B Billaud
email: bbillaud@u-cergy.fr

T.-T Truong

Lamb shift of interactive electron-hole pairs in spherical semiconductor quantum dots

Keywords: spherical semiconductor Quantum Dot, Lamb shift PACS: 12.20.Ds, 71.35.-y, 73.22.Dj

The ground state Lamb shift of a semiconductor spherical Quantum Dot is computed in the effective mass approximation. It appears to be significant enough to be detectable for a wide range of small quantum dots synthesized in semiconductors. A possible way to observe it, via the Casimir effect, is suggested.

Introduction

The investigation of atomic physics properties in semiconducting Quantum Dots (QDs) is a popular topic for fundamental as well as applied physics. Such nanostructures restrict the motion of charge carriers to a confined region of space. Two QDs are never identical, because of the crucial role of phonons, surface effects or bulk disorder on their electronic structure, but they may be advantageously considered as artificial giant atoms. Thanks to their adjustable quantized energy spectrum, controlled only by their size, they may be used, for example, as semiconductor lasers [1], single-photon sources [2], qubits [3], single-electron transistors [4], artificial fluorophores [5]. In the early 1980s, the so-called Quantum Size Effects (QSE), characterized by a blue-shift of their optical spectra, has been observed in a large range of strongly confined systems [6][7][8][9][10]. It comes from a widening of the semiconductor optical band gap, due to the increase of the charge carriers confinement energy [11]. Modern methods as well as empirical pseudo-potential methods or ab initio approaches, such as the Density Functional Theory (DFT), which are appropriate for numerical determination of semiconductor bulk band structures or confinement energies, are discussed in [12,[START_REF] Delerue | Nanostructures: Theory and Modelling[END_REF]. However, to the best of our knowledge, no comprehensive approach, which offers a significant analytic treatment, seems to exist at present.

In this work, to explore their analogy with real atoms, we look at the interaction of a spherical semiconductor QD with a quantized electromagnetic field and investigate the Lamb shift of its energy levels. Discovered and actively studied in the late 1940s [START_REF] Lamb | [END_REF][15][16], the Lamb shift still remains nowadays an area of intense research, for example within dielectric materials [18], on the excitation modes of an electromagnetic cavity [19], or on the coupling of electromagnetic modes to the semiconductor QD surface [20]. Recent works also present experimental protocols, in which Lamb shifts enhanced by an electromagnetic cavity are measured in nanostructures like transmon [21] or studied in semiconducting QDs coupled to a planar slab of negative-index material [22]. However, the Lamb shift, well understood for real atoms, seems not to be approached neither theoretically nor experimentally in semiconductor QDs.

The object of this paper is to fill this gap by offering a consistent theoretical treatment leading to the evaluation of the shifts for a wide range of realistic spherical semiconductor QDs. To describe a confined electron-hole pair, it seems appropriate to use the popular effective mass approximation (EMA) model [23][24][25][26]. The electron-hole Coulomb interaction is to be taken into account subsequently by the Ritz variational principle. We first review the tools needed to work out the Lamb shift in section 2, and the basic properties of a spherical semiconductor QD in section 3. From there on, we deduce, in sections 4 and 5, the Lamb shift for a massive charged particle confined by an infinite potential well. In section 6, we finally treat the case of a spherical semiconductor QD, and show that there exists a possibility to experimentally detect the Lamb shift, at least in a so-called strong confinement regime and for an judiciously chosen semiconductor. We also suggest a protocol, built on the Casimir effect, which might allow to observe of this predicted Lamb shift, despite the non-degeneracy of semiconducting QD levels. A concluding section summarizes our main results and indicates possible future research perspectives.

Considerations on Lamb shift

In 1947, investigations on Zeeman effect in hydrogen atom showed that its fine structure does not agree with the predictions of Dirac theory [START_REF] Lamb | [END_REF]. The 2s-level, which should be degenerated with 2p-levels, is actually shifted by an energy ≈1057MHz, the so-called Lamb shift. Theoretical work demonstrated that it arises from the coupling of the electron motion with the surrounding quantized electromagnetic field [15,16]. Consider a non-relativistic spinless massive particle of mass m * and of charge qe in interaction with a quantized electromagnetic field1 . It is represented by the Pauli-Fierz Hamiltonian in the Coulomb gauge [29], here written in units where = c = 1,

H PF = H 0 + eH int + e 2 H int + H em .
The particle Hamiltonian in absence of electromagnetic field H 0 = p 2 2m * +V (r) is supposed to be diagonal in an orthonormal basis of its eigenvectors {|n } n with energy eigenvalues

{E n } n . H int = -q A•p m * and H int = q 2 A 2 2m *
are interaction Hamiltonians, and

H em = d 3 r E 2 +B 2 2
is the free electromagnetic field Hamiltonian, A being the vector potential, E and B the electric and magnetic fields. The electromagnetic field is second quantized in the Coulomb gauge [START_REF] Ballac | Physique quantique 2 e dition[END_REF], neglecting the interaction Hamiltonian H int on the basis of weak intensities of light sources involved 2 . We now review the results of the two main approaches to the Lamb shift.

The Bethe approach to the Lamb effect is purely pertubative. The quantum second order time independent degenerate perturbation theory is applied to the Pauli-Fierz Hamiltonian H PF , where Hamiltonians H int and H int are treated as perturbations in the weak field limit [15]. Using renormalization arguments, the Lamb shift for an energy eigenstate |n is found to be given by

∆E n = 2α 3π q 2 m * 2 log m * |E m -E n | m | m|p|n | 2 {E m -E n } = α 3π q 2 m * 2 log m * |E m -E n | n|∇ 2 V (r)|n , (1)
where |E m -E n | is the mean value of energy level difference absolute values. Historically, this predicts a Lamb shift for the hydrogen atom 2s-level in excellent agreement with experimental values [15,17].

In the Welton approach, the Lamb shift is interpreted as a fluctuation effect on the particle position due to its interaction with the surrounding electromagnetic field. These fluctuations ∆r can be described as a continuous random variable, whose probability density is a three-dimensional centered isotropic Gaussian distribution of variance

(∆r) 2 = 2α π q 2 m * 2 log m * κ , ( 2
)
where κ is a IR cut-off, and m * is used as a natural UV cut-off consistent with non-relativistic assumption, discarding fluctuations modes of order of the particle Compton wavelength [16]. The particle then evolves in a new effective potential V (r + ∆r) , avegered on the fluctuation distribution, which can be written as3

V (r + ∆r) = 1 + (∆r) 2 6 ∇ 2 + . . . V (r) = V (r) + ∆V (r).
where ellipsis dots ". . . " denote terms of order higher than first order in α. The corrective term ∆V (r) of the first order in α is precisely the term giving rise to the Lamb shift

∆E n = d 3 r | r|n | 2 ∆V (r) = α 3π q 2 m * 2 log m * κ n|∇ 2 V (r)|n .
Comparison of this result with Eq. (1) shows that the IR cut-off κ can be identified to |E m -E n | = κ in the Bethe result. This also means that κ should not depend on the quantum numbers n.

Physics of a spherical semiconductor QD

It is admitted that, under some physical assumptions, semiconductor QDs may be described by EMA models as a first approximation. We use the standard EMA model presented in [26], as a first attempt to apprehend the Lamb effect in semiconductor QDs. Electrons and holes are assumed to be non-relativistic spinless particles of effective masses m * e,h , confined in an infinite spherical potential well

V ∞ (r) = 0 if 0 ≤ r ≤ R, region I ; ∞ if r > R, region II.
They are consequently isolated from the insulating surrounding of the QD. The electromagnetic field amplitude should not then exceed some threshold, so that charge carriers would not leak out by tunnel effect. This working assumption is known as the weak field limit. Such model leads to an overestimation of the electron-hole pair ground state energy for small QDs, which can be corrected by restoring a confining finite potential step of experimentally acceptable height [START_REF] Thoai | [END_REF]. Other QD models with parabolic confinement [33,34] or parabolic potential superimposed to an infinite potential well [35] exist. But, the concept of QD size is no longer well-defined since their eigenfunctions are delocalized.

Interactive electron-hole pair EMA model

The electron-hole Coulomb interaction V C (r eh ) is taken into account in the total Hamiltonian

H 0 = H e + H h + V C (r eh ),
where κ = 4πε, ε denotes the semiconductor dielectric constant, r eh the electron-hole relative distance, and

H e,h = - ∇ 2 e,h
2m * e,h + V ∞ (r e, ) the electron and hole confinement Hamiltonians. Without loss of generality, the semiconductor energy band gap E g may be set equal to be zero for convenience. Electron and hole, as decoupled particles, have wave functions of the form

ψ ∞ lnm (r e,h ) = R I∞ ln (r e,h )Y m l (θ e,h , ϕ e,h ) = 2 R 3 χ [0,R] (r e,h ) j l+1 (k ∞ ln ) j l k ∞ ln R r e,h Y m l (θ e,h , ϕ e,h ).
where l ∈ N, n ∈ N {0} and m ∈ [[-l, l]], Y m l (θ, ϕ) are spherical harmonics and j l (x) spherical Bessel functions of the first kind, χ A (r) = 1 if r ∈ A 0 otherwise the radial characteristic function of the set A. Finally, {k ln } ln is the wave numbers set, defined as the n th non-zero root of j l (x), from the continuity conditions at r = R [23]. The respective energy eigenvalues for electron and hole is expressed in terms of {k ln } ln as

E e,h∞ ln = (k ∞ ln ) 2 2m * e,h R 2 .
The interplay between quantum confinement energy, scaling as ∝ R -2 , and Coulomb interaction, scaling as ∝ R -1 , can be described by the ratio R a * of the QD radius R over the Bohr radius of the bulk Mott-Wannier exciton a * = κ e 2 µ , µ being the reduced mass of the exciton. As an exact analytic solution is beyond reach, two regimes are discussed in [25].

