≈ 3 ω respectively. In other words, the typical shell structure of the response observed in atomic nuclei is preserved in the inner crust, as can be seen in Fig. 4, where the octupole strength calculated in the Random Phase Approximation (RPA) is displayed. It is found (see ref. [9]) that the strength evolves in a rather continuous fashion going from atomic nuclei into the inner crust. These results should be contrasted with those obtained making use of the operator r λ , which heavily weight the free neutron part of the cell. In that case, the strength associated with the response of bound or resonant states is overwhelmed by the response of the free neutrons [10,11].

-20 ). The dashed and solid curves (in units of MeV fm -2 ) are obtained folding the unperturbed and the RPA discrete response with a Lorentzian curve having a width of 1 MeV.

-

Pairing correlations in the Wigner-Seitz cell

On the basis of BCS theory and using standard nucleon-nucleon potentials, one finds that neutron matter in the 1 S 0 channel is superfluid at the densities and at the temperatures (T ≈ 0.01 MeV) typical of the inner crust. If one ignores its inhomogeneities, the inner crust can be considered as a macroscopic piece of neutron matter, where superfluid flow can fully develop. At a density n ≈ 0.01 fm -3 (corresponding to a Fermi momentum k F ≈ 0.65 fm -1 ) one would estimate a value ξ ≈ 2 k F /mπ∆ ≈ 6.5 fm for the coherence length, using the value ∆ ≈ 2 MeV calculated in uniform matter at this density with the bare nucleon-nucleon force. This is smaller than the coherence length in the atomic nucleus (ξ ≈ 20 fm, using ∆ ≈ 1 MeV and k F ≈ 1.5 fm -1 ), where however superflow is limited by the small value of the nuclear radius.

The typical values of the coherence length in the inner crust are several times larger than the diffusivity of the nuclear potential, and the application of the results obtained in neutron matter to the actual case of the inner crust requires a careful consideration of its inhomogeneous character. Pethick and Ravenhall wrote in 1995 that since 'the spatial variation of superfluid gaps is not well understood theoretically, even in the absence of superfluid flow, the properties of vortices in such an inhomogeneous medium are even less well understood from basic theory' [12]. Much progress has been made since then concerning pairing correlations. The spatial dependence of the pairing gap has been studied by several groups at the mean field level, using parametrized potentials or selfconsistent Hartree-Fock-Bogoliubov (HFB) theory [13,14,15,16,17,18]. According to the Local Density Approximation (LDA), one expects that the gap should be suppressed in the interior of the nucleus, because there the local Fermi momentum is usually larger than in the external neutron gas. The effect is reduced in actual HFB calculations, because LDA does not take into account proximity effects: close to the nuclear surface, a Cooper pair feels both the interior of the nuclear impurity and the external free neutron superfluid. As a result, the value of the gap close to the Fermi energy is decreased by a few hundred keV compared to the value in neutron matter, leading to an increase of the specific heat of the crust which can affect the earliest phases of neutron star cooling, at least in some astrophysical scenarios [15,17,19]. Nonetheless, the influence of the inhomogeneities on the pairing gap at the mean field level is considerably smaller than typical uncertainties adscribed to the existing calculations of medium polarization effects.

Within this context one can mention the fact that two 'ab initio' calculations of the gaps in uniform neutron matter carried out recently with the help of Montecarlo techniques [20,21,22] are found to be in rather poor agreement with each other. Furthermore, one is still lacking a critical comparison and assessment of the differences existing between these calculations and studies based on the renormalization group [23] or on Brueckner and Nambu-Gorkov theory [START_REF] Sedrakian | Pairing in fermionic systems[END_REF][START_REF] Cao | [END_REF], in particular concerning the role played by collective modes.

In spite of quantitative differences, all these calculations predict a reduction of the pairing gap as compared to the BCS results. This is because polarization effects in neutron matter are dominated by the repulsion induced by spin fluctuations. This is different from what is found in atomic nuclei, in which the exchange of collective, low-lying surface modes between Cooper pair partners induces an attractive pairing interaction which enhances the gap. Also in this case, the quantitative aspects of such results are lively debated [26,27,28,29].

The inhomogeneous WS cell poses a particularly interesting challenge, due to the interplay of the free neutron and of the nuclear phase. Some calculations have used a density functional obtained interpolating between a functional adopted in neutron matter and a functional used in atomic nuclei [30]. Only exploratory calculations have directly addressed medium renormalization effects in the inner crust [31,32]. They have explicitly included the coupling with the vibrations of the systems in the WS cell (cf. Section 2), and indicate that the screening of the gap due to spin fluctuations is counterbalanced to some extent by density fluctuations associated with the dynamics of the surface of the nucleus. This leads to characteristic changes in the spatial dependence of the pairing gap close to the nuclear surface. Of notice that the effect of the vibrations of the Coulomb lattice on the pairing induced interaction has never been considered [33].

It would be important to have evidence of superfluidity in the crust directly associated with astronomical observations. As we have already mentioned, superfluidity affects the thermal properties of the crust. This can influence neutron star cooling, but simulations generally show that cooling is much more sensitive to the properties of the core [34]. However, recent observations of rapid cooling of the crust after accretion have open exciting perspectives, because this phenomenon directly depends on the properties of the crust. Simulations indicate that neutron superfluidity in the crust is indeed needed to obtain agreement with the data [35,36].

Vortex structure and pinning to nuclei

In keeping with the ansatz that neutron star matter is superfluid, the rotation of the star should induce the formation of vortices in the inner crust.

While there is a rich literature concerning vortices in condensed matter, only few studies have been devoted to the quantum structure of vortices in neutron matter in general [37] and in neutron stars in particular [38,39,40,41]. It turns out that shell structure can play an important role in a microscopic study of a vortex in the inner crust. This is because vortices are excitations based on Cooper pairs made out of single-particle states having different parity.

In what follows we shall only consider the axially symmetric case, in which the vortex axis coincides with the z-axis. The vortex can be characterized by a pairing field

∆(ρ, z, φ) = ∆(ρ, z)e iνφ , (1)
where the second factor depends only on the azimuthal angle φ about the z-axis. The index ν is the vortex number. It indicates the number of quanta of angular momentum carried by each Cooper pair. In particular, if we set ν = 0 we obtain the usual HFB equations (no vortex). Setting instead ν = 1, we can describe a vortex excitation, in which each Cooper pair carries one unit of angular momentum along the z-axis.

Experiments on superfluids indicate that it is energetically more favorable to develop an array of ν = 1 vortices rather than a few vortices carrying many quanta of angular momentum. In the following we shall assume that this is also the case in neutron stars.

When the vortex axis goes through the center of one of the nuclei building the Coulomb lattice, it is said that the vortex is pinned. The associated pairing field is axially symmetric (cf. Fig. 5), in keeping with the spherical symmetry of the nuclear mean field.

For a given value of m, the projection of the orbital angular momentum on the z-axis, and indicating the other quantum numbers by the index q, the quasiparticle amplitudes have the form

v qm (ρ, z, φ) = V qm (ρ, z)e imφ u qm (ρ, z, φ) = U qm (ρ, z)e i(m-ν)φ
(2)

with the property V q,m (ρ, z) = V q,-m+ν (ρ, z), which for ν = 1 leads to an azimuthal current given by

J(ρ, z, φ) = - i mρ qm v * qm (ρ, z, φ) ∂v qm (ρ, z, φ) ∂φ , (3)
with the associated velocity field J/n. For large values of ρ Eq. ( 3) leads to the well known Onsager profile, v = J/n = /2mρ. Aside from axial symmetry, the system has mirror symmetry with respect to the x -y plane (see Fig. 5), so that ∆(ρ, z) = ∆(ρ, -z). Applying the parity operator Π (φ → φ + π and z → -z) to the pairing field one obtains

Π∆(ρ, z)e iνφ = e iνπ ∆(ρ, z)e iνφ , (4)
implying that for ν = 1 the Cooper pairs are constructed out of single-particle levels of opposite parity. As discussed in Section 2, the persistence of the shell structure in the WS cell implies the existence of resonant states of different parity, which are separated by an energy ω ≈ 8 MeV. This hinders the formation of a vortex within the nuclear volume as compared to the uniform situation. The pairing gap associated with a vortex pinned on a spherical nucleus, obtained solving the HFB equations in the WS cell [41],

is shown by the solid line in the right panel of Fig. 6, as a function of the distance from the vortex axis in the equatorial plane of the nucleus. Also displayed are the gaps calculated: (a) in the WS cell without a vortex (ν = 0), (b) in the ground state of uniform neutron matter (ν = 0), (c) for a vortex in uniform matter (ν = 1). In all cases the same pairing force was used, namely, a density-dependent, zero-range interaction whose parameters are tuned to reproduce the gap obtained with the bare force.

It is seen that the gap of the pinned vortex is strongly suppressed not only in the interior of the nucleus, but also in a rather thick layer at the interface between the nucleus and the neutron gas. The vortex radius can be estimated as the distance from the z-axis at which ∆ reaches a value equal to 90% of its asymptotic value. In the case displayed in Fig. 5, it has a value of about 12 fm, that is, about 6 fm larger than in uniform matter (cf. the solid and the short-dashed curve).

