Kokou M Robert Afansounoudji

Rabiou Issa

Komi Sodoga
email: ksodoga@univ-lome.tg

David Lauvergnat
email: david.lauvergnat@universite-paris-saclay.fr

Vibrational transitions of H 2 Cl + : potential energy surface and anharmonic computations

A new three-dimensional potential energy surface of the electronic ground state of the chloronium ion, H 2 Cl + , based on the explicitly correlated coupled cluster method with a triple zeta basis set adapted to this method has been expanded in an analytical representation. This potential energy surface is later incorporated into our home-made Fortran code to compute variationally the vibrational levels, zero-point ground average structural parameters and the rotational constants of the chloronium ion and several isotopologues. Our results show a good agreement with experimental data and that our results will help to detect H 2 Cl + isotopologues in the interstellar medium.

Introduction

Recently, the chloronium ion H 2 Cl + was detected for the first time in the interstellar medium (ISM) by the Herschel Space Observatory using the Heterodyne Instrument for the far-Infrared (HIFI) [START_REF] Lis | Herschel/HIFI discovery of interstellar chloronium H 2 Cl +[END_REF][START_REF] Pilbratt | Herschel Space Observatory-An ESA facility for far-infrared and submillimetre astronomy[END_REF]. The dominant ionization state of each element in diffuse interstellar gas clouds is determined by its ionization potential. Chlorine has a slightly lower ionization potential than that of a hydrogen molecule, and is therefore mostly ionized individually in the form of Cl + . The latter ion can react with the dominant constituent of the interstellar medium H 2 to form the ion HCl + [START_REF] Lis | Herschel/HIFI discovery of interstellar chloronium H 2 Cl +[END_REF] which, in turn, reacts with hydrogen to form the chloronium ion H 2 Cl + . Chemical modeling suggests that the chloronium ion could be very abundant around the diffuse ISM [START_REF] Neufeld | The chemistry of interstellar molecules containing the halogen elements[END_REF]. However, prior to the launch of Herschel, H 35 Cl and H 37 Cl were the only Cl-bearing molecules detected in and around the ISM [START_REF] Blake | Chlorine in dense interstellar clouds-the abundance of HCl in OMC-1[END_REF][START_REF] Zmuidzinas | HCl absorption toward sagittarius B2[END_REF][START_REF] Schilke | Hydrogen chloride in OMC-1[END_REF][START_REF] Salez | Hydrogen Chloride Detection in Orion A and Monoceros R2 and Derivation of the H 35 Cl/H 37 Cl Isotopic Ratio[END_REF]. This can be explained by the fact that the H 2 Cl + ion transforms into hydrogen chloride, HCl by dissociative recombination or by proton transfer to carbon monoxide CO. In addition to the detection by Herschel, another study reported by Neufeld et al [START_REF] Neufeld | Herschel observations of interstellar chloronium[END_REF], also detected absorption and emission of H 2 Cl + from two interstellar cloud sources. The chemical reactions that give rise to the deuterated forms HDCl + and D 2 Cl + are similar to those giving the chloronium ion, H 2 Cl + , with the same orders of magnitude of the reaction rate coefficient. It follows that, like the chloronium ion, its deuterated isotopologues should be abundant in the ISM. Surprisingly, to date, the existence of these deuterated isotopologues (HDCl + and D 2 Cl + ) has not been observed in the ISM.