Strong confinement regime

In this regime, valid for R 2a * , the electron-hole relative motion is so affected by the infinite potential well, that exciton states should be considered as uncorrelated electronic and hole states. The Coulomb potential is considered as a perturbation with respect to the infinite confining potential well. A variational approach is used to obtain the electron-hole pair ground state, with the following trial function

φ ∞ (r e , r h ) = ψ ∞ 010 (r e )ψ ∞ 010 (r h )φ rel (r eh ), (3)
which is a product of non-interacting charge carriers ground state wave functions with an interaction wave function of the form φ rel (r eh ) = e -σ 2 r eh , σ being a variational parameter. The electron-hole pair energy minimization selects the value σ 0 = 4B a * , and yields the ground state energy4

E strong eh = E ∞ eh -A e 2 κR -4B 2 E * ,
where E ∞ eh = E e∞ 01 + E h∞ 01 is the electron-hole pair ground state confinement energy, and E * = 1 2µa * 2 the binding exciton Rydberg energy [25].

Weak confinement regime

In this regime, valid for R 4a * , the exciton retains its character of a quasi-particle of total mass M = m * e + m * h . Its center-of-mass motion is confined, and should be quantized. The Coulomb interaction remains a perturbation to the infinite confining potential well, and the variational function φ ∞ (r e , r h ) should be kept. However, the QD size allows for a partial restoration of the long range Coulomb potential between the charged carriers, such that the Coulomb energy and the kinetic energy in electron-hole relative coordinates are of the same order of magnitude. Then, the leading contribution to the ground state energy of the exciton should be -E * , which may be viewed as the ground state energy of a hydrogen-like atom of mass µ. The total translational motion of the exciton, should be restored and contribute to the exciton total energy by a term π 2 2M R 2 , i.e. the ground state energy of a free particle trapped in a region of size R. The exciton is treated as a rigid sphere of radius η(λ)a * , where

η(λ) is a phenomenologically determined function of λ = m * h m * e [25]
. The exciton center-of-mass cannot reach the infinite potential well surface unless the electron-hole relative motion undergoes a strong deformation. On this basis, to improve the description of the excitonic ground state, we let the center of mass motion in the variational procedure be represented by a plane wave φ G (r G ) = e i π R σG•rG , where r G is the center-of-mass coordinates and σ G is vector quantum number of unit norm |σ G | 2 = 1. The trial function φ ∞ (r e , r h ) takes now the form

ψ ∞ (r e , r h ) = φ ∞ (r e , r h )φ G (r G ). ( 4
)
This leaves the exciton probability density unchanged as well as the Coulomb potential matrix element, whereas the confinement Hamiltonian H e + H h mean value gets an appropriate further contribution π 2 2M R 2 . The variational calculation yields σ 0 ≈ 2a * -1 , and a ground state variational energy of

E weak eh = -E * + π 2 6µR 2 + π 2 2M (R -η(λ)a * ) 2 , ( 5
)
where terms of second and third order terms in a * R 1 are neglected. An analytical expression for η(λ) can now be extracted from Eq. ( 5), as shown in [26].

Pseudo-potential-like method

The exciton ground state energy E weak eh of [25], differs from Eq. ( 5) by the term π 2 6µR 2 , interpreted as a kinetic energy term in the relative coordinates. As the virial theorem in this set of coordinates should be satisfied, this energy should be already contained in the Rydberg energy term -E * , and therefore should be removed. An elegant way to do this consists in introducing a pseudo-potential W (r eh )

W (r eh ) = W r 2 eh R 2 e -2 r eh a * = - 32π 2 9 E * r 2 eh R 2 e -2 r eh a * .
Inspection shows that it makes contributions to the second order of the exciton total energy but not to the third one, while leaving the zeroth and first order terms. Higher order contributions are interpreted as higher order corrections to the kinetic energy of the exciton π 2 2M R 2 . While the amplitude W is to be fixed to get the correct kinetic energy ψ|W (r eh )|ψ

ψ|ψ = -π 2 6µR 2 1 + O a * 2 R 2
, the pseudo-potential form is not arbitrary. It is attractive at distances ≈ a * to promote excitonic state with typical size around its Bohr radius, repulsive at short distances to penalize excitonic state with small size, and exponentially small for large distances in order not to perturb the long range Coulomb potential [26].

Adding the pseudo-potential W (r eh ) to the Hamiltonian H 0 implies a significant decrease of the expected value of the exciton energy in the strong confinement regime φ|W (r eh )|φ φ|φ = -64π 2 9 CE * 1 + O R a * , which is now only valid for 2R a * . The excitonic energy computed in presence of the pseudo-potential gets a better fit to experimental results in this validity domain, than those calculated without this tool. Nevertheless, the divergence for very small QD size still persists as a relic of the infinite potential well assumption.

Lamb shift of a particle confined in a spherical potential well

This section deals with calculational steps of the Lamb shift for a particle of mass m * and of charge ±e, i.e. q = ±1, confined in a theoretical spherical QD with a spherical infinite potential well V ∞ (r) 5 . The difficulty then resides in the determination of the Poisson equation satisfied by this singular potential. To overcome this issue, we will replace it by a potential step of finite constant height V ≥ 0

V (r) = V χ ]R,∞[ (r) = 0 if 0 ≤ r ≤ R, region I ; ∞ if r > R, region II.
When V → ∞, the Lamb shift undergone by any energy level of the particle in the infinite potential well is supposed to be the finite part, independent from V , of the expansion in powers of V of the Lamb shift undergone by the level with same quantum numbers of the particle in the finite potential step, given by Eq. ( 1)

∆E lnm = α 3πm * 2 log m * κ ψ lnm |∇ 2 V (r)|ψ lnm ,
where, for fixed quantum numbers l ∈ N, n ∈ N {0} and m ∈ [[-l, l]], the wave function ψ lnm (r, θ, ϕ) is a linear combination of the spherical Bessel function j l k ln R r and of the spherical Hankel function h

(1) l i K ln R r ψ lnm (r) = R I ln (r)χ [0,R] (r) + R II ln (r)χ ]R,∞[ (r) Y m l (θ, ϕ) = A ln j l k ln R r χ [0,R] (r) + C ln h (1) l i K ln R r χ ]R,∞[ (r) Y m l (θ, ϕ).
Because of the limit V → ∞, it is possible to adjust the step potential height V , so that the energy eigenvalue E ln is written as function of the wave numbers k ln and K ln in regions I and II

E ln = k 2 ln 2m * R 2 and V -E ln = K 2 ln 2m * R 2 ≥ 0.
The normalization coefficients A ln and C ln , and the wave numbers k ln and K ln are subjected to boundary conditions at r = R, i.e.

A ln j l (k ln ) = C ln h (1) l (iK ln ) A ln k ln j l±1 (k ln ) = iC ln K ln h (1)
l±1 (iK ln ) and by orthonormalization conditions. In region I, the radial wave function R I ln (r) has the same form as the radial wave function R I∞ ln (r), which is then really the limit of R I ln (r), when V → ∞. The same remark should be made for the radial wave function

R II ln (r) --→ V ∞ 0, since, by construction, K n ∼ V ∞ 0. Nevertheless,
the introduction of the potential step allows the particle to get out the confining potential by tunnel effect, even if the probability for such event is exponentially small.

Lamb shift via the Poisson equation

The potential step V (r) now satisfies the Poisson equation

∇ 2 V (r) = V 2 R δ(r -R) + δ (r -R)
, δ(r) being the Dirac distribution and δ (r) its derivative. In our regularization method, the unique term independent of V of the diagonal matrix element of the Laplacian of the potential step V (r) expansion in powers of V should be retrieved when V → ∞, since it is the only non-divergent and non-vanishing one. Expressions

ψ lnm |∇ 2 V (r)|ψ lnm = - 4V π |A ln | 2 k 2 ln j l (k ln )j l (k ln ) = 2 R 3 2V m * (k ∞ ln ) 2 - 8 R 2 E ∞ ln + o(1).
suggest that

∆E ∞ ln = - 8 3π α m * 2 E ∞ ln R 2 log m * κ . ( 6
)
As expected, because of the spherical symmetry of the potentials V (r) and V ∞ (r), the Lamb shift is independent of the azimuthal quantum number m. Furthermore, it is negative, since, as we shall see later, the IR cut-off satisfies the constraint κ ≤ m * . Hence, the Lamb effect brings down the energy levels of the particle, instead of raising them up, as in real atoms. This is a remarkable characteristic of the Lamb shift which, to the best of our knowledge, is noted for the first time.