An important issue is whether the vortex lines tend to be pinned to the nuclei forming the Coulomb lattice. If for some reason vortices were catastrophically unpinned from the nuclei, they could release angular momentum to the neutron star surface and the angular velocity of the star would show a sudden increase [42]. This might explain the irregularities (glitches) observed in the period of the radiation emitted by a large number of pulsars [43]. Consequently an important quantity which can be obtained from a microscopic study of the vortex structure is the pinning energy, namely, the difference between the energy of the system when the axis of the vortex goes through the center of the nucleus and when the axis of a vortex line is placed far from the nuclei.

According to semiclassical models [44,45,46], the pinning energy reflects a competition between the kinetic energy flow and the condensation energy. In such a competition, the size of the vortex core, defining the region in which the gap vanishes, is generally smaller than the nuclear radius; then the pinning energy may be negative, vortices being attracted by nuclei. This is in keeping with the fact that the loss of condensation energy associated with the vortex core can be less severe if the vortex goes through the nucleus, where the gap was already small in the absence of the vortex, as compared to the outer neutron gas.

The large vortex radii found in the quantum calculations strongly influence the contribution of the condensation energy and of the flow energy to the pinning energy, invalidating the semiclassical approximation, and changing the dependence of the pinning energy on density in a qualitative way. In spite of the progress accomplished in the understanding of the basic vortex structure, at present it is still not possible to calculate a reliable value of the pinning energy. This is because of the uncertainties adscribed to the effective interactions, concerning not only the influence of medium polarization effects on the pairing interaction, but also the effective mass associated with the interaction used to calculate the mean field, a quantity which affects, among other properties, the spacing between resonances.

The strength of the interaction between a segment of a vortex line and a nucleus is one of the basic elements determining the vortex dynamics in the crust. The latter constitutes a complex problem (cf. Sect. 8.3 in [4] for a review), although it is simplified to some extent because vortices can safely assumed to be isolated, their radius (of the order of the coherence length) being much smaller than the intervortex distance (of the order of 10 -4 cm for a millisecond pulsar). An attempt towards a detailed description of the interaction of a vortex line with the Coulomb lattice has been made recently [47]. The motion of the line is determined by the equations of classical mechanics including the force exerted by the nuclei of the lattice, the vortex tension, the hydrodynamical Magnus force which tends to push the vortex towards the surface of the star as well as a dissipative force that accounts for electron scattering and for the generation of vortex waves (Kelvin modes). In particular, the Magnus force depends on the relative velocity v rel between the vortex line and the macroscopic superfluid flow produced by the ensemble of the vortices. For values of v rel below a critical value, which is determined by the absolute value of the pinning energy, one finds that the vortex line is immobilized within the lattice. Above this velocity the vortex can move through the lattice towards the surface of the star where it disappears transferring its angular momentum to the non superfluid, rigid components of the star. Pinning energies of the order of 1 MeV correspond to critical velocities of the order of 10 6 -10 7 cm/s, which allow the system to store enough angular momentum to drive a giant glitch.

The axial symmetry adopted in the previous discussion is appropriate for the interaction of a vortex with isolated, spherical nuclei. However, according to several calculations based on the semiclassical Thomas Fermi approximation, on quantum HF theory, or on molecular dynamics, in the deeper layers of the crust the stable nuclear configuration changes with increasing volume fraction of the nuclei, going through a sequence of shapes, from spheres to cylinders (spaghetti), slabs (lasagna), tubes and bubbles, until the homogeneous phase becomes favoured and the core of the star is reached (cf. Section 3.3 in [4]). The interaction between a vortex and one of these structures has never been computed microscopically (with the partial exception of the cylinder, cf. [39]). A systematic analysis would require full threedimensional calculations, which should explore different configurations, considering different alignments of the vortex lines with respect to the various shapes.

Conclusions

We have presented an overview of recent progress made in the study of the quantal structure of the inner crust of neutron stars. We have shown that quantum microscopic calculations are needed, to reveal qualitative effects associated with the inhomogeneous nature of the inner crust and neutron shell effects in the continuum. We have also indicated some of the areas where much work remains to be done, in particular concerning a quantitative treatment to deal with the effects associated with band structure, with the influence of medium polarization on superfluidity and with the connection between the microscopic structure of vortices and their dynamics in the star.

Figure 1 .

 1 Figure 1. Neutron and proton mean field potentials (a) and densities (b) in the 1364 Sn Wigner-Seitz cell containing 50 protons and 1314 neutrons. The thin solid and dashed lines in (a) indicate the neutron and proton Fermi energies.

Figure 2 .

 2 Figure2. Phase shifts calculated as a function of energy for different values of the total and orbital angular momentum of the neutron single-particle states in the exrtapolated HF potential of the 1364 Sn WS cell. The phase shifts associated with spin down (j = l -1/2) and spin up (j = l + 1/2) neutron single-particle orbitals are reported in (a) and (b) respectively. The continuum spectrum starts at E ≈ -3.5 MeV (cf. Fig.1).

3 -Figure 4 .

 34 Figure 4. Strength functions associated with the octupole operator dU/drY 30 in the 1364 Sn WS cell. The discrete unperturbed and RPA responses are shown by the open and filled histograms (in units of MeV 2 fm -2 ). The dashed and solid curves (in units of MeV fm -2 ) are obtained folding the unperturbed and the RPA discrete response with a Lorentzian curve having a width of 1 MeV.

Figure 5 .

 5 Figure 5. (left) Geometry used to calculate the pairing gaps shown in the right panel. (Right) Pairing gaps as a function of ρ in the equatorial plane (z = 0) associated with four different configurations (vortex pinned on a nucleus in the WS cell : solid curve; vortex in uniform neutron matter: short dashed curve; nucleus in the WS cell: dashed curve; uniform neutron matter: dotted curve) calculated for an asymptotic neutron density 0.01 fm -3 at large values of ρ.

Table 1 .

 1 Total

	l	j	E res lj [MeV] Γ res lj [MeV]
	4 9/2	-2.1	0.08
	4 7/2	-0.2	1.0
	6 11/2	0.2	0.04
	7 15/2	2.4	0.08
	7 13/2	8.7	2.5
	8 17/2	9.9	1.7
	8 15/2	20.3	8.0
	9 19/2	18.3	4.5
	10 21/2	28.7	9.0

and orbital angular momentum lj, energies E res and widths Γ res of the resonant neutron single particle states calculated extrapolating the HF potential of the 1364 Sn WS cell up to infinity (cf. Fig.

1 and 2

).

  Neutron particle-hole transitions associated with the largest values of the strength of the operator dU/drY 20 and dU/drY 30 in the WS cell 1364 Sn. Neutron single-particle levels of even and odd parity are drawn by solid and dashed lines, and are labeled by the usual spherical quantum numbers {nlj}. Solid and dashed arrows refer to 2 + and to 3 -transitions. The neutron Fermi energy is represented by the thick dashed line.

		30	9l17/2 9m21/2
			1364 Sn
		20	8k15/2 7l19/2
	E (MeV)	0 10	4g7/2 2i11/2 3j15/2 5j13/2 5k17/2
			1i13/2
		10	1h9/2
			1h11/2
			1g7/2
	Figure 3.

Acknowledgment

This paper is based on work carried out in collaboration with P. Avogadro, S. Baroni, A. Pastore and F. Raimondi. F.B. acknowledges partial support from the Spanish Education and Science Ministry Project Nos. FPA2006-13807-c02-01, FIS2005-01105, and INFN08-33.


J Dudek

B Szpak

M-G Porquet

H Molique
email: herve.molique@ires.in2p3.fr

K Rybak

B Fornal

Nuclear hamiltonians: The question of their spectral predictive power and the associated inverse problem

Keywords: numbers: 21.10.-k, 21.10.Pc, 21.60.-n, 21.60.Jz, 02.30.Zz, 02.60.Ed ¨¨¨¨¨¨¨¨¨¨©

In this article we introduce a subjective notion of the predictive power of nuclear Hamiltonians (an objective one does not exist) and examine it in the particular context of the single-nucleon energy spectra. We consider various types of uncertainties originating both from the experiment and theory stressing the dominating character of the theoretical errors. The latter originate from the complexity of the nuclear many body systems that is not matched adequately by the formalism behind the present day nuclear Hamiltonians. The related inverse problem is formulated and the presence of errors (ignorance, lack of knowledge) is parametrized in terms of the associated probability distributions. Various hypotheses concerning the input uncertainties ('numerical noise') are formulated and the impact of the input-uncertainties in the adjustment procedures down to the final parameter values and theoretical spectra is illustrated and discussed. A number of Open Problems are formulated and listed at the end of the article.

Introduction: Predictive power and the inverse problem

In this article we formulate and discuss the problem of predictive power of nuclear theories focusing on the underlying properties of nuclear structure Hamiltonians. The notion of predictive power associated with a given theory, similarly to the very notion of probability, is subjective ‡. According to the common sense, the adjective 'predictive' signifies theory's capacity of providing realistic i.e. experiment-comparable outcome prior to the effectively performing the measurement. Again, the qualifying terms such as 'realistic' and 'comparable with experiment' are left to physicist's judgement.