The availability of accurate spectroscopic data on chloronium and its deuterated forms could greatly facilitate their identification in interstellar space. Despite their importance, very few spectroscopic data on this ion and even less on its deuterated forms HDCl + and D 2 Cl + are available in the literature. Experimentally, the first structural information on the chloronium ion was reported in 1986 by Kawaguchi and Hirota [START_REF] Kawaguchi | Magnetic field modulated infrared laser spectroscopy of the chloronium ClH 2 + ion ν 2 band[END_REF], who studied its ν 2 band with infrared diode laser spectroscopy. Saito et al. [START_REF] Saito | The microwave spectrum of the H 2 Cl + ion[END_REF] detected microwave spectral lines of H 2 35 Cl + in 1988. In the same year, Lee et al [START_REF] Lee | Difference-frequency laser spectroscopy of the ν 1 and ν 3 fundamental bands of H 2 Cl + : Determination of the equilibrium molecular structure[END_REF] worked on the difference-frequency laser spectroscopy of the chloronium. These latter authors have observed the gas phase of the ν 1 and ν 2 fundamental bands of H 2 Cl + and determined its equilibrium structure. The rotational spectra of H 2 Cl + and HDCl + were measured by Araki et al [START_REF] Araki | Submillimeter-wave spectra of H 2 Cl + and its isotopic species: Molecular structure[END_REF] using a submillimeter-wave spectrometer. Furthermore, theoretically, Botschwina [START_REF] Botschwina | Anharmonic potential-energy surfaces, vibrational frequencies and infrared intensities calculated from highly correlated wavefunctions[END_REF] computed a potential energy surface (PES) at coupled electron-pair approximation (CEPA) level and the vibrational frequencies for H 2 Cl + and its isotopologues HDCl + and D 2 Cl + . More recently, after the detection of the chloronium ion, Majumdar et al [START_REF] Majumdar | Formation of different isotopomers of chloronium in the interstellar medium[END_REF] explored the possibility of finding the isotopologues of H 2 Cl + in and around the ISM. These authors used Moller-Plesset perturbation theory (MP2) along with a quadruple-zeta correlation consistent (aug-cc-pVQZ) basis set to optimize the geometry of chloronium and computed the harmonic vibrational frequencies of H 2 Cl + and its two deuterated isotopologues HDCl + and D 2 Cl + . Within their chemical modeling, they found that HDCl + could be efficiently formed in the gas phase and that the emission line should be strong enough to be observed.

The lack of availability of laboratory data, in particular in the case of reactive species (transition molecules, radicals, ions) which remain more difficult to study in the laboratory than their stable counterparts, is the main factor currently limiting astronomical molecular detection. In order to facilitate the detection of the chloronium isotopologues HDCl + and D 2 Cl + , in this work, we provide information on the spectroscopic properties of these isotopologues. We use the explicitly correlated F12, coupled-cluster method, CCSD(T)-F12b, approach to optimize the geometric parameters of the ion and to generate the 3D-potential energy surface. This allows the computation of vibrational energy levels of chloronium H 2 Cl + , its two isotopologues HDCl + and D 2 Cl + (with 35Cl and 37Cl chlorine isotopes), using the variational approach implemented in our home-made Fortran code. We compare our theoretical spectra with those available in the literature, experimental ones with Lee et al [START_REF] Lee | Difference-frequency laser spectroscopy of the ν 1 and ν 3 fundamental bands of H 2 Cl + : Determination of the equilibrium molecular structure[END_REF] and theoretical ones with Botschwina [START_REF] Botschwina | Anharmonic potential-energy surfaces, vibrational frequencies and infrared intensities calculated from highly correlated wavefunctions[END_REF]. Our results show good agreements with those of the literature.

The present paper is organized as follows. In Section 2, the methods for the determination of the kinetic energy operator, the electronic structure, the potential energy surface and the computational details are presented. The results are discussed and concluding remarks are given in Section 3. In the Appendix, the fitting procedure of PES is given.