Infrared cut-off κ

The evaluation of the IR cut-off κ = |E ∞ ln -E ∞ ij | is also problematic. In the Bethe approach, the IR cut-off explicitly depends on the considered quantum state, which is not the case in the Welton approach. In real atoms, it has been shown that the IR cut-off is of about the same order of magnitude independently of the energy levels [17]. To introduce a relevant and appropriate IR cut-off κ, we suggest to define it by the following average

κ = |E ∞ ln -E ∞ ij | = 1 2m * R 2 ijln (2i + 1)(2l + 1)|(k ∞ ln ) 2 -(k ∞ ij ) 2 | ijln (2i + 1)(2l + 1) . ( 7
)
As quantum numbers are not limited from bellow, and are infinite, this sum is infinite. To confer a rigorous meaning to Eq. ( 7), a regularization method should be prescribed. First, the number of terms in the sum is finite, since the trapped particle, being excited through its interaction with the electromagnetic field, can have access only to a finite number of states. The highest accessible level is limited by the finite field energy

E lim = κ 2 lim 2m * R 2
, which allows to define a UV cut-off κ lim for authorized wave numbers. In practice, the Feynman regularization method is used instead of this regularization method. A dimensionless parameter > 0 is introduced, and the regularized sum

κ = 1 2m * R 2 ijln (2i + 1)(2l + 1)|(k ∞ ln ) 2 -(k ∞ ij ) 2 |e -i e -j e -l e -n ijln (2i + 1)(2l + 1)e -i e -j e -l e -n
. should be expanded in powers of . The finite part of this expansion should be kept and identified with the IR cut-off κ. Considering the Bessel functions asymptotical behavior and their roots repartition at infinity [START_REF] Gradshteyn | Table of Integrales, Series and Products[END_REF], for sufficiently large quantum number q ∈ N {0}, but for all p ∈ N, we can assume that

k ∞ pq ≈ k ∞ p1 + (q -1)π ≈ k ∞ 01 + (q -1) = qπ.
While this approximation is not valid for any quantum numbers q ∈ N, in the limit → 0, the most important contribution to κ corresponds to high quantum numbers, i.e. it should be used for all wave numbers. Then,

κ = π 2 2m * R 2 jn |n 2 -j 2 |e -j e -n jn e -j e -n = π 2 2m * R 2 4 2 + 3 + 7 6 + o(1) ,
which leads to the choice

κ = 7π 2 12m * R 2 .
Thus, κ is of the order of magnitude of the ground state energy of the confined particle, contrary to the case of real atoms [17]. Because of the state degeneracy, such behavior is a posteriori expected. Terms, whose relative importance is the largest in Eq. (7), are terms of the order of magnitude

≈ |(k ∞ ln+1 ) 2 -(k ∞ ln ) 2 |.
The asymptotical behavior of the Bessel functions roots, for sufficiently large λ * ≈ 2, 399λ * , of the same order of magnitude of the particle reduced Compton wave lengthλ * = m * -1 . Thus in a theoretical QD of radius R smaller than R * min , the confined particle should acquire at least a confinement energy E ∞ 01 of the order of magnitude of its rest mass energy

n ∈ N {0}, |(k ∞ ln+1 ) 2 -(k ∞ ln ) 2 | ≈ π 2 (
m * ∝ π 2 2m * R * 2 min ≤ π 2 2m * R 2 = E ∞ 01 ,

Alternative route to the Lamb shift: a direct calculation

An alternative calculation of the Lamb shift undergone by a particle confined in the infinite potential well V ∞ (r) is presented here. Starting with the regularized version by the finite potential step V (r) of Eq. ( 1), but for which summations over the quantum numbers i, j and k are not performed

∆E lnm = 2 3π α 2 m * 2 log m * κ ijk | ψ ijk |p|ψ lnm | 2 {E ij -E ln }, we directly evaluate the sum ijk | ψ ijk |p|ψ lnm | 2 {E ij -E ln } = -ψ lnm |∇ • H 0 ∇ -E ln ∇ 2
|ψ lnm , and expand it in powers of V , as already mentioned in subsection 4.1. The obtained expansion does not have divergent terms when V → ∞, such as terms scaling as ∝ √ V of subsection 4.1. So, obtaining the Lamb shift for a state confined by the infinite potential well from the one confined by the finite potential step defined by the same quantum numbers is rigorously justified. Actually, it can be shown that

{∇ • H 0 ∇ -E ln } ψ lnm (r, θ, ϕ) = R I ln (R) -R II ln (R) 2m * δ (r -R) + 2 R δ(r -R) Y m l (θ, ϕ), from which one gets ijk | ψ ijk |p|ψ lnm | 2 {E ij -E ln } = R 2 2m * R I∞ ln (R)R I∞ ln (R) + o(1) -4 E ∞ ln R 2 + o(1).
As stated, the first term in this expansion, and therefore also in the Lamb shift expansion, is the finite term. Despite an identical regularization method, no divergent terms appear in the limit V → ∞, which is most satisfactory from the conceptual point of view. This corroborates that the fact that such a term is not physical, when the trapped particle is confined by the infinite potential well V ∞ (r). As we will see later, this property appears to be fundamental. Finally, this second calculation validates the regularization of the confining potential by an intermediate finite potential step, since in the end it leads to a Lamb shift, identical to Eq. ( 6).

Gauge invariance and Lamb shift

As an observable effect, the Lamb shift is indeed gauge invariant. Welton argument is clearly gauge invariant, since the electromagnetic field only intervenes through its energy density, a gauge invariant quantity. Bethe argument depends however on the gauge, since the interaction Hamiltonian H int ∝ A • p in the Pauli-Fierz Hamiltonian H PF , used as perturbation, is written in the Coulomb gauge. Therefore, it is useful to check gauge invariance in calculations of the Lamb shift of the energy levels for a theoretical QD. To this end, we study the Lamb effect in the gauge imposed by the electric dipole approximation.

In this approximation, the electric field spatial variation is negligible over typical particle distancesi.e. E = E(t, 0) and the related scalar potential is A 0 (t, r) = -r • E(t, 0) -, and that there is no magnetic field B = 0, hence no vector potential A = 0. The system Hamiltonian is then of the form

H PF = H 0 + H em + eH int ,
where H int = -qr • E(t, 0) is the new interaction Hamiltonian of the particle with the electromagnetic field. In particular, it is obvious to observe that H int and H int contribute in the same way to the classical action of the particle-electromagnetic field system, especially when terms scaling as ∝ A 2 are neglected 6 . Thus, quantization in the gauge fixed by the electric dipole approximation makes sense.

Bethe approach in the electric dipole approximation

We now go over Bethe argument with the dipole approximation. The second order correction is

∆E n = - 2α 3π q 2 m | m|r|n | 2 dk k 2 -(E m -E n )k + (E m -E n ) 2 -dk (E m -E n ) 3 E m -E n + k = - 2α 3π q 2 m | m|r|n | 2 m * 3 3 -(E m -E n ) m * 2 2 + (E m -E n ) 2 m * -(E m -E n ) 3 log m * κ .
The two integrals in this expression have respectively a cubic and a logarithmic UV divergence. Thus, the second integral is the one which gives rise to the Lamb shift in the dipole approximation. It is natural to introduce m * as UV cut-off and κ as IR cut-off to regularize the integrals. Renormalization arguments of the original Bethe article are used to get the corrective term

∆E n = - 2α 9π q 2 m * 3 n|r 2 |n + α 2π q 2 m * + α 3π q 2 m * 2 log m * κ n|∇ 2 V (r)|n .
The third term in ∆E n is recognized as the Lamb shift for a state |n , as expected from the Welton approach or the Bethe approach in the Coulomb gauge. The second term is simply a numerical constant, which can be omitted using an additive renormalization argument. The first term is explicitly gauge dependent, since

n|r 2 |n ∝ n|H 2 int |n = m | m|H int |n | 2 .
So, the Lamb shift in the gauge fixed by the electric dipole approximation is identical to the one computed in the Coulomb gauge.

Alternate route to Lamb shift: a direct calculation

Applying the reasoning of subsection 4.3 to the dipole approximation, we obtain once again the Lamb shift given by Eq. ( 6). Using expressions

∆E lnm = 2α 3π log m * κ ijk | ψ ijk |r|ψ lnm | 2 (E ij -E ln ) 3 = 2α 3π log m * κ ψ lnm |[H 0 , r] † •[H 0 , [H 0 , r]]|ψ lnm ,
and computing the following commutators

[H 0 , r]ψ lnm (r, θ, ϕ) = - 1 m * ∂ r r∂ r + ∂ θ r ∂ θ r 2 + ∂ ϕ r ∂ ϕ r 2 sin 2 θ ψ lnm (r, θ, ϕ),
and

[H 0 , [H 0 , r]] ψ lnm (r, θ, ϕ) = - R I ln (R) -R II ln (R) 2m * 2 δ(r -R) r R ,
we come up with 1). Hence, we can conclude that this Lamb shift is identical to the one given by Eq. ( 6). Let us also note that this method does not generate divergent terms in the limit V → ∞, as in subsection 4.3.

ψ lnm |[H 0 , r] † . [H 0 , [H 0 , r]]|ψ lnm = - 4 m * 2 E ∞ ln R 2 + o(

Lamb shift in spherical semiconductor QDs

In a spherical semiconductor QD, the interaction of the electron-hole pair with the quantized electromagnetic field generates a Lamb shift in its energy levels. In the Welton approach, electron and hole positions are fluctuating. Consequently, the Lamb shift of the electron-hole pair ground state consists of the sum of the contributions of Lamb shift undergone by the electron and by the hole, when the pair is in its ground state. It will be evaluated for each confinement regime of section 3. The electron and hole respective position fluctuation variance

(∆r e,h ) 2 = 2α πε 1 m * 2 e,h log m * e,h κ * e,h
, are given by expressions analogous to Eq. ( 2), where κ * e,h are electron and hole IR cut-offs. Then, in a semiconductor for which the fluctuations are large enough, i.e. for which the effective masses m * e,h are significantly smaller than the electron bare mass m e , there exists a possibility to detect experimentally the Lamb shift for a range of reasonable QD sizes.

In this section, we perform Taylor expansions of the Lamb shift in both confinement regimes to the second order terms to account for the pseudo-potential W (r eh ). As the contributions to the electron-hole pair confinement energy come only from second order terms, diagonal matrix elements of the action of the Laplacian on potentials are to be evaluated to this order. Therefore, the Lamb shift undergone by the electron or by the hole has three contributions. The first one is a boundary effect due to the confinement step-potential V ∞ (r e,h ), which is of the same nature as the one experienced by a confined particle in section 4. The second one is due to the Coulomb interaction V C (r eh ) between the electron and the hole, analogous to the Lamb shift in real atoms. And, finally, the third comes from the pseudo-potential W (r eh ).