We arbitrarily distinguish between two regimes of theory's functioning and the two related forms of predictive power called extraneous, from the Latin 'extraneus' (outside) and intraneous, from the Latin 'intraneus' (inside). The former adjective refers to the capacity of predicting the experimental results outside of the realm of the experimental input used to determine parametrisation of the theory, e.g. predictions for the super-heavy or exotic nuclei based on the parameter adjustments to the properties of, say, β-stable nuclei. Alternatively, the latter adjective refers to the capacity of predicting the properties of nucleus '2' using the parameter adjustments to the 'neighbouring' nuclei '1' and '3'.

The very word 'prediction' refers to our lack of knowledge: We do not know what the binding energy of the nucleus 322 126 Xx 196 is before we actually succeed in measuring it and thus in the best case we may try to 'predict it' before experimenting. However, we may try to quantify our lack of knowledge. Based on our experience with hundreds of other binding energies we may be able to estimate the probability within which, we believe, our theoretical prediction should apply. This does not offer any guarantee (as it is always the case when working within the realm of the probability calculus) -but diminishes our ignorance and possibly strengthens our (subjective) confidence §. In what follows we will illustrate these issues using realistic (phenomenological) mean field Hamiltonians and realistic experimental data.

The science of obtaining statistically significant result 'for the whole' out of the 'limited data-samples' is a sub-field of applied mathematics called inverse problem: the latter is usually formulated as follows. Given the experimental results called data {d} and a set of parameters {p} that must be determined by these data. Next, let all the operations leading within the considered theory from the set of parameters to the theoretical prediction of the data be abbreviated by Ô: we write symbolically {d} = Ô{p}. In what follows, the data will represent the energies generated by a certain quantum Hamiltonian while {p} -the set of its parameters. Calculating the energies out of the known parameter set is referred to as solving the direct problem, whereas determining the parameters out of predefined (usually experimental) energies -the related inverse problem. Since the need for the inverse problem solutions is ‡ Indeed, the word probability does not have any unique/consistent definition, there exist two most frequently involved approaches dealing with probability, the 'frequentist' and the 'Bayesian' ones, not to forget a couple of others (Jeffrey's the so-called logical and Fischer's fiducial), together with over a dozen probability interpretations. In short: by entering the territory of probability distributions we introduce inevitably subjective criteria; in this text we follow the frequentist notion of probability.

§ In their Introduction to Bayesian Scientific Computing, Calvetti and Somersalo [START_REF] Calvetti | Introduction to Bayesian Scientific Computing[END_REF] use the following dialogue to illustrate the similar context. [Origin: Umberto Ecco's 'The Name of the Rose'].

"So you don't have a unique answer to your questions?" "Adson, if I had, I would teach theology in Paris." "Do they always have a right answer in Paris?" "Never", said William, "but there they are quite confident of their errors".

overwhelmingly present in practically all the modern sciences ranging from geology, sociology, politics and medicine to physics and mathematics, there exists rich literature related to the underlying mathematical tools. The use of them was relatively seldom practiced in low-energy sub-atomic physics but we believe that our field of research has numerous possibilities to profit from this evolution.

Our analysis aims at Hamiltonians of contemporary microscopic theories such as Density Functional Theories including various realisations of the Nuclear Mean-Field Methods, Nuclear Shell-Model and Nuclear Phenomenological Model Hamiltonians. Despite an apparent diversity all these complex nuclear theories depend on a common factor: their performance and especially their predictive power may considerably profit from the modern advanced mathematical theories of statistically significant parameter determination. The performance of realistic theories depends crucially on the adequate selection of theory's adjustable parameters. The latter depends on the choice and the quality of the experimental-data input, and, what is strongly related, on the statistical significance of the resulting parameterisation.

The parameter determination is strongly influenced by theoretical modelling since extraction of the experimental information often depends on model dependent procedures (examples will be given below). Moreover, theories themselves attempt a description of only a part of the physical reality whereas both the already established and yet undiscovered mechanisms have their impact on the experimental result. In this article we pose the problem of quantifying the unknown through probability distributions within the framework of what could be dubbed elements of theory of theoretical errors. Presenting our selection of Open Problems, we hope to contribute to intensifying the interest in this field and hopefully accelerating the associated research.

We have chosen to limit our discussion to the spectral properties of the nuclear systems i.e. the energy eigenvalues of the underlying Hamiltonians; other observables can be included in the discussion with no major difficulty. The illustrations (but not the considerations) are purposely limited to the problem of reproducing the singlenucleonic energies in spherical doubly-magic nuclei. This will allow us to illustrate examples of problems encountered when extracting the experimental information before injecting it into the parameter determination algorithms. The discussion of the parameter determination techniques is greatly facilitated by using a realistic context, but simple enough to be able to construct exactly soluble test-cases with the rapid solution possibilities. For that purpose we will select a toy-model Hamiltonian containing the spherical Woods-Saxon central potential together with the spin-orbit mean-field components. The reader is warned not to be mislead by the simplicity of the chosen toy model as opposed to complexity of the microscopic theories aimed at. Since the functioning of the inverse problem techniques is to far an extent not dependent on their practical realisations, we will try to profit from the presented examples to pose important, fundamental questions for the research to come.

The issue of statistical significance is not void of interdisciplinary interest as the authors of [START_REF] Press | Numerical Recipes in FORTRAN 77: The Art of Scientific Computing[END_REF] observe with sarcasm: "Unfortunately, many practitioners of the parameter estimation never proceed beyond determining the numerical values of the parameter fit. They deem a fit acceptable if a graph of data and model 'look good'. This approach is known as chi-by-the-eye. Luckily, its practitioners get what they deserve". Expressed more explicitly: parametrisations obtained in this way risk to be unstable, statistically worthless and thus useless for extrapolations of any kind and in particular killing the chances for an extraneous predictive power.

Experimental information, built-in modelling and related uncertainties

The following discussion shows clearly, that the so-called experimental single particle energies in spherical nuclei are highly complex, model dependent objects. The possibly precise extracting of the energy information from experiment requires well designed instrumentation, involves weak signals whose analysis deserve increased statistics data -conditions that are not easy to meet especially for the exotic nuclei such as a very important 132 Sn nucleus. Its importance consists in the fact that together with 208 Pb these two spherical doubly-magic nuclei form the last strongholds when exploring the exotic heavy and/or super-heavy nuclei.

The experimental information about the single-particle level-structure of eveneven doubly-magic nuclei with the proton number Z 0 and neutron number N 0 originates mainly from the (Z 0 ± 1) and (N 0 ± 1) nuclear neighbours. Population of the single-particle or single-hole nucleonic levels in the (A o ± 1) residual nuclei is favoured through the light-projectile direct nucleon-transfer reactions such as (d,p), ( 3 He, d), (t,d) or (p,d), (d, 3 He), (d,t) on mass-A 0 doubly-magic nuclei. In principle, the results of such experiments directly reveal the j-shell structure around the magic energy gaps, since each measured level does correspond to one single-particle or singlehole energy state and the reactions of this type are capable of providing the excitation energy, angular momentum and parity of the states in question. It is well known that all the j π characteristics of the low-lying states measured in the (A o ± 1) residual nuclei close to the stability line are in agreement with the expected sequence of singleparticle energy-levels in a spherical mean field containing the spin-orbit potential. Nevertheless the separation energies of these low-lying states are not necessarily the single-particle energies we are looking for. There are often other levels with the same j π in the same energy region, arising from excited core configurations or from coupling to the collective vibration of the surface, which mix (i.e. strongly couple among themselves through nuclear interactions thus giving rise to big matrix elements) with the unperturbed single-particle state. As a result, even though the single-nucleon strength may be maximum for the first state with the given j π , it is distributed among several other nuclear levels as well. To recover the single-nucleon energy, ε κ , one computes the centre of gravity of the j π group using weight factors proportional to the excitation strengths observed in the transfer reaction, viz.:

ε κ = k S(κ, k) e exc k with the sum rule k S(κ, k) = 1. (1)
Above, e exc k is the measured energy of the k th level, and S(κ, k) is its spectroscopic factor, i.e. the probability that the wave-function of the measured level k contains the single particle state κ. This procedure has been justified by M. Baranger [START_REF] Baranger | [END_REF], from a theoretical point of view. One has to stress that the appropriate use of (1) to calculate the single-particle energy ε κ requires that the sum rule is fulfilled i.e. that (i) All the k-levels to which the single-particle state κ contributes are identified, and: (ii) Spectroscopic factors have been determined precisely i.e. without any major bias.

The weighted-average procedure may be modified through a normalisation

ε κ = k S(κ, k) • e exc k / k S(κ, k) (2)
with the hope that even if, in view of the large experimental errors on the spectroscopic factors, the sum rule k S(κ, k) = 1 is poorly approximated, the impact of the implied uncertainties is diminished in this way.