Methods

Coordinates and kinetic energy operator

The vibrational motion of N atom molecular system, moving on a potential energy surface, V(q) can be described by n = 3N -6 internal coordinates, q = [q 1 , q 2 , . . . q n ]. Considering the body-fixed (BF) Cartesian coordinates expressed as functions of the internal coordinates, x = x (q), i.e x α = x α (q 1 , . . . , q n ), α = 1, . . . 3N, the standard expression of the deformation part of the kinetic energy operator (KEO) in terms of the contravariant components of the metric tensor G ij (q) and with the Euclidean volume element, dν E , reads as [START_REF] Nauts | Momentum, quasi-momentum and hamiltonian operators in terms of arbitrary curvilinear coordinates, with special emphasis on molecular hamiltonians[END_REF]:

TE de f (q, ∂ q ) = - h2 2 n,n ∑ i,j=1 J(q) -1 ∂ i J(q)G ij (q)∂ j , ∂ i ≡ ∂/∂q i , (1)
where J(q) = det(g), [G ij (q)] = [g ij (q)] -1 . The covariant components of the metric tensor, g ij (q), are split in several contributions, g = S n×n C t n×3 C 3×n I 3×3 where S n×n , I 3×3 and C 3×n are the deformation part, the rotation part or the inertia tensor and the Coriolis part of the metric tensor, respectively. Furthermore, dν E de f = J(q)dq 1 • dq 2 • • • dq n is the deformation part of the Euclidean volume element. By expanding the kinetic energy operator (1) and changing the volume element, it can be rewritten in the form : Tρ

de f (q, ∂ q ) = ∑ i≤j f ij 2 (q)∂ 2 ij + ∑ i f i 1 (q)∂ i + V e (q), ∂ 2 ij ≡ ∂ 2 /∂q i ∂q j , (2)
where V e (q) is the extrapotential term [START_REF] Lauvergnat | Exact numerical computation of a kinetic energy operator in curvilinear coordinates[END_REF][START_REF] Mátyus | Toward black-box-type full-and reduced-dimensional variational (ro) vibrational computations[END_REF][START_REF] Chapuisat | N-body quantum-mechanical hamiltonians: Extrapotential terms[END_REF] resulting from the use of a non-Euclidean volume element dν ρ = ρ(q)dq 1 dq 2 . . . dq n . ρ(q) is a real positive function which has to be adapted so that the basis set be orthonormal. The quantities f ij 2 (q), f i 1 (q) and V e (q) in eq. ( 2) are expressed as follows:

f ii 2 (q) = - h2 2 G ii (q), (3)
f ij 2 (q) = -h 2 G ij (q) (i ̸ = j), (4)
f i 1 (q) = - h2 2 ∑ j G ij ∂ j ln ρ(q) + ∂ j ln G ij (q) , (5)
V e (q) = J(q) ρ(q)

TE de f (q, ∂ q ) ρ(q) J(q) . ( 6
)
The formalism of the derivation of the kinetic energy operator (KEO) Eq. ( 1) is well known [START_REF] Nauts | Momentum, quasi-momentum and hamiltonian operators in terms of arbitrary curvilinear coordinates, with special emphasis on molecular hamiltonians[END_REF][START_REF] Meyer | General internal motion of molecules, classical and quantum-mechanical hamiltonian[END_REF][START_REF] Wilson | Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra[END_REF][START_REF] Brocks | Quantum dynamics of non-rigid systems comprising two polyatomic fragments[END_REF]. The first numerical implementation of the KEO operator dates back to the work of Laane et al. [START_REF] Laane | Vector representation of large-amplitude vibrations for the determination of kinetic energy functions[END_REF][START_REF] Harthcock | Calculation of kinetic energy terms for the vibrational hamiltonian: Application to large-amplitude vibrations using one-, two-, and three-dimensional models[END_REF]. Later, other groups proposed numerical approaches to determine the kinetic energy operator in the context of quantum dynamics [START_REF] Smeyers | Ab initio theoretical study of the methyl and phosphine torsion modes in ethylphosphine[END_REF][START_REF] Senent | Determination of the kinetic energy parameters of non-rigid molecules[END_REF][START_REF] Rush | Ab initio CBS-QCI calculations of the inversion mode of ammonia[END_REF]. M. Ndong and coworkers [START_REF] Ndong | Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach: General