General considerations

Let the non-normalized wave functions of the electron-hole pair ground state, confined by the step potential V (r e,h ) in both confinement regimes of Eqs. ( 3) and (4) be φ(r e , r h ) = ψ e 010 (r e )ψ h 010 (r h )φ eh (r eh ) and ψ(r e , r h ) = ψ(r e , r h )φ G (r G ).

Here, electron and hole ground state wave functions confined individually ψ e,h 010 (r e,h ) depend on the quantum numbers k e,h 01 and K e,h 01 , subjected to k e,h2 01 + K e,h2 01 = 2m * e,h R 2 V . The function φ G (r G ) is a pure phase factor and can be dropped. Since the variational parameter σ appears in φ ∞ (r e , r h ), it should be replaced by its variational value σ 0 , proper to each confinement regime. Then, the electron or the hole Lamb shift is

∆E ∞ e,h = α 3πε 1 m * 2 e,h log m * e,h κ * e,h φ ∞ |∇ 2 V ∞ (r e,h )|φ ∞ φ ∞ |φ ∞ + φ ∞ |∇ 2 V C (r eh )|φ ∞ φ ∞ |φ ∞ + φ ∞ |∇ 2 W (r eh )|φ ∞ φ ∞ |φ ∞ ,
where

∇ 2 V (r e,h ) = V 2 R δ(r e,h -R) + δ (r e,h -R) , ∇ 2 V C (r eh ) = 4πα ε δ (3) (r eh ),
and

∇ 2 W (r eh ) = W R 2 6 -8 r eh a * + 4 r 2 eh a * 2 e -2 r eh a * .
The previous mean values are indeed confinement regime dependent, as well as the Lamb shift, because of the IR cut-offs κ * e,h . Thus, we shall examine them in each confinement regime. To this end, we give some general results on diagonal matrix elements of the action of the Laplacian on potentials.

First, the case of Coulomb potential V C (r eh ) is trivially obtained since integrals over the electron (or the hole) coordinates are carried out over δ (3) (r eh ). This eliminates dependence on σ, so that

φ ∞ |∇ 2 V C (r eh )|φ ∞ = e 2 ε d 3 r ψ ∞4 010 (r) = 8e 2 κR 3 D
is independent from the confinement regime. Following section 3, we have for the pseudo-potential W (r eh )

φ ∞ |∇ 2 W (r eh )|φ ∞ = -W 8 R 4 6∂ Σ - 8 a * ∂ 2 Σ + 4 a * 2 ∂ 3 Σ 1 Σ 1 0 y 0 dx x dy y sin 2 (πx) sin 2 (πy) sinh(ΣRx)e -ΣRy ,
where Σ = σ + 2 a * accounts for the exponential dependence of the pseudo-potential W (r eh ). The determination of the formal quantities φ ∞ |∇ 2 V ∞ (r e,h )|φ ∞ is more cumbersome. As before, this is done using the regularization method of section 4, i.e. we expand the diagonal matrix element of the Laplacian of the finite potential step φ|∇ 2 V (r e,h )|φ in powers of V , and assume that φ ∞ |∇ 2 V ∞ (r e,h )|φ ∞ corresponds to the unique term independent from V . This calculation yields a term scaling as ∝ √ V , as in subsection 4.1. The results of subsections 4.3 and 5.2 suggest to drop it, because it is not physical, so that

φ ∞ |∇ 2 V ∞ (r e,h )|φ ∞ = 4 R 3 E e,h∞ 01 ∂ σ e -σR 3 σR -1 1 0 dx x sin 2 (πx) sinh(σRx) .
We now give the behavior of the Lamb shift of the confined electron-hole pair ground state in both confinement regimes for CdS 0.12 Se 0.88 and InAs quantum dots, with the numerical data collected in Table 1.

Strong confinement regime

Here R a * , Taylor expansions of the diagonal matrix elements of the action of the Laplacian on various potentials give the following expressions

                   φ ∞ |∇ 2 V ∞ (r e,h )|φ ∞ φ ∞ |φ ∞ = - 8 R 2 E e,h∞ 01
1 -F R a * + F R 2 a * 2 + O R 3 a * 3 , φ ∞ |∇ 2 V C (r eh )|φ ∞ φ ∞ |φ ∞ = 8e 2 κR 3 D + F R a * + O R 2 a * 2 , φ ∞ |∇ 2 W (r eh )|φ ∞ φ ∞ |φ ∞ = - 64 3 E ∞ eh a * 2 1 + O R a * .
In the strong confinement regime, the electron and the hole are essentially uncorrelated, so that the IR cut-offs κ * e,h should be chosen independently as, according to subsection 4.2,

κ * e,h = 7π 2 12m * e,h R 2 .
Omitting terms of the third order or higher in R a * and dropping off the exponent " ∞ ", we conclude that the Lamb shift undergone by the ground state of the electron-hole pair in the strong confinement regime is

∆E strong Lamb = ∆E strong e + ∆E strong h , where ∆E strong e,h E ∞ eh = - 16α 3πε -λ * 2 e,h R 2 log R R e,h min 1 - µ m * e,h F + 2 π 2 D R a * + µ m * e,h F - 2 π 2 F + 8 3 R 2 a * 2 , -λ * e,h = m * -1 e,h and R e,h min = π 2 7 3
λ * e,h are respectively the reduced Compton wave lengths and minimal radii of the electron and of the hole in the considered semiconductor.

Figure 1 shows the behavior of the Lamb shift of the electron-hole pair ground state in spherical CdS 0.12 Se 0.88 or InAs QDs. Their order of magnitude suggests a possible detection for experimentally accessible QD radii 7 . This is confirmed by table 2 a.-b., since the energy orders of magnitude are equivalent to those theoretically predicted and experimentally observed in hydrogen atom, at least in semiconductors with material parameters of the same order of magnitude than those of InAs, e.g. GaAs. Here, we do not identify the finite potential step V (r eh ) used for the regularization of φ ∞ |V ∞ (r eh )|φ ∞ , which is a calculational intermediate tool, with the real potential step, which confines the charge carriers inside the QD and is a physical quantity of the problem. Results of sections 4 and 5 impose that electron and hole masses inside or outside the QD should be identical. This means that the charge carriers are first excited, and then confined. However, in reality, electrons and holes are excited in a semiconducting microcrystallite. In this situation, electrons and holes should have different masses inside as well as outside of the QD [START_REF] Thoai | [END_REF].

If we want to identify the effective confining potential step at the surface of the microcrystal with V (r eh ), then the possibility for the electron, and to a lesser extent the hole -the notion of hole is not clearly defined in the surrounding insulating matrix -, to escape of the semiconducting QD by tunnel conductivity should be taken into account rigourously. Since sections 4 and 5 show that the only explicit dependence on the step potential appears as a non-physical term in the Lamb shift expression, this refinement is superfluous. This reflects the coherence of the reasoning made.

Finally, in the strong confinement regime, the proposed analytical expression for the Lamb shift ∆E strong Lamb displays a linear dependence on the electron-hole pair confinement energy E ∞ eh . This does not contradict the assumption of charge carriers confined by a infinite potential well. Table 1 shows its order of magnitude exceeds the typical height of the effective potential step at the surface of the QD. As shown by analytical study of Stark Effect in spherical semiconductor QDs [39], the confinement energy itself should not be a relevant quantity, as far as energy shifts are concerned. Then, to validate the modeling of the confining potential by the infinite potential well, it is sufficient to compare the order of magnitude of Lamb shifts to the height of the real potential step. This is confirmed by table 2 a.-b., since the examples of Lamb shifts it presents are of the order of magnitude at most of tens µeV, a negligible value compared to the effective confinement potential, which is of the order of eV [START_REF] Thoai | [END_REF].

Weak confinement regime

For R πa * , Taylor expansions of the Laplacian action diagonal matrix elements yield

         φ ∞ |∇ 2 V C (r e,h )|φ ∞ φ ∞ |φ ∞ = 8 E * a * 2 1 + 2 3 π 2 a * 2 R 2 + O a * 3 R 3 , φ ∞ |∇ 2 W (r eh )|φ ∞ φ ∞ |φ ∞ = 8 E * a * 2 - 2 3 π 2 a * 2 R 2 + O a * 3 R 3 .
In this weak confinement regime, the contribution of the pseudo-potential W (r eh ) to the electron-hole pair ground state Lamb shift cancels the second order correction of the contribution of the Coulomb potential V C (r eh ). Since they both scale as

∝ π 2 E * R 2 = E ∞ eh a * 2
, the presence of W (r eh ) allows the removal of contributions proportional to the electron-hole pair kinetic energy in relative coordinates, which are still contained in the exciton Rydberg energy. Finally, it is important to explain why it is not necessary to account for contributions coming from the confinement potential well V ∞ (r eh ) in the weak confinement regime. Direct calculations show that such contributions arise as a fifth order correction term in a

* R to E * a * 2 φ ∞ |∇ 2 V ∞ (r e,h )|φ ∞ φ ∞ |φ ∞ ∝ E 0∞ e,h R 3 a * ∝ E * a * 2 a * 5 R 5 .
Omitting terms of the third order or higher in R a * and dropping off the exponent " ∞ ", we conclude that the Lamb shift undergone by the ground state of the electron-hole pair in the weak confinement regime is contributes to the exciton ground state Lamb shift. Then, after some trivial manipulations, the Lamb shift of the ground state of an hydrogen-like atom of reduced mass µ in a dielectric medium characterized by its dielectric constant ε is recognized. These observations show that electron-hole pair states behave as excitonic bound states in this regime. Using known results for the hydrogen-like atom [17], they suggest to take an acceptable approximate value for the IR cut-off as κ * e,h ≈ 19.8E * .