Extracting single-nucleon energies and associated uncertainties

A typical experiment on one-nucleon transfer reaction provides the spectrum of the light ejectiles, i.e. the number of counts as a function of the ejectile energy, for many detection angles in the laboratory system. These raw data are analyzed in order to deduce, for each state populated in the residual nucleus, the differential cross section, dσ dΩ , as a function of θ cm , the ejectile emission angle in the centre-of-mass system. Depending on the experimental conditions, the functions dσ dΩ (E k ) = f (θ cm ) are not always determined with enough precision. For instance the detection efficiency may be too small to measure some weakly populated states, the knowledge of detection efficiency may be too fragmentary to compute the absolute value of dσ dΩ , or the angular range of detection may be too narrow to obtain the full angular distribution.

The above measurements provide the information about the single-nucleon stateenergies. Firstly, the measured angular distribution determines the orbital angular momentum l of the 'stripped' or 'picked-up' nucleon and this assigns the angular momentum and the parity to each level. The remaining ambiguity, whether the level under consideration corresponds to j = (l + 1/2) or j = (l -1/2) configuration, can be removed when using polarized beams. Secondly, the absolute value of the cross section gives a measure of the proportion in which a particular single-particle state contributes to the given level j π . More precisely, whereas the former aspect is well under control for standard one-nucleon transfer reactions, the extraction of the socalled 'experimental' value of the spectroscopic factor needs precise modelling of the reaction process, since it is obtained through dividing the experimental cross section by the theoretical value calculated under the assumption that the populated state is a pure single-particle/hole state,

dσ dΩ (θ, E i ) exp = S(κ, i) • dσ dΩ (θ, E κ ) th . (3)
From the above expression, the spectroscopic factor should be a constant independent of the angles, implying that the theoretical cross section would reproduce experiment throughout the entire angle range. In practice this is not the case and to minimize the error, the value of S is determined at the angle where the experimental cross section has its maximum. This is clearly one of the aspects contributing to the final uncertainties of the extracted level-energies.

The simplest useful reaction model, the Distorted Wave Born Approximation, assumes a direct, one-step transfer process in which the transfers to specific states are individually weak and may be treated using perturbation theory. The distorted waves are provided by optical model potentials that describe the appropriate elastic scattering in the entrance and exit channels. It is important to emphasize that various types of optical potentials, while fitting the elastic scattering data equally well, may give very different results for the shape of the angular distribution of the ejectile as well as the values of the spectroscopic factors of the populated states. Moreover, two potentials are needed, the one of which binds the transferred nucleon in the exit channel as well as the one binding it in the entrance channel. The geometric parameters of the binding potentials are rather poorly constrained so that they are subject to rather large parametric uncertainties. Since the theoretical cross section scales with the radius parameter, the derived spectroscopic factor can vary up to 30% depending of the chosen parameter values.

The fact that the fit to the measured angular dependence of the cross-section, dσ dΩ (θ, E i )| exp , is not satisfactory in the whole angular range signifies that in many cases the reaction model is too simple to describe the reality satisfactorily. One may expect that a part of the discrepancies originates form the nuclear structure effects not taken into account in the model as well as the influence of two-step transfer processes including inelastic excitation of the target, projectile or residual nucleus. Using more elaborate models for the transfer reactions such as the Coupled Channel Born Approximation or Coupled Reaction Channels produces better fits whereas the values of spectroscopic factors are often strongly altered. It follows that the ambiguities related to the spectroscopic factors may be as large as 30% in absolute values as a result of otherwise careful analyses. Thus even though a level strongly populated in one-nucleon transfer reaction has been interpreted as a pure single-particle state (because the value of its spectroscopic factor has been found close to unity) such a result has still to be counter-checked by verifying whether other states with the same value of spin and parity are not populated as well using the same reaction. This is a very difficult task, dealing with large energy ranges and very low values of the cross sections making the identification of the states a very tough issue. In addition, it is worth recalling that the transfered angular momentum strongly depends on the projectile energy. Therefore in order to obtain a complete picture of all the states with all the expected j π values, one would ideally need to combine experimental results obtained in a large range of bombarding energies and using various projectiles. Unfortunately, such complete studies are seldom available in the literature and remain an Open Problem in this type of research.

Important textbook case: Single-nucleon energies in 208 Pb

The doubly-magic, spherical nucleus 208 Pb is considered to be one of the best studied doubly-magic nuclei. It will be instructive to discuss the analysis of its single-particle structure especially in view of an attempt of extending the analysis results to the much poorer known nucleus of 132 Sn, or even lighter doubly magic nuclei. 208 Pb core. The lowest excited state of the 208 Pb nucleus, at 2.614 MeV, is the well-known collective-oscillation octupole state of B(E3)=40 W.u. At higher excitation energies corresponding to about 3-to-4 MeV, there are several states interpreted mainly in terms of combinations of one-particle one-hole (1p-1h) excitations, some of them having negative parity (such as the 5 -level at 3.198 MeV) or positive parity (such as the 2 + level at 4.085 MeV). The octupole vibration of 208 Pb is the most important in the fragmentation of the single-particle strengths in the neighbouring odd-A nuclei because of both its low energy and its strong collectivity, nevertheless the other states may also contribute to the fragmentation.

Excited states of the

2.2.2.

Deconvolution of the single-nucleon states around gaps at Z 0 =82 and N 0 =126. The four odd-A nuclei close to 208 Pb viz. 209 83 Bi 126 , 207 81 Tl 126 , 209 82 Pb 127 and 207 82 Pb 125 , had been studied using several one-nucleon pick-up or stripping reactions, some thirty years ago. At that time, the experiments had only focused on the dominant singleparticle or single-hole states, lying in a restricted range of excitation energies. Except for a few cases, all the observed states were considered as pure single-nucleon states. Their spectroscopic factors were assumed to be close to unity and used to test the Distorted Wave Born Approximation procedures.

The resulting, lowest-lying excited states, denoted e exc 1 , are presented in table 1. . When written in bold, this state is usually considered as a pure single-nucleon state, i.e. with a spectroscopic factor close to unity [START_REF]ENSDF data base[END_REF]. The energies εκ of the single-nucleon states are computed with (2), taking into account the N fragments which have been measured in the quoted references. The shift in energy is the difference between the energy e exc Some years later, by using other transfer reactions at higher bombarding energies, many cases of fragmentation have been observed. This allows us to calculate several values of ε κ using (2), as shown in table 1. Unfortunately, these detailed results are not available for all the average energies associated with the whole set of single-nucleon levels of interest. The proton levels were studied in quite some detail, except for the highest-excited ones lying above Z 0 =82. It turns out that because of their low orbital momentum, the population of the latter is not favoured in the proton-stripping reaction ( 4 He,t) at 80 MeV bombarding energy [START_REF] Galès | [END_REF]. The fragmentation of the neutron levels is not very well documented; for instance, the fragments of the νj 15/2 level in 209 Pb were sought because of the expected mixing with the νg 9/2 ⊗ 3 -coupling, whereas none of the high-lying-level fragments were precisely studied. Similarly, no fragment of the low-lying νp 3/2 level in 207 Pb could be identified in the reactions ( d,t) at 200 MeV bombarding energy [6] or ( 3 He,α) at 70 MeV [7], because of the strong mismatch in transfered angular momentum. Those, together with previously mentioned uncertainties prolongate the list of experimental Open Problems whose solutions may contribute to narrowing the error distributions as discussed below.

2.2.3. Impact of collective excitations: particle-vibration coupling. The differences between e exc 1 and ε κ amount to several hundreds of keV (see table 1). Such shifts can be partly attributed to the octupole coupling, as in each of the four odd-A nuclei, there are couples of single-particle levels with ∆l=3 (and ∆j=3):

πf 7/2 ↔ πi 13/2 πh 11/2 ↔ πd 5/2 and νg 9/2 ↔ νj 15/2 νi 13/2 ↔ νf 7/2 . ( 4
)
Within the particle-vibration-coupling approach [10], we expect two fragments for each level [one of them related to the coupling between the single-particle lower-l wave-function and the octupole 3 -phonon wave-function coupled to the wave-function with the higher-l, say l , (l ⊗ 3 -) l ] and the implied shift in energy is expected to be a fraction of 2.614 MeV, the energy of the collective octupole state of the core (see figure 10 of [10]). Even though the actual number of fragments is usually greater than 2, the mean value of the shifts (0. The above estimates and the information in table 1 have been used to model the typical error probability distributions, cf. Sects. 3-5. But we wish to emphasize, anticipating the conclusions from the results presented below, that the experimental information concerning not only the error estimates but also simply the too low a number of the experimental levels is a vital Open Problem in the determination of the nuclear Hamiltonians especially for lighter doubly-magic nuclei for which there are much fewer single-particle levels known at present (compared to 208 Pb).

In-exact theories, random variables and error probability distributions

In what follows, we arbitrarily oppose two, as it seems extreme types of modelling: a. Academic modelling designed to clarify functioning of certain physical mechanisms in abstraction from the detailed experimental data, as for instance over-viewing certain general features of multi-phonon oscillations, and: b. Realistic modelling, optimised to predict and/or to reproduce the experimental data related to real physical systems, as for instance the energy spectra of atomic nuclei. The first family of such theories is sometimes referred to as exact-, while the other one -due to a number of factors discussed in some detail below -may lead to what is referred to as in-exact theories.