Thus, the Lamb shift undergone by the exciton ground state does not depend on the QD radius, at least up to the third order in a * R . This is a strong argument in favor of the validity of the pseudo-potential W (r eh ), while it was introduced as a phenomenological effective potential.

Table 2 c. gives values of the exciton ground state Lamb shift in CdS 0.12 Se 0.88 or InAs microcrystals. They are not experimentally accessible at present.

Observability of the Lamb shift in spherical semiconductor QDs

As known, the experimental observability of the Lamb shift in hydrogen atom is due to the s-and p-level degeneracy, when the principal quantum number is n ≥ 2, in absence of interaction with the quantized electromagnetic field. The Lamb shift arises as a separation of the ns-spectral band from np-spectral band, while they should stay merged in absence of Lamb effect. So, how would an energy level Lamb shift be detected for quantum systems displaying no spectral band degeneracy such as a QD? In such systems each non-degenerate energy level is dressed by the quantum zero-point fluctuations of the electromagnetic field, forbidding the detection of the corresponding bare level. This may be the reason for the lack of articles addressing the Lamb effect in such structures.

In Quantum Field Theory, the summation of the zero-point energy fluctuations yields a divergent ground state energy. In the absence of gravity, this divergence is subtracted off in an additive renormalization scheme. However, a careful analysis on its volume dependence (via boundary conditions) shows the occurrence of a finite and observable force, known as Casimir force [40,41]. This effect can be intuitively understood as follows. In vacuum, two parallel perfectly conducting squared plates of linear size L are placed at a separation d L. Since the zero-point energy fluctuations are more important outside than inside the plates, they are subjected to an attractive the Casimir force.

If we adopt the Welton framework, we may view the Lamb shift as due to the particle position fluctuations induced by the zero-point energy fluctuations energy of the electromagnetic field. So, by placing a QD in two different quantized electromagnetic surroundings (vacuum and inside the Casimir pair of conducting plates), one would be able to detect an energy level difference between two Lamb shifted levels. An experimental protocol according to which the energy levels dressed by the zero-point fluctuations energy with or without the two Casimir plates are to be compared. It should allow to overcome the need of degenerate energy levels, or of exactly computed energy levels.

There exist some theoretical works dealing with Lamb effect of real atoms confining in a Casimir device, but they are based on a relativistic Bethe approach [42,43]. They predict an additional shift to the standard Lamb shift, which depends on the separation distance between the mirrors such that it goes to zero in the limit d → ∞. On the other hand the Welton approach suggests, in this limit, that the Lamb shift in presence of the Casimir device should go to the Lamb shift in absence of such device. While the predicted additional shifts should be experimentally measured, they seem to not be available in the literature. The protocol suggested here should be also able to discriminate between the Bethe and Welton approaches. This is the reason why, as a continuation of this work, we shall investigate a generalization of the Welton approach to evaluate the particle position fluctuations, in the presence of two Casimir plates.

Conclusion

Following the conventional approaches for computing the Lamb shift in real atoms, we have worked out, in this paper, an analytical expression of the Lamb shift for an arbitrary energy level of a spherical semiconductor QD. An explicit expression is obtained in the framework of the EMA model augmented by the Coulomb interaction, for both strong and weak confinement regimes. We find that, in the strong confinement regime and for sufficiently small but experimentally realistic QD sizes, this Lamb shift has an order of magnitude comparable to the Lamb shift in atoms.

However, the observability of such a Lamb shift is put to question since the energy levels are not degenerate as in the hydrogen atom. A way out, based on the Welton approach, is suggested. It consists in comparing the energy levels of the same spherical semiconductor QD in two different quantized electromagnetic field environments: the vacuum and the Casimir cavity. It is amusing to see that the two foremost effects which have served to validate Quantum Electrodynamics over the years may be now reunited to demonstrate the existence of the Lamb shift in semiconductor QDs.

A. Constants

In the following tables, we sum up all appearing constants and give their approximate values. The function Si(x) = x 0 dt t sin(t) denotes the standard sine integral. Table 3 a. presents analytical expressions and approximate values of constants occurring in section 3. Table 3 b. presents analytical expressions and approximate values of constants occurring in section 6.

  2n + 1). By regularizing sums over n, since sums over l do not play any role, we deduce that κ ≈ π 2 m * R 2 . Because of the non-relativistic condition in the IR and UV cut-offs κ(R) ≤ m * , there exists an lower potential radius bound R * min =

  which will explicitly contravene to the non-relativistic assumption. Finally, there exists another particular potential radius R * max = √ eR * min ≈ 3, 956λ * , for which the Lamb shift of any energy level represents a maximal fraction of it, i.e.

Figure 1 :

 1 Figure1: Lamb shift undergone by the electron (--), the hole (-• -) and the exciton (-), when the exciton occupies its ground state in the strong confinement regime in a. CdS 0.12 Se 0.88 and b. InAs microcrystallites as a function of the QD radius.

.-

  In the limit of infinite hole mass, i.e. in the limit λ * h in the weak confinement regime, only the electronic term ∆E weak e

Table 1 :

 1 Numerical values of material parameters in CdS 0.12 Se 0.88 et InAs, where the electron bare mass is me ≈ 9.1110 -31 kg. a 0 ≈ 0.529 Å and E I ≈ 13.6eV are the Bohr radius and the ionization energy of the hydrogen atom.

	Semiconductor in units of CdS 0.12 Se 0.88 m * e m e 0.13 m * m e 0.46 h ε 9.3 a * 91.8 a 0 E * E I 1.17 10 -3	InAs 0.026 0.41 14.5 595 1.16 10 -4

Table 2 :

 2 Lamb shift undergone by the electron-hole pair ground state in CdS 0.12 Se 0.88 or InAs microcrystals a. for R = 10 Å or for R = 30 Å in the strong confinement regime and c. in the weak confinement regime.

	a. b.	Semiconductor CdS 0.12 Se 0.88 E ∞ eh (eV) 3.71 ∆E strong Lamb E ∞ -5.52 10 -9 eh E ∞ eh (eV) 0.412 ∆E Lamb E ∞ eh -5.13 10 -9 strong	InAs 15.38 -6.17 10 -7 1.71 -8.64 10 -8	c.	Semiconductor CdS 0.12 Se 0.88 E * (meV) 15.9 ∆E weak Lamb E * 4.56 10 -7	InAs 1.58 1.70 10 -10

Relativistic corrections can generally be calculated[27,

28], but they are not relevant in the comprehension of Lamb effect, particularly in spherical semiconductor QDs, where confined charged carriers are considered as non-relativistic particles.2 For more details, one can refer to complement A XIII of chapter XIII[START_REF] Cohen-Tannoudji | Mécanique quantique[END_REF].

Even if this Taylor expansion seems to converge only for sufficiently smooth potential V (r), it is summable in general. However, a fully quantum treatment, i.e. both particle and electromagnetic field dynamics are quantized, is needed. It has been then shown that the averaged effective potential V (r + ∆r) can be obtained from the bare potential V (r) by applying the well-defined differential operator[16] V (r + ∆r) = e (∆r) 2

All constants appearing in the text and formulas are listed in appendix A.

The notation X ∞ = lim V ∞ X is adopted, for all physical quantities. This is the reason why we will choose to note the infinite potential well V ∞ (r), and the finite potential step V (r), in sections 4 and 5.

By a simple integration by parts, one can show that dt H int = dt H int .

There exists a large number of references on spherical CdS 0.12 Se 0.88 or InAs microcrystallites with nm radius which are synthesized and experimentally used, see e.g. respectively[START_REF] Nomura | [END_REF] or[38].


T T Truong

M K Nguyen

On new V-line Radon transforms in R 2 and their inversion

Radon transforms on piecewise smooth curves in R 2 are rather unfamiliar and have not been so far widely investigated. In this paper we consider three types of Radon transforms defined on a pair of half-lines in the shape a V-letter, with a fixed axis direction. These three Radon transforms arise from recently suggested tomographic procedures. Our main result consists in obtaining their analytic inverse formulas, which may serve as mathematical foundation for new imaging systems in engineering and physics.

Introduction

Radon transforms in the plane have been extensively studied in the past. Besides the cases of the straight line [START_REF] Radon | Über die Bestimmung von Funktionnen durch ihre Integralwerte längs gewisser Mannigfaltikeiten Ber[END_REF], the circle intersecting a fixed point, Radon transforms on other classes of curves have been also treated (see among other works e.g. [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF][START_REF] Quinto | Radon transforms on curves in the plane[END_REF][START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF]). The universality of Radon transform is known in imaging science ranging from medicine [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF][START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF], seismic [START_REF] Chapman | Generalized Radon transforms and slant stacks[END_REF], astrophysics [START_REF] Bracewell | Strip integration in radio astronomy[END_REF], non-destructive industrial testing [START_REF] Kershaw | The Determination of the Density Distribution of a Gas Flowing in a Pipe from Mean Density Measurements[END_REF], etc. It also plays a crucial role in pure mathematics [START_REF] John | Planes Waves and Spherical Waves Applied to Partial Differential Equations[END_REF][START_REF] Gelfand | Integral geometry and its relations to the theory of group representation Russian Math[END_REF][START_REF] Militello | Radon transform as a set of probability distributions[END_REF][START_REF] Saygili | Trkalian fields and Radon transformation[END_REF] and recently has become increasingly a powerful investigation tool in numerous fields of physics, e.g. [START_REF] Manuel | Construction of the Fermi surface from positron-annihilation methods[END_REF][START_REF] Behringer | A demonstration of ocean acoustic tomography[END_REF][START_REF] Mane | Radon reconstruction in longitudinal space[END_REF][START_REF] Asorey | Radon transform on the cylinder and tomography of a particle on the circle[END_REF][START_REF] Florescu | Single scattering optical tomography[END_REF].

In 1997 R Basko introduced a Radon transform on a pair of half-lines forming a Vletter in an attempt to model image formation in the so-called one-dimensional Compton camera [START_REF] Basko | Analytical reconstruction formula for the one-dimensional Compton camera[END_REF]. The axis of this V-line swings around a point of the plane such that its vertex lies on a straight line (representing a "scattering" detector) and the opening angle of the two half-lines (representing a Compton scattering angle) is an imaging data variable. No other cases of Radon transforms on piecewise lines (or curves) seem to have been discussed in the literature until we have put forward recently the idea of V-line Radon transforms with a fixed axis direction [START_REF] Morvidone | On the V-line Radon transform and its imaging applications[END_REF]. An akin concept of broken ray Radon transform has just now appeared [START_REF] Florescu | Inversion formulas for the broken-ray Radon transform[END_REF].

In this work we consider the class of V-line Radon transforms with a fixed symmetry axis direction. Such Radon transforms may be of theoretical interest in integral geometry in the sense of Gel'fand [START_REF] Gelfand | Integral geometry and its relations to the theory of group representation Russian Math[END_REF]. Three types of V-line Radon transforms with fixed axis direction will be treated. The first one is mathematically the most important and arises from a coupled transmission-reflection tomographic process while the two others are its † To whom correspondence should be addressed (truong@u-cergy.fr) variants corresponding respectively to a collimated one-dimensional Compton camera and to scattered radiation emission imaging using a one-dimensional gamma camera. For all three cases an exact inverse formula is established. These results are of theoretical nature: they support the feasibility of three new tomographies. An illustration of these inversion formulas in two-dimensional medical and industrial imaging can be found in [START_REF] Morvidone | On the V-line Radon transform and its imaging applications[END_REF]. The paper closes on a short conclusion with some forthcoming topics of investigation.

The V-line Radon transform

Short review of the standard Radon transform

Before going into the subject, we briefly go over the standard Radon transform. It maps a

L 1 (R 2 )-function f (x, y) onto R f (p, ϕ) via the integral R f (p, ϕ) = ∫ R 2 d x d y f (x, y) δ(p -x cos ϕ -y sin ϕ),
in which p is the distance from the coordinate system origin to the straight line and ϕ is the angle of the normal unit vector to the line with the Ox axis, see Fig. From the structure of the delta function kernel, it follows that

R f (p, ϕ) = Rf (-p, ϕ + π).
There exists an explicit inverse formula [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF]

f (x, y) = 1 2π 2 ∫ π 0 d ϕ ( P.V. ∫ R d p R ′ f (p, ϕ) (p -x cos ϕ -y sin ϕ) ) (1)
where

R ′ f (p, ϕ) = ∂ ∂p R f (p, ϕ).
In eq. ( 1), the p-integral is a Cauchy principal value (P.V.). An alternative form in Fourier space of this inverse formula is

f (x, y) = ∫ π 0 d ϕ ∫ R d k |k| e 2i πk(x cos ϕ+y sin ϕ) ∫ R d q e -2i πqk R f (q, ϕ),
which expresses a summation image of the "ramp"-filtered Radon data [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF].

Definition

Consider a pair of half-lines meeting at a point M of abscissa ξ on the Ox-axis, each of which making an angle ω (0 < ω < π/2) with the O y-axis, see Fig. 2. Let f (x, y) be a L 1 (R 2 ) nonnegative continuous function with compact support in {R 2 |y > 0}. We call Eq. ( 2) may be put under the form of an integral transform in R 2 with a delta function kernel concentrated on the V-line

V f (ζ, τ) = ∫ ∞ 0 d r ( f (ζ + r sin ω, r cos ω) + f (ζ -r sin ω, r cos ω) ) , (2)
V f (ζ, τ) = ∫ R 2 d x d y K V (ζ, ω|x, y) f (x, y), with K V (ζ, ω|x, y) = ( δ((x -ζ) cos ω -y sin ω) + δ((x -ζ) cos ω + y sin ω) ) .
Thus the image of a given point source at (x, y) ∈ R 2 , is the sum of two Dirac delta distributions, which have a support in the upper (τ, ζ)-plane, consisting of two half-lines, with τ > 0, meeting at ξ = x on the ξ-axis and having a slope ± y -1 , i.e.

τ = | x -ζ y | or ω = arctan | x -ζ y |.
In a (ω, ζ)-representation, this is just the arctan-curve, which is the support of the pointsource image. Thus the support of V f (ζ, τ) is not compact in this representation.

Clearly the V-Radon transform can also be written in the language of the standard Radon transform as

V f (ζ, τ) = ∫ R d x ∫ ∞ 0 d y f (x, y) ( δ(u -x cos ω -y sin ω) + δ(u -x cos ω + y sin ω) ) ,
where u = ζ cos ω, the distance of the coordinate origin to the two half-lines forming the letter V. Note also that the unit normal vectors to these half-lines make an angle ±ω with the Ox-axis. So the result appears as the sum of two standard Radon transforms of f (x, y), except for the y-integration range.

Physical implementation

In standard Computed Tomography, a calibrated X-ray pencil emitted from a point-source S traverses a two-dimensional slice of an objet along a straight line before being absorbed by a point-like detector D. In this way the attenuation of the radiation beam is expressed as an integral of the linear attenuation function of the object to be imaged over a straight line. This is a Radon's problem which is solved by eq. ( 1).

The V-line Radon transform may be conceived as representing the measurement of a X-ray pencil attenuation along a broken path in the shape of a V-letter, which is physically realized by putting a mirror at the vertex M of the V-line as shown in Fig. 2. Measurements requiring various angles ω may be performed by varying the distance SD, but in practice one may set a series of point-sources and a series of point-detectors respectively on the left and on the right of a given axis of symmetry of the V-line. To vary the parameter ζ, one may put the object on a conveyor belt in a scanning process. Such a device is clearly a new type of X-ray scanner which could prove to be more efficient wherever the rotational motion of a standard X-ray scanner cannot be used and should be replaced by a translational motion. Such a situation would arise when tomographic images of series of objects on a conveyor belt are to be taken successively. As possible applications, one may think of luggage control in airports as well as continuous imaging of series of biological objects in medical research.

Inversion formula

We now establish a relation between Radon transform and Fourier transform which is used to obtain the inverse of the V-line Radon transform. For the standard Radon transform, such a relation tells how this transform appears in Fourier spaces. It is known in imaging science as the central-slice theorem [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF]. As V-lines of given opening angle ω can be deduced from each other by x-translations, we may introduce the x-Fourier transform f (p, y) of f (x, y) by

f (x, y) = ∫ ∞ -∞ d p e 2i πpx f (p, y).
Eq. ( 2) can be now rewritten as

V f (ζ, τ) = ∫ ∞ -∞ d p e 2i πpζ ∫ ∞ 0 dr f (p, r cos ω) 2 cos (2πpr sin ω). ( 3
)
To proceed further we set z = r cos ω. Recalling that 0 < ω < π/2, we have

z > 0 and cos ω = 1 1 + τ 2 > 0.
Then eq. ( 3) appears as

V f (ζ, τ) = 2 √ 1 + τ 2 ∫ ∞ -∞ d p e 2i πpζ ∫ ∞ 0 d z f (p, z) cos (2πpτz). (4)
Extracting the p-Fourier component V f (p, τ) of V f (ζ, τ) on the left-hand-side of eq. ( 4)

gives ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2 ) = 2 ∫ ∞ 0 d z f (p, z) cos (2πpτz).
We now multiply both sides of this equation by cos(2πτv) and integrate on τ > 0. Using the identity

∫ ∞ 0 d τ cos (2πτ v) cos(2πkz τ) = 1 4 (δ(kz + v) + δ(kz -v)) , (5)
we get

∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2 ) = 1 2 ∫ ∞ 0 d z f (p, z) ( δ(pz + v) + δ(pz -v) ) ,
with (p, v) ∈ R 2 and z > 0. We now perform the τ-integration

• if v/p > 0, then only δ(v -pz) = |p| -1 δ(z -v/p) contributes, ∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2 ) = 1 2|p| f (p, v/p), • if v/p < 0, then only δ(v + pz) = |p| -1 δ(z + v/p) contributes, ∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2 ) = 1 2|p| f (p, -v/p),
Hence we can regroup these two cases into a single formula as

f (p, |v/p|) = 2|p| ∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2
)

.

Putting y = |v/p| > 0, we can reconstruct f (x, y) by inverse Fourier transform in p and f (x, y) is expressed as a triple integral on the V-line Radon data

f (x, y) = ∫ ∞ -∞ |p| d p e 2i πpx ∫ ∞ 0 d τ 2 cos (2πpτy) 1 + τ 2 ∫ ∞ -∞ d ζ e -2i πζp V f (ζ, τ), (6)
where we have made use of the evenness of the cos-function.

Inspection of convergence shows that exchange of integration order is possible and eq. ( 6) may now be rewritten as

f (x, y) = ∫ ∞ 0 d τ 1 + τ 2 ∫ ∞ -∞ |p| d p e 2i πpx ( e 2i πpτy + e -2i πpτy ) ∫ ∞ -∞ d ζ V f (ζ, τ) e -2i πζp . (7)
Thus analogously to the Radon transform, we can write

f (x, y) = 1 2π 2 ∫ ∞ 0 d τ 1 + τ 2 ( P.V. ∫ R d ζ ( V ′ f (ζ, τ) (ζ -x -yτ) + V ′ f (ζ, τ) (ζ -x + yτ) )) , ( 8
)
where

V ′ f (ζ, τ) is the ζ-derivative of V f (ζ, τ).