Inexact theories pertain to the description of physical systems whose complexity exceeds our present-day theoretical knowledge. For instance, an insufficient knowledge of elementary interactions may not allow to built up a description of adequate manybody interactions. Alternatively, the complexity of physical systems may require that complications of the formalism go beyond the present-day level of human motivation (and capacities) -in particular if confronted with the lack of adequate contemporary mathematical tools or numerical capacities. Both types of discussed theories do not include all of the already discovered elements of knowledge and must not pretend to any completeness in describing the full reality. Therefore they must contain inadequacies leading us directly to the notion of the theoretical errors.

From the mathematical point of view theoretical and measurement errors are seen as random variables. It then follows that the observables which in the exact theories would have been represented just by real numbers -in the inexact theories turn into random variables (within certain intervals) and the associated probability distributions.

This article focuses on the combined theory and experimental uncertainties and errors from the point of view of the associated probability distributions.

Realistic in-exact theories and underlying postulates

Realistic modelling and associated theories, aim at describing with the possibly highest fidelity a selected class of physical systems and phenomena. Such theories are not only based on certain experimental input but also aim at predicting and/or interpreting the outcome of measurements. We do not suggest that realistic theories are necessarily inexact. However, in the nuclear structure theory context this will be manifestly the case: in what follows the term realistic theory will be synonymous with inexact.

3.1.1. Realistic theories and the inverse problem. The number of experiments on any class of processes or systems is necessarily finite and, moreover, 'infinitesimally small' as compared to the number of all similar processes or systems that can be found in nature. Nevertheless, humans wish to model infinitely many physical situations using the experience with the very restricted sets of data below referred to as samples. This is what, among others, the inverse problem is about: using the results of observations to infer the values of the parameters characterizing the system under investigation. By doing so, equipped with a mathematical model, we may proceed to retrieve the information about the unknown quantities of interest by an indirect measurement. For instance by directly measuring the positions of say, 1s 1/2 and 1d 5/2 levels in a certain nucleus we may hope to be able to infer through modelling the energy of the 1d 3/2 level thus obtaining some information indirectly.

Realistic theories and errors of modelling.

From now on we consider quantum Hamiltonian systems i.e. the physical systems whose states can be described in quantum mechanics with the help of the realistic energy operator Ĥ = T + V (x, p) :

Ĥψ n = e n ψ n → e n = e n (p). (5)
Above, x represents observables (operators) such as position, momentum, spin, isospin etc. and p -the full set of Hamiltonian parameters. Obviously the results of the theory, here the n b bound-state discrete energies e n , for n = 1, 2, . . . n b , are known functions of parameters provided we have determined the parametric dependence of Ĥ on p from the data. However, the data are known only within experimental uncertainties which may include in addition to the usual instrumental errors also the errors coming from the model-dependence in the data extraction-procedures as discussed in Sect. 2. Moreover, working with inexact theories introduces inadequacies between the experimental data and the theory predictions. Indeed, all mechanisms present in the physical system, known or still unknown up to date, contribute to the measurement's outcome while only part of those mechanisms are modelled within the Hamiltonian and thus only this part contributes to the theoretical spectra. Let the experimental data be represented by the set of discrete energies {ε k } with k = 1, 2, . . . k ε ; observe in passing that in general k ε < n b , more precisely, k ε n b , but it may happen that k ε is comparable to n p , the number of parameters: p ↔ {p} ≡ {p 1 , p 2 . . . p np }. By taking into account the uncertainties and using obvious notation we may write:

Theory → e n = e true n + δe n and ε k = ε true k + δε k ← Exp. (6)
Above, e true n and ε true k are abstract 'exact' theory and 'exact' measurement results that are introduced only to facilitate the discussion of the principles but they do not appear explicitly it the final results. The theoretical and the experimental errors, denoted δe n and δε k , respectively, represent our lack of knowledge and thus, from the point of view of mathematics, they can be seen as continuous random variables. Undesired effects of the missing knowledge can be diminished through experience: for instance we may learn that a certain experimental outcome is more probable than the other. By introducing explicitly the implied probability distributions that quantify our lack of knowledge, we explicitly introduce subjective elements into considerations.

In this way we arrive at what is known in applied mathematics as statistical inference (or induction) problem and the idea of random sampling to infer certain unknown aspects of the studied system (statisticians would say: population).

The usual approach when determining Hamiltonian parameters from the set of data {ε k } is to minimize the generalized distance between the theory predictions and the data points. Such a distance can be, up to a normalisation, defined as

χ 2 (p) ∼ k exp k=1 w k {[ε true k + δε k ] experiment -[e true k + δe k ] theory } 2 → ∂χ 2 ∂p j = 0, (7)
where w k are physicist's choice positive weight factors. Let us shorten the notation and write δe n ≡ ēn and δε k ≡ εk . Let the experimental and theoretical probability distributions associated with each of these variables be εk ↔ P e k (ε k ) and ēn ↔ P t n (ē n ). In (7) there appear sums (alternatively differences) of two random variables, viz. ηk ≡ εk + ēk , that are mutually independent. In such a case the minimized function will be a weighted sum of quadratic expressions of the form w k {[ε k -e k (p)] + ηk } 2 depending on the probability distributions D k (η k ), in the following referred to as 'noise', for the variables ηk ; the latter are given through the k-dependent convolutions

D k (η k ) = (P e k * P t k )(η k ) = Z +∞ -∞ P e k (ε k -ēk ) P t k (ē k )d ēk = Z +∞ -∞ P e k (ε k ) P t k (ē k -εk )d εk . ( 8
)
One can see that any further consideration must depend on what we know (or what we agree to accept as reasonable assumptions about) the distributions P e k (ε k ) and P t k (ē k ). This step is clearly one of the Open Problems that, as we believe, will need to be addressed in detail the future. Here we wish to focus on possibly simplified but effective solutions and thus we reformulate the problem in (7) as follows.

Uncertainty distributions and their dynamical transforms

The idealized, 'true model energies', e true n , although conceptually important, remain in practice unknown elements of the inexact theories, and since finding the 'exact' measurement result is a void issue, we may try to simplify the formulation. Let us replace distributions {D k (η k )} by a new set {D k (η k )} of the new random variables η k . Doing so we modify the minimisation in (7) by introducing a new function

χ 2 (p) ∼ k w k {[ε k + η k ] -e k (p)} 2 . ( 9
)
Let us also introduce a set of samples η i ≡ {η 1 , η 2 , . . . η k max } i for i = 1, 2, . . . N , each variable sampled according to its D k (η k ) distribution. Next by solving (here) the Schrödinger equation we perform N minimisations to find the sets of the optimal parameters pi ≡ {p 1 , p2 , . . . pnp } i , the sets of implied eigenvalues ēk (i) ≡ {e k (p i )} and the corresponding χ 2 ({e k (p i )}. For the ensemble of η i samples sufficiently large (below we use N ∼20 000) we may collect the occurrence histograms such as e.g.

N ( χ,δχ) = N (χ ∈ [ χ -1 2 δχ, χ + 1 2 δχ]) (10)
illustrating the quality of the generated spectra in terms of their r.m.s distribution or

N ( pj ,δ pj ) = N (p j ∈ [p j -1 2 δ pj , pj + 1 2 δ pj ]), (11)
showing the parametric uncertainties reflecting the ones in the input data and finally

N [ē k (i),δē k (i)] = N (ē k ∈ [ē k (i) -1 2 δē k (i), ēk (i) + 1 2 k (i)]), (12)
illustrating, in which manner the uncertainties of the individual energy levels are affected. Distributions in Eqs. (10)(11)(12) depend on the dynamics of the system through interaction Hamiltonians underlying the equations of motion such as Schrödinger, Dirac, Hartree-Fock etc. For that reason we refer to them as dynamical transforms of the input uncertainty distributions of the inverse problem. In the detailed illustrations we distinguish among the variants such as: the r.m.s. transforms represented by (10), induced parameter-transforms by (11), energy transforms by ( 12), etc. It will be instructive to illustrate the above notions using a simple but realistic phenomenological model. We chose for that purpose the neutron single-particle levels in 208 Pb using the phenomenological Woods-Saxon Hamiltonian with parameters given in table 2. The spectrum generated with the latter is then used for the noise testing purposes replacing the true experimental input: we refer to such a spectrum as 'pseudo-experimental'. This offers the advantage of allowing to vary the number of levels to fit in excess of those known experimentally (similar type analysis will need to be performed in the future for other, microscopic theories).

Table 2. Woods-Saxon potential parameters for the neutrons in 208 Pb used to generate the pseudo-experimental levels, among others for the purposes of the noise testing. These parameters reproduce the 'true' experimental levels in Table 1 with the r.m.s. deviation of 0.164 MeV and the maximum error of 0.353 MeV.

V c o r c o a c o λ
r so o a so -39.520 1.371 0.694 26.133 1.255 0.500 Let us illustrate the impact of both the number of data points taken for the fit and the broadness of the noise distribution of the data. Illustrations in figure 1 show that if the number of data points is not large enough, the Hamiltonian is capable of adapting itself and of reproducing any noise feature of the experimental input so that the maximum of the distribution corresponds to χ 2 =0 (error-less fit to 'any' input !). Only when the number of data points is sufficiently large (actually larger than the number of single-particle energies known from today's experiments) can a consistency between the resulting fit distribution and the input noise distribution be (only partially !) achieved. Indeed, observe that (figure 1, right) the 'noisy' input and output distributions lie close, but correspond to the maxima at χ = 0. Recall that the noise-less (exact) input generates by definition of the pseudo-experimental spectrum, the exact χ = 0 solution always. Our observation can be viewed as intuitive: applying a noise with the distribution width σ to an exact, pseudo-experimental spectrum, shifts the centroid of the calculated distribution by an increment related to σ. In all cases an improvement is achieved: the maximum probability is shifted towards smaller χ.

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Let us emphasize that the capacity of a Hamiltonian to reproduce a number of energy levels exactly i.e. with χ = 0, is not only the question of the number of adjustable parameters vs. the number of data points but also the question of adequacy. If the data points form a parabolic dependence in function of a certain parameter while the theory predicts a trigonometric one, even with non-trivially few data points it will not be possible to reproduce the data exactly -unless infinite series are involved. " " " " " " " " " " " " " " " " " " " " " " 2. For comments on the double-maximum structure cf. Refs. [11,12].

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Examples of the Schrödinger parameter-transforms for a few least-bound neutron levels are shown in figure 2. Observe the double maximum structure, the so called 'compact' (r so o ≈ 0.92 fm) and 'normal' or 'non-compact' (r so o ≈ 1.25 fm) geometries. This mechanism is interesting and known to the experts: here, not entering into details, it will be sufficient to observe that the single-maximum distribution reflecting the single-maximum distribution of the input noise is obtained only at sufficiently large number of pseudo-experimental levels. A few examples of the single-nucleon Schrödinger energy transforms defined through (12) are illustrated in figure 3; observe that the widths corresponding to various levels differ despite the fact that the input uncertainty distribution is common for all the levels in the tests illustrated.

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Theoretical-level errors caused by experimental-level uncertainties

It may be (and often is) very useful to know e.g. for the experiment programing, what are the uncertainties in theoretically predicted positions of various levels caused by a possibly large uncertainty in the inverse problem input information about specific levels. For instance, it may happen that a certain experimental error on, say, fitinput 1i 11/2 -level, may have no major impact on the results of the fitting algorithm, whereas the same uncertainty on, say 2g 7/2 -level, may influence the positions of the fitted levels considerably. Should a similar or any alternative scenario be established, special measurement precision would be required on specific levels and possibly much less so, on some others. This information will usually be very practical to know before establishing the priorities of the related research program -and we give numerous arguments in this article to indicate that the present-day experimental information on the single-nucleon levels in spherical doubly-magic nuclei remains largely an Open Problem for the future.

Information about a 'level-to-level' sensitivity can be obtained through a simple simulation which consists in repeating the adjustment algorithm in function of the displacement of the individual input-level energies (one level at the time). To illustrate the problem we have varied the positions of the input levels of the adjustment algorithm for the neutrons in 208 Pb. The illustration of differences between the effects caused by high-j orbitals (here: 1i 11/2 ) and low-j orbitals (here 3d above results with the ones for 2f 7/2 and 2g 9/2 levels, figure 5. Observe in particular a strong mutual coupling between the 2f 7/2 and 2g 7/2 levels (same sign) and the strong but opposite sign couplings between 3p 3/2 and 2f 7/2 , on the one hand, and between 3p 3/2 and 2g 7/2 on the other. The relative parity of orbitals does not seem to have any direct role: observe that the same parity orbitals 1i 11/2 and 1i 13/2 couple with opposite signs to the two considered orbitals in the left-and right-hand side diagrams. One may conclude that the 'level-to-level' correlations through the adjustment algorithm, if needed e.g. for the predictions and/or extrapolations of the level positions in nuclei away from the tested ones (cf. Sect. 4), or for the experiment planning, may be very precious but will need to be examined individually, i.e. one level after another. Moreover: these results suggest that the schematic treatment of the noise parametrisation in terms of a common normal distribution is certainly not optimal and will need to be replaced by both experimental and theoretical level-by-level analysis of the uncertainties in order to construct a more intrinsically-consistent procedure.

Predictive power: Examples of neutron spectra in 132 Sn nucleus

The 132 Sn nucleus exhibits all the usual characteristics of the doubly-magic nuclei, first of all the high energies of its first excited states. Indeed, the first excited 2 + state lies at 4041 keV and the 3 -state at 4351 keV. Moreover, there is only a weak quadrupoleexcitation probability associated with the first 2 + state, with B(E2) = 7(3) W.u. accompanied by the collective octupole-excitation probability with B(E3) > 7 W.u.

The four odd-A nuclei close to 132 Sn have not been yet studied using one-nucleon transfer reactions. This prevents us to characterize any single-particle content of their states so far identified using γ-spectroscopic studies following β-decay, and in some cases by the prompt γ-decay of nuclei produced as fission fragments.

Considering the lowest excitation energies, denoted e exc 1 , as in table 1, we would like to evaluate the energies ε of all the single-neutron levels around N 0 =82. For that purpose we will use: (i) The conclusions formulated in Sect. 2 for 208 Pb, and: (ii) The energy-shifts due to the fragmentations which have been measured, for each level of interest, in the neighbouring isotopes and isotones lying closer to the valley of β-stability. Single-particle quantities εκ have the similar meaning as the quantities denoted ε κ in the preceding Sections except that in the absence of the adequate experimental information about the neighbours of 132 Sn, certain information "extrapolated" from the area of 208 Pb will be employed. N=83 isotones Figure 6. Excited states having j π =7/2 -and 9/2 -identified in the N =83 isotones (from [START_REF]ENSDF data base[END_REF]). The average energies (asterisks) are calculated using (2) from the data of Z=56-62 nuclei listed in the text.

Let us begin with the example of the two neutron levels lying above the N 0 =82 gap: νf 7/2 and νh 9/2 . In fact two j π =7/2 -states and two j π =9/2 -states have been measured in the close-lying N =83 isotones, 6, together with the related average energies calculated according to (2): the latter give approximated positions of the expected single-particle neutron-energies. We observe that the fragmentation of the νf 7/2 levels results in an energy shift of, approximately, 0.2 MeV, whereas the shift associated with the νh 9/2 level amounts to, approximately, 0.1 MeV. In a similar fashion many states measured in the N =81 isotones and the Z=51 isotopes that lie between 132 Sn and the β-stability valley in the (Z, N )-plane can be used to obtain the shifts in energy due to the fragmentations of the single-nucleon levels. Some of the results of this type have been illustrated in [13]; all the results for which the data exist today collected in the third column ('shift 1') in table 3.

Table 3. Energies of the first states measured in 133 50 Sn 83 and 131 50 Sn 81 , usually considered as pure single-nucleon states [START_REF]ENSDF data base[END_REF]. Excitation energies ε of the singlenucleon states are calculated from the e exc 1 energy and the shift in energy due to the fragmentation. The empirical values of the binding energies (B.E.) take into account the values of the total binding energies of the involved nuclei [14]. All the energies are given in MeV. The data written in parentheses are tentative, and since the identification of the corresponding states is rather questionable (new experiments have to be performed to confirm both their energies and their spinparity assignments) these states are ignored in the following discussion. Another possibility of the order of magnitude estimates of the energy-shifts is to use the same proportions of the 2 + and 3 -excitation energies as the ones found in the 208 Pb nucleus (cf. Sect. 2). Using these proportions we obtain the shifts of the order of 0.6 MeV (15% of 4 MeV), for both the single-particle states coupled through ∆l=2 (and ∆j=2) and those coupled through ∆l=3 (and ∆j=3). These estimates are given in the fourth columns ('shift 2') in table 3, together with the estimated errors. The errors have been determined in such a way that the values of 'shift 1' and 'shift 2' are consistent within the error bars. Finally the empirical values of binding energies (B.E.) of the nucleon levels in the potential well of 132 Sn are calculated using the total binding energies of the involved nuclei [14] and the ε values. We have performed the calculations of the neutron levels in 132 Sn using the parameters of the 'well studied' nucleus 208 Pb together with the uncertainties of the theoretical predictions represented in the form of the histograms that resemble the ones presented earlier. The results are illustrated in figure 7. Let us observe that except for the orbitals 1g 7/2 and 2f 7/2 , the ones that were discussed in the preceding Section with the conclusion that they respond rather sensitively to the input errors all other levels lie close to the experimental positions. Moreover, the newly estimated positions of the levels from table 3 (marked with the dashed lines) lie closer to the theory predictions. Another conclusion at this step of the analysis is that the initial input uncertainties with the width σ of 0.1 MeV was most likely slightly too optimistic: a value of the order of σ ∼0.15-to-0.2 MeV corresponds better to the estimated uncertainties.

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨-
Figure 7 can be seen as a prototype of the future illustrations of theory results that include the cumulative theory and experiment uncertainties of the modelling. Such diagrams, completed by the experimental uncertainty distributions that would replace the dashed-and full-bars in the upper part of the figure would offer a much more satisfying representation of our ignorance. The corresponding information (let us emphasize: subjective, as the very notion of probability) would be nevertheless the precious starting element for the planing of improvements. The levels identified in this way as poorly known, with too broad uncertainty distributions, should be re-examined with priority. Similarly, the levels manifesting an 'unusual' structure of their histograms like the one for the 1g 7/2 level in figure 7, should be re-examined from the point of view of the quantum origin by, first of all, identifying the responsible interaction: central, spin-orbit, tensor etc. Studies of the spectral properties of the nuclear Hamiltonians and their spectroscopic predictive power, as illustrated above, represent a series of, in our opinion fascinating, Open Problems for the near future.

Instabilities of adjustment algorithms and ill-posed inverse problems

The problem of stability of the parameter optimisation ¶ depends not only on the form of the Hamiltonian but also on the number and the characteristics of the data points. In applied mathematics this problem is related to the notion of ill-posedeness of the inverse problem and it will be instructive to present its basic elements briefly.

Consider a normalised χ 2 -function obtained from the one in ( 9) by removing the error (noise) factor η and fixing the normalisation in accordance with the literature, cf. e.g. [START_REF] Björck | Numerical Methods for Least-Squares Problems[END_REF]:

χ 2 (p) = 1 (k ε -n p ) • kε k=1 w k [ε k -e k (p)] 2 ; ( 13
)
as before, k ε is the number of experimental levels and n p the number of parameters. Assuming that we are not too far from the solution we can linearize the problem

e k (p) ≈ e k (p 0 ) + np j=1 ∂e k ∂p j p=p0 (p j -p 0,j ) (14)
according to the standards of the Newton algorithms. Abbreviating the notation

J kj ≡ w k ∂e k ∂p j p=p0 and b k = w k [ε k -e k (p 0 )], (15)
we may transform (13) into the form

χ 2 (p) = 1 (k ε -n p ) • kε k=1 np j=1 J kj • (p j -p 0,j ) -b k 2 . (16)
It is straightforward to show that the necessary conditions for the minimum of χ 2 are equivalent to the matrix relation (cf. e.g. [START_REF] Vogel | Computational Methods for Inverse Problems[END_REF]):

∂χ 2 ∂p i = 0 → (J T J) • (p -p 0 ) = J T b with [J kj ] ≡ J. (17)
Thus the inverse problem takes the algebraic form with the schematic structure

J • (p -p 0 ) = b ↔ p -p 0 = J -1 b. ( 18
)
Recall that J is in general rectangular k ε × n p matrix whereas p ∈ R np and b ∈ R kε .

To illustrate the problem of ill-conditioning we will remind the reader of the socalled Singular Value Decomposition. Recall that any rectangular matrix A can be decomposed as a product of three matrices (cf. e.g. [START_REF] Calvetti | Introduction to Bayesian Scientific Computing[END_REF][START_REF] Press | Numerical Recipes in FORTRAN 77: The Art of Scientific Computing[END_REF] and references therein)

A = U • D • V T with U ∈ R m×m , V ∈ R n×n , D ∈ R m×n , (19)
where the diagonal matrix D can be represented with the help of the eigenvalues {d j ; j = 1, 2, . . . min(m, n)} and written down symbolically in the form:

D = diag{d 1 , d 2 , . . . d min(m,n) }. (20)
Applying the above decomposition to a real, orthogonal matrix J for which

J -1 = J T : (p -p 0 ) = J T b ↔ J T = V • D T • U T (
21) ¶ Ill-posedeness of the inverse problem leads to instability of the parameter adjustment procedures: narrow distributions of the input noise leading to broad distributions in terms of outcome parameters. The instability manifests itself also through the divergence of the parameter correlation matrix and that of confidence intervals -obviously highly undesired effects.

where

D T = diag 1 d1 , 1 d2 , . . . 1 dp ; 0, 0, . . . 0 . (22)
In other words, if the diagonal matrix D in the decomposition of J has very small or zero eigenvalues, actually quite a common case, then J -1 in (18) is getting singular and no stable solution for the Hamiltonian parameters can be obtained. We say that the inverse problem is ill-posed.

It follows from linear algebra that the above singularity manifests a correlation: one or more components of p are functions of the others. The singularities and thus the possible destabilizing inter-dependence among the parameters depend in turn on the number and the characteristics of the data points, and as we will point out shortly, even one parameter-divergence of this type causes the divergence of the whole correlation matrix. Indeed, it can be shown [START_REF] Björck | Numerical Methods for Least-Squares Problems[END_REF] that the correlation matrix takes the form

(p i -p i ) • (p j -p j ) ∼ χ 2 (p) (J T J) -1 ij , (23)
whereas from Eqs. ( 20)-( 22) it is possible to show that

[(J T J) -1 ] ij = r =1 V i 1 d 2 [V T ] j . (24)
If in (22) one or more d k → 0, then (J T J) -1 tends to infinity and generally, the confidence intervals (p i -p i ) of all parameters diverge. This conclusion shows once again the importance of analysis of the statistical significance of the parameter adjustment procedures if the predictive power of the resulting Hamiltonian is aimed at. We believe that this is one of the important Open Problems in today's progress in the physically meaningful applications of the nuclear microscopic theories.

- It can easily be shown that the presence of inter-parametric correlations manifest itself in the form of deep valleys of the χ 2 -test plots that can be illustrated in terms of projections onto certain hyper-planes in the space of p ∈ R np . Illustration of this type is shown in figure 8 (left) for the correlation between the radius and the potential depth parameters of the central potential in the case of the Woods-Saxon Hamiltonian with all other remaining parameters fixed. The generalisation of this test for the case of minimisation over all adjustable parameters does not change the conclusion: the two parameters are strongly, but not fully correlated; indeed, the full, exact correlation would imply that the bottom of the valley rest at the zero-value. Let us emphasize that the details of the behaviour change with the number of the experimental levels taken into considerations.

We may remark at this point that: a. Examining of the possible presence of the parametric correlations is a prerequisite; only after having established the stability of the adjustment, cf. Eqs. (22-23), we may decide about the next steps of the procedure; b. Fortunately, the methods of regularization of the unstable algorithms are copiously described in the textbooks and specialized literature -it is usually sufficient to select the optimal approach. But the solution of the problem begins with the first item above rather than proceeding according to the 'chi-by-the-eye' approach mentioned in the third footnote of this article.

Summary -Predictive power of nuclear Hamiltonians: Open Problems

We have discussed the question of the statistically-significant determination of the nuclear physics Hamiltonians through conscious adjustment of its free parametersthe method known in applied mathematics as the inverse problem. In contrast to sometimes formulated opinions, these are the inherent theory and modelling errors that lead to the most important parametric uncertainties of the Hamiltonians and not necessarily the experimental errors. In our analysis we have distinguished between the general discussion of the impact of theoretical and experimental errors on the theory's predictive power and a specific realm of applications: Illustrations using the example of properties of single-nucleon levels of doubly-magic spherical nuclei.

The article advocates the new way of formulating and interpreting the theory predictions: not only provide the theory's numerical results (the present day situation) but provide also their statistical significance represented e.g. by probability(ies) with which the theoretical predictions are expected to apply in nature.

The 1 st series of Open Problems consists then in constructing the error probability distributions that would quantify in the best possible, even though always a subjective way, the degree of the missing knowledge. In the application to the single-nucleon spectra, it would be important to extend the analysis to all doubly-magic spherical nuclei, gradually and systematically including the impact of pairing correlations, coupling with the surface oscillations (quadrupole and octupole), as well as the coupling with the continuum for the weakly bound states. The generalized modelling would include both the Hartree-Fock and phenomenological mean field theories as the first step.

For a given Hamiltonian, the functioning of the parameter adjustment algorithms depends both on the number of data points as well as on their intrinsic characteristics (such as e.g. deeply bound nucleonic states, high-j or low-j orbitals etc.) vs. the number of parameters. Hamiltonians that can be called adequate in the context are able to exactly reproduce the non-trivial number of data points, for instance typically 6-to-8 single-nucleon levels in all the doubly magic nuclei when using Woods-Saxon potentials. We have demonstrated that for these low numbers of data points, the discussed Hamiltonians are also capable of reproducing an infinite number of sets of data points obtained from a selected initial set by a random noise perturbation. Such Hamiltonians reproduce then (within relatively broad limits) an arbitrary noise exactly. It is therefore necessary to establish the lower limit for the numbers of levels in the fit-input data set so that the Hamiltonian can be really constrained. On the basis of the Woods-Saxon Hamiltonian tests we could establish that this number exceeds the number of experimentally known single-particle levels even in the relatively well studied 208 Pb nucleus.

The 2 nd series of Open Problems, specifically addressing the single-nucleon subfield of the project, would consist in as extended as possible an experimental study of the new average one-nucleon energies introduced in section 2. Alternatively, a similar type of information could be obtained from studying the so-called band-head energies in the deformed nuclei; those also constrain specifically the mean-field Hamiltonians and combining the two approaches seems a valuable complementary counter-checking research to perform.

The 3 rd series of Open Problems concerns an important general aspect of the proposed new strategy and techniques: examining the inter-dependences among some parameters of the Hamiltonian that remain 'hidden' i.e. we can discover them only after having performed an analysis of the type discussed in section 5. This mechanism, as discussed in the article, depends not only on the structure of the Hamiltonian but also on the samples of the data used to solve the inverse problem -and thus should be discussed on the case-by-case basis. Its importance consists in the fact that in the presence of inter-relations, the parametric confidence intervals diverge rendering the obtained Hamiltonians unsuited for extrapolations into the new areas of nuclei.

1

  and the average energy, εκ. All the energies are given in MeV.

Figure 2 .

 2 Figure 2. Histograms representing the Schrödinger parameter-transforms of (11) obtained by fitting to 6, 11 and 21 pseudo-experimental neutron levels in 208 Pb, assuming a common Gaussian noise distribution with σ=0.1 MeV, left, and 0.4 MeV, right. Vertical dashed lines give the position of the exact parameter values from table2. For comments on the double-maximum structure cf. Refs.[11,12].

- 5 . 5 Neutron

 55 0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.

- 5 . 5 NeutronFigure 3 .

 553 Figure 3. Schrödinger energy-transforms, (12), for a few neutron levels in 208 Pb when performing fits to 11 pseudo-experimental levels with the Gaussian noise distributions of the width σ=0.1 MeV (left) and 0.4 MeV (right).

Figure 5 .

 5 Figure 5. Similar to that in figure 4 but for levels 2f 7/2 (left) and 2g 7/2 (right).

( a )

 a Shifts in energy due to the fragmentation measured in neighbouring nuclei.(b) The values obtained through analogy by extrapolating from the data on 208 Pb. The numbers in parentheses give errors in the last digit. (c) From[15] .

Table 1 .

 1 Proton and neutron levels around the two magic gaps at Z 0 =82 and N 0 =126. Symbol e exc 1 represents the energy of the lowest state of each spin value measured in 209 83 Bi, 207 81 Tl, 209 Pb 127 , and 207 Pb 125

  [START_REF]ENSDF data base[END_REF] MeV) might be linked to2.614 MeV what would give about 15% in terms of energy. Noteworthy is the fact that the shifts calculated in 207 Pb are markedly larger, close to 30% of the 3 -energy; but in that case the two levels of interest are very far from the Fermi level, lying in the energy range in which the mixing with many other configurations is expected.Similarly, the quadrupole vibration can be at the origin of the fragmentation involving couples of single-nucleon states, characterized by ∆l=2 (and ∆j=2), such as πh 9/2 ↔ πf 5/2 , πs 1/2 ↔ πd 5/2 , and πd 3/2 ↔ πg 7/2 . Once more, we can rate the mean shift in energy (around 0.6 MeV, see table1) in proportion of the 2 + energy, what gives about 15%, in terms of energy.

  Histograms representing the r.m.s transforms of(10) obtained by fitting to the pseudo-experimental data. In this case we assumed a common 'noise' Gaussian distributions D k (η k ) → N (η k ), with the width parameters σ=0.1 and 0.4. Left: the results for 6 levels distributed symmetrically around the Fermi level for the neutrons in 208 Pb; Right -similar but using 21 highest-bound levels. The curves are normalized so that the histogram area gives 1, but scaled, for better legibility so that the highest maximum is at 1. The label 'noise' refers to the input noise distributions, prior to fitting.

	Probability Distribution	0.2 0.4 0.6 0.8 1.0			6 levels σ = 0.1 MeV; χ σ = 0.1 MeV; noise σ = 0.4 MeV; χ σ = 0.4 MeV; noise	Probability Distribution	0.2 0.4 0.6 0.8 1.0				21 levels σ = 0.1 MeV;χ σ = 0.1 MeV; noise σ = 0.4 MeV;χ σ = 0.4 MeV; noise
		0.0							0.0
		1	0.2	0.3	0.4	0.5	0.6	0.7	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7
	208 82 Pb		Function χ			208 82 Pb			Function χ
				Figure 1.

  The energy differences as measures of the fit-output uncertainty in the single-particle levels listed -caused by the displacement (as measures of the input-level uncertainty) of the energy of the 1i 11/2 -level, left, and the 4s 1/2 -level, right. The displacement window of 4 MeV is centred on the original positions of these levels as obtained with the parameters in table 2. Fits using in total 11 pseudo-experimental levels. figure 4. One may notice that the forms of the over-all dependence are different, e.g. the strongest variation is approximately represented by the interval of only [-0.3, +0.3] MeV for 1i 11/2 and [-0.6, +0.6] MeV in the case of the 3d 3/2 -levels. Moreover, the strength and the sign of the coupling depend significantly on the quantum numbers of the concerned states: whereas the state 2f 5/2 practically does not couple with 3d 3/2 it is among the strongest-coupling with the level 1i 11/2 . It is instructive to compare the

	m						m
		0.8							0.8
	Energy Difference [MeV]	-0.4 0.0 0.4	3p 1/2 2f 7/2 2g 9/2 1i 13/2 3p 3/2 4s 1/2 3d 3/2 2g 7/2 3d 5/2 2f 5/2		1i 11/2		4s 1/2 1i 13/2 3p 1/2 3d 3/2 3p 3/2 3d 5/2 2f 7/2 2g 7/2 2f 5/2 2g 9/2	Energy Difference [MeV]	-0.4 0.0 0.4	2f 5/2 3p 1/2 4s 1/2 3p 3/2 1i 11/2 3d 5/2 2g 9/2 2g 7/2 1i 13/2 2f 7/2	3d 3/2	1i 11/2 2f 7/2 2g 9/2 3p 3/2 1i 13/2 2f 5/2 3p 1/2 3d 5/2 4s 1/2 2g 7/2
		-0.8							-0.8
			-2.0	-1.0	0.0	1.0	2.0			-2.0	-1.0	0.0	1.0	2.0
	208 82 Pb	Energy Displacement [MeV]	208 82 Pb	Energy Displacement [MeV]
	Figure 4. m		m
		0.8							0.8
	Energy Difference [MeV]	-0.4 0.0 0.4	2f 5/2 3p 1/2 4s 1/2 3d 5/2 2g 9/2 1i 11/2 3d 3/2 2g 7/2 3p 3/2 1i 13/2		2f 7/2		2g 7/2 3d 3/2 3d 5/2 4s 1/2 1i 11/2 2g 9/2 1i 13/2 2f 5/2 3p 1/2 3p 3/2	Energy Difference [MeV]	-0.4 0.0 0.4	3d 5/2 1i 11/2 1i 13/2 3p 1/2 2g 9/2 3d 3/2 2f 5/2 4s 1/2 3p 3/2 2f 7/2	2g 7/2	2f 7/2 2g 9/2 3p 3/2 3d 5/2 3p 1/2 1i 13/2 4s 1/2 1i 11/2 2f 5/2 3d 3/2
		-0.8							-0.8
			-2.0	-1.0	0.0	1.0	2.0			-2.0	-1.0	0.0	1.0	2.0
	208 82 Pb	Energy Displacement [MeV]	208 82 Pb	Energy Displacement [MeV]
											3/2 ) is presented in

  139 56 Ba 83 , 141 58 Ce 83 , 143 60 Nd 83 and 145 62 Sm 83 , using one-neutron transfer (d, p) reaction on stable targets. The corresponding neutron levels are illustrated in figure

  Results of the extrapolation from the 208 Pb to the 132 Sn nucleus for the neutrons. With a sample of N =20 000 sets η i , cf. the comments around equations(10)(11)(12), the parametric histograms resembling those in figure2have been obtained (using 11 level simulation); the noise width σ=0.1MeV. With each of the so obtained N =20 000 sets of parameters the results for the neutrons in 132 Sn nucleus have been obtained. Solid bars give positions of experimetal levels from table 3, dashed bars give positions of experimentally uncertain levels.

	Probability Distribution	0.2 0.4 0.6 0.8 1.0	2d 5/2	1g 7/2 1h 11/2 3s 1/2	2d 3/2				2f 7/2		3p 3/2	1h 9/2 3p 1/2
		0.0
			10	-9	-8	-7	-6	-5	-4	-3	-2	-1	0
	132 50 Sn					Neutron Energy [MeV]
				Figure 7.

  Left: Illustration of the χ 2 -test in function of the depth and radius parameters of the central Woods-Saxon potential for the neutrons in 208 Pb, evaluated for 11 levels close to the Fermi energy. The remaining four parameters of the potential are fixed at the values from table 2. Right: Results of the N=20 000 minimisations over all the six parameters of the potential to fit the same 11 pseudo-experimental levels as in the left-hand side, each time changing the noise hypothesis (cf. discussion around Eqs.(10)(11)(12)). Observe the same parametric correlations visible from both types of tests.

	Radius Parameter r c o [fm]	1.3 1.4 1.5 1.6		2 .4 3 .2 1 .6	3 .2	0 . 8	1 . 6 2 . 4		4.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6	Radius Parameter r c o [fm]	1.6 1.3 1.4 1.5
		1.2	50	-45	-40	-35	-30	0.0		-50 1.2	-45	-40	-35	-30
	208 82 Pb	Depth Parameter V c o [MeV]	χ[MeV]	208 82 Pb	Depth Parameter V c o [MeV]
					Figure 8.

Acknowledgements. Support within the collaboration project TetraNuc through the IN 2 P 3 , France and through exchange program between IN 2 P 3 and COPIN, Poland, is acknowledged.