Alternative form of the inversion formula

Reintroducing u = ζ cos ω, which is the common distance from the coordinate system origin to the two branches of the V-line, and redefining f (ζ, τ) as F (u, ω), eq. ( 8) becomes

f (x, y) = 1 2π 2 ∫ π/2 0 d ω ( P.V. ∫ R d u [ V ′ F (u, ω) (u -x cos ω -y sin ω) + V ′ F (u, ω) (u -x cos ω + y sin ω) ]) , (9)
where V ′ F (u, ω) is the derivative of VF (u, ω) with respect to u. This expression is a sum of two ω-integrals which, as functions of (x, y) respectively take a constant value along the half-line of unit normal vector n = (cos ω, sin ω) and along the half-line of unit normal vector n ′ = (cos ω,sin ω). A similar feature occurs in the standard Radon transform but with a double angular range [0, π]. Thus we may view the V-line Radon transform as the sum of two half-line Radon R-transforms, as suggested by the defining equation ( 2). In imaging science equation ( 9) is called a "summation image" of back-projected Radon data [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF], the ζ-derivative of the data being back-projected on the left and on the right branches of the V-line in the ζ-integrand of eq. ( 9).

Extension to the range π/2 < ω < π

Up to now, we have consider L 1 0 (R 2 )-functions with compact support. By appropriate translations, their support can be shifted into the upper half-plane. But when the support of f (x, y) is extended over all R 2 , we need to collect data pertaining to the part of its support below the real axis. This can be achieved using the angular range π/2 < ω < π, for which cos ω and τ are negative. The V-line is now up-side down. In this case, we set z = -z ′ and τ = -τ ′ in eq. (2.4), to have

∫ R d p e -2i π pζ ( V f (ζ, -τ ′ ) 1 + τ ′2 ) = 2 ∫ ∞ 0 d z ′ f (p, -z ′ ) cos (2πpz ′ τ ′ ), since zτ = z ′ τ ′ , d z = -d z ′ and because of cos ω = -1/ 1 + τ ′2
. This shows that again we have to invert a cosine-Fourier transform to reconstruct f (x, y) for y < 0. Following the previous steps we obtain

∫ ∞ 0 d τ ′ cos(2πτ ′ v) ∫ R d ζ e -2i πζp ( V f (ζ, -τ ′ ) 1 + τ ′2 ) = 1 2 ∫ ∞ 0 d z ′ f (p, -z ′ ) ( δ(pz ′ + v) + δ(pz ′ -v) ) .
with (p, v) ∈ R 2 and z ′ > 0. We now perform the z ′ -integration on the right-hand-side of this last equation.

• if v/p > 0, then only δ(v -pz ′ ) = |p| -1 δ(z ′ -v/p) contributes, ∫ ∞ 0 d τ ′ cos(2πτ ′ v) ∫ R d ζ e -2i πζp ( V f (ζ, -τ ′ ) 1 + τ ′2 ) = 1 2|p| f (p, -v/p), • if v/p < 0, then only δ(v + pz ′ ) = |p| -1 δ(z ′ + v/p) contributes, ∫ ∞ 0 d τ ′ cos(2πτ ′ v) ∫ R d ζ e -2i πζp ( V f (ζ, -τ ′ ) 1 + τ ′2 ) = 1 2|p| f (p, v/p),
Analogously to the case with 0 < ω < π/2 (see eq. ( 6)), we can write a single formula

f (p, -|v/p|) = 2|p| ∫ ∞ 0 d τ ′ cos(2πτ ′ v) ∫ R d ζ e -2i πζp ( V f (ζ, -τ ′ ) 1 + τ ′2
) .

Putting y = -|v/p| > 0, we can reconstruct f (x, y) by inverse Fourier transform in p for y < 0. Using again the evenness of the cos-function, f (x, y) is then expressed as a triple integral on the V-line Radon data

f (x, y) = ∫ ∞ -∞ |p| d p e 2i πpx ∫ ∞ 0 d τ ′ 2 cos (2πpτ ′ y) 1 + τ ′2 ∫ ∞ -∞ d ζ e -2i πζp V f (ζ, -τ ′ ).
And hence for y < 0, we obtain

f (x, y) = 1 2π 2 ∫ ∞ 0 d τ ′ 1 + τ ′2 P.V. (∫ R d ζ ( V ′ f (ζ, -τ ′ ) (x + τ ′ y -ζ) + V ′ f (ζ, -τ ′ ) (x -τ ′ y -ζ) )) . ( 10
)
To sum up, formulas [START_REF] Bracewell | Strip integration in radio astronomy[END_REF] give the reconstruction of a function f (x, y) with compact support, which can be always, by an appropriate choice of the coordinate origin, positioned in the upper half-plane (y > 0). But when the support of f (x, y) is infinitely extended and f (x, y) decreases fast at infinity, then the reconstruction formula ( 10) for (y < 0) should be used. We can put ( 8) and [START_REF] John | Planes Waves and Spherical Waves Applied to Partial Differential Equations[END_REF] together in one single formula as

f (x, y) = 1 2π 2 ∫ ∞ 0 d σ 1 + σ 2 P.V. (∫ R d ζ ( V ′ f (ζ, (sgny)σ) (x + σy -ζ) + V ′ f (ζ, (sgny)σ) (x -σy -ζ) )) ,
where (sgny) is the sign of y. This completes the proof of the inversion of the V-line Radon transform with fixed axis direction for all values of y ∈ R.

Adjoint transform

The kernel K V (ζ, ω|x, y) may be used to define an adjoint V-line Radon transform V + . For an integrable function g (ζ, ω), V + g (x, y) is given by

V + g (x, y) = ∫ π/2 0 d ω ∫ R d ζ K V (ζ, ω|x, y) g (ζ, ω).
This is also a V-line Radon transform in (ζ, τ)-space on the V-line given by eq. (2.2). After performing the ζ-integration, the following form appears

V + g (x, y) = ∫ π/2 0 d ω cos ω ( g (x -yτ, ω) + g (x + yτ, ω) ) ,
where τ = tan ω as before. We observe that g (xyτ, ω) is constant on the left half-line of equation (yy 0 )τ = (xx 0 ), whereas g (x + yτ, ω) is constant on the right half-line of equation -(yy 0 )τ = (xx 0 ). The inverse formula of eq. ( 8) has been viewed already as a "summation image" of back-projected V-line Radon data (see eq. ( 9)). Here it may be viewed now as the V-line adjoint Radon transform of the data with the identifications

∫ ∞ 0 d τ 1 + τ 2 = ∫ π/2 0 d ω,
and

g (x -yτ, ω) = P.V. 1 2π 2 ∫ R d ζ V ′ f (ζ, τ) (ζ -x + yτ) , g (x + yτ, ω) = P.V. 1 2π 2 ∫ R d ζ V ′ f (ζ, τ) (ζ -x -yτ) .
This behavior shows up also in the standard Radon transform [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF].

The V * -line Radon transform

Definition

A variant of the V-line Radon transform arises when the integration measure on the halflines is performed with a non-uniform density along the branches of the V-line. Technically this occurs in the so-called one-dimensional special Compton camera for flat radiating objects [START_REF] Morvidone | On the V-line Radon transform and its imaging applications[END_REF], which is a special version of an imaging device investigated by [START_REF] Cree | Towards direct reconstruction from a gamma camera based on Compton scattering[END_REF] in three dimensions, see Fig. 3. Here a radiation emitting object is described by its radio-activity emission density f (x, y). Emitted radiation from a point-source N will Compton scatter electric charges, under a scattering angle ω, in the scatter detector before entering a collimated absorbing detector. The amount of scattered gamma radiation detected is thus given by the V * -line Radon transform of f (x, y), over an integration measure m σ (r ), which accounts for radiation dispersion from object emission site to scattering site on the scatter detector. Upon close analysis, the integration measure is given by the two-dimensional photometric law as

m σ (r ) = 1 πσ tan -1 σ 2r ,
where σ is the linear size of scatterers. As in the previous section, we shall assume that the support of f (x, y) is strictly in the upper half-plane y > 0. To keep the discussion simple and concentrate on the nature of this transform, we have ignored the Compton kinematic factor, which includes the Klein-Nishina scattering probability density.

V * f (ζ, τ) = ∫ ∞ 0 d r m σ (r ) ( f (ζ + r sin ω, r cos ω) + f (ζ -r sin ω, r cos ω) ) .
For σ ̸ = 0, an inversion formula cannot be worked out at present. But for vanishingly small target size σ → 0, it has a simpler form

V * f (ζ, τ) = ∫ ∞ 0 d r 2πr ( f (ζ + r sin ω, r cos ω) + f (ζ -r sin ω, r cos ω) ) , (11)
which can be inverted with the procedure of section 2. Since the support of f is "above" the Ox-axis, the integral in equation ( 11) is well defined.

The integral kernel has the form

K V * (ζ, ω|x, y) = 1 2π y ( δ((x -ζ) cos ω -y sin ω) + δ((x -ζ) cos ω + y sin ω)
) .

It differs from the kernel K V (ζ, ω|x, y) by a factor 1/2πy due to the presence of 1/2πr in the r -integration. Its support is the same as the one given by eq. (2.2).

Inversion formula

We follow now the steps and the same notations as before. Eq. ( 11) becomes now

∫ R d ζ e -2i πζp V * f (ζ, τ) = 1 π ∫ ∞ 0 d z ( 1 z f (p, z) ) cos (2πτpz).
We apply now on both sides of this equation the integral operator

∫ ∞ 0 d τ cos(2πτv),
and use the identity [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF] to have

∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp V * f (ζ, τ) = ∫ ∞ 0 d z 2πz f (p, z) 1 2 ( δ(v + pz) + δ(v -pz) ) .
As (p, v) ∈ R 2 and z > 0, there are two types of contributions

• if p v > 0, only δ(v -pz) = |p| -1 δ(z -v/p) contributes, hence ∫ ∞ 0 d z 2πz f (p, z) 1 2|p| δ(z - v p ) = 1 4πv sgn(p) f (p, v p ) = f (p, v p ) 4π|v| , since sgnp = sgnv. • if p v < 0, only δ(v + pz) = |p| -1 δ(z + v/p) contributes, hence ∫ ∞ 0 d z 2πz f (p, z) 1 2|p| δ(z + v p ) = 1 2|p| 1 2π(-v p ) f (p, - v p ) = f (p, -v p ) 4π|v| , since |p| ( - v p ) = sgn(p) (-1) |v| sgn(v) = |v|, because p v < 0.
We can put these two cases together with y = |v/p| to have

f (p, |v/p|) = 2π|v| ∫ ∞ 0 d τ 2 cos(2πτv) ∫ R d ζ e -2i πζp V * f (ζ, τ). ( 12
)
The reconstruction formula is then for y = |v/p| > 0, thanks to the evenness of the cosfunction

f (x, y) = 2πy ∫ R d p |p| e 2i πx ∫ ∞ 0 d τ 2 cos(2πτy p) ∫ R d ζ e -2i πζp V * f (ζ, τ).
It can be rewritten also as a Barrett "summation image" as

f (x, y) = 2πy ∫ ∞ 0 d τ ∫ R d p |p| e 2i πx ( e 2i πp yτ + e -2i πp yτ ) ∫ R d ζ V * f (ζ, τ) e -2i πζp .
The p-integration is the same as in eq. ( 7) and yields

f (x, y) = y π ∫ ∞ 0 d τ ( P.V. ∫ R d ζ ( V ′ * f (ζ, τ) (ζ -x -yτ) + V ′ * f (ζ, τ) (ζ -x + yτ)
)) ,

where

V ′ * f (ζ, τ) is the ζ-derivative of V * f (ζ, τ).
This expression can also be represented as a summation image on back-projected data on a V-line in (x, y)-space, with f (ζ, τ) = F (u, ω)

f (x, y) = y π ∫ π/2 0 d ω cos ω P.V. ∫ R du [ V ′ * F (u, ω) (u -x cos ω -y sin ω) + V ′ * F (u, ω) (u -x cos ω + y sin ω) ] . (13)
Note that the apparent singularity in the angular integration at ω = π/2 does not exists thanks to the assumption on the support of f (x, y).

Extension to the range π/2 < ω < π

Repeating the approach of subsection 2.5, we obtain the complement of eq. ( 12)

f (p, -|v/p|) = 2π|v| ∫ ∞ 0 d τ ′ 2 cos(2πτ ′ v) ∫ R d ζ e -2i πζp V * f (ζ, -τ ′ ),
and the corresponding reconstruction formula for y = -|v/p| < 0

f (x, y) = 2πy ∫ R d p |p| e 2i πx ∫ ∞ 0 d τ ′ 2 cos(2πτ ′ y p) ∫ R d ζ e -2i πζp V * f (ζ, -τ ′ ), or f (x, y) = ( y π ) ∫ ∞ 0 d τ ′ ( P.V. ∫ R d ζ ( V ′ * f (ζ, -τ ′ ) (ζ -x -yτ ′ ) + V ′ * f (ζ, -τ ′ ) (ζ -x + yτ ′ ) )) . ( 14
)
Formula [START_REF] Saygili | Trkalian fields and Radon transformation[END_REF] gives the reconstruction of f (x, y) when its support is compact for y > 0 and when it is not compact one must add [START_REF] Manuel | Construction of the Fermi surface from positron-annihilation methods[END_REF]. Note these formulas differs from that of the V-transform. We can put (3.2) and ( 14) together in a single formula as

f (x, y) = ( y π ) ∫ ∞ 0 d σ ( P.V. ∫ R d ζ ( V ′ * f (ζ, (sgny)σ) (ζ -x -yσ) + V ′ * f (ζ, (sgny)σ) (ζ -x + yσ) )) ,
where (sgny) is the sign of y. This completes the inversion of the V * -line Radon transform by a factor y/π and by the form of the τ-integration. An interpretation of equation ( 14) in terms of back-projected data or in terms of its adjoint transform can also be made along the lines of the previous section.

The V ♯ -line Radon transform

A third version of the V-line Radon transform is a so-called "compounded" or integrated form of the V * -Radon transform over a given integration measure. It arises in scattered radiation emission imaging, a concept which has been extensively investigated in recent years [START_REF] Nguyen | On an integral transform and its inverse in nuclear imaging[END_REF][START_REF] Nguyen | Radon transforms on a class of cones with fixed axis direction[END_REF][START_REF] Nguyen | On a novel approach to Compton scattered emission imaging[END_REF][START_REF] Truong | On a class of generalized Radon transform and its role in imaging science[END_REF]. If the Compton camera exploits the scattering of object emitted radiation by an external scatterer in the form of a scatter detector for imaging purpose, this time it is the object itself (in fact its electric charges) which serves as natural scatterer. Thus the outgoing radiation from the object itself is registered now by a motionless collimated gamma camera (as opposed to a moving collimated linear detector as in Single Photon Emission Tomography (or SPECT)). The data is collected at various values of the scattering energies -or equivalently at various values of the scattering angles ω), -see Fig. 4. Hence the detected radiation flux density appears as the V ♯ -line Radon transform of the object radioactivity emission density f (x, y), i.e.

V ♯ f (ζ, τ) = ∫ ∞ 0 d η ( 1 πσ ′ tan -1 σ ′ 2η ) ∫ ∞ 0 d r ( 1 πσ tan -1 σ 2r ) × [ f (ζ + r sin ω, η + r cos ω) + f (ζ -r sin ω, η + r cos ω) ] , (15)
where σ ′ has the same meaning as in subsection 3.1 ‡. A successful inversion of this V ♯ -line Radon transform would allow to reconstruct the radio-activity emission density f (x, y) with a stationary linear collimated detector collecting series of "images" at different scattering angles ω. This imaging process presents a very important advantage over the existing ones: it avoids the motion of the detector around the object during data acquisition, hence does not require the heavy bulky rotational mechanics to be included.

This type of V-line transform is called "compounded" V-line Radon transform or V ♯line Radon transform, by analogy to the three-dimensional case [START_REF] Nguyen | On an integral transform and its inverse in nuclear imaging[END_REF]. Now let us define

∫ ∞ 0 d η ( 1 πσ ′ tan -1 σ ′ 2η ) f (ξ ± , η + r cos ω) = h(ξ ± , r cos ω), (16)
where ξ ± = (ζ ± r sin ω). Then we see that

V ♯ f (ζ, τ) = V * h(ζ, τ).
Hence the inversion problem is that of V * for h, followed by the extraction of f from eq. ( 16).

Eq. ( 15) may be rewritten as an integral transform with the kernel

K V ♯ (ζ, τ|x, y) = √ 1 + τ 2 ( Y (x -ζ) Y (y -τ(x -ζ)) m σ ((x -ζ) √ 1 + τ 2 ) m σ ′ (y -τ(x -ζ))+ Y (-x + ζ) Y (y + τ(x -ζ)) m σ (-(x -ζ) √ 1 + τ 2 ) m σ ′ (y + τ(x -ζ))
) ,

where Y (x) is the Heaviside unit step function. Here the kernel is not a Dirac delta function.

Section 3 shows how to reconstruct h(x, y) from the data V ♯ f (ζ, τ) for all (x, y) ∈ R 2 .

Thus given h(x, y), f is the solution of the convolution equation ∫ R d z e -2i πzq h(x, z), [START_REF] Florescu | Single scattering optical tomography[END_REF] as h(x, z) is known for all z ∈ R. As an example, in the limit σ → 0, using the Fourier table [START_REF] Lavoine | Transformation de Fourier des Pseudo-fonctions avec Tables de Nouvelles Transformées[END_REF], we have an exact expression for m σ (q) = -

( ln 2π|q| -γ + i π 2 sgnq
) , to be substituted in equation [START_REF] Florescu | Single scattering optical tomography[END_REF] to reconstruct f (x, y). We observe that the inversion of the V ♯ the product of the inversion of V ⋆ times an inverse Fourier transform.

Conclusion and perspectives

In this article we have presented for the first time three types of Radon transforms in R 2 which are defined on a piecewise continuous curve having the shape of a letter V with a fixed axis direction. The main theoretical result is their analytic inverse formulas. These transforms are instrumental in new possible tomographic processes involving scattering, coupled transmission-reflection and emission phenomena. Numerical simulations on test phantoms to ascertain the viability of these imaging processes will appear in a separate publication.

We also envisage to study Radon transforms on more general piecewise continuous curves consisting of connected pieces of arcs, e.g. arcs of circle, as well as Radon transforms on V-line with an axis swinging around a fixed point. Their inversion would open the way to new advantageous imaging processes. These topics are the object of future research.

Figure 1 .

 1 Figure 1. Parameters of the Radon transform

Figure 2 .

 2 Figure 2. Parameters of the V-line Radon transform

Figure 3 . 1 0 (R 2 +

 312 Figure 3. The V ♯ -line Radon transform and its parameters

h(x, y) = ∫ ∞ 0 dFigure 4 .

 04 Figure 4. Setup and parameters of the V ♯ -line Radon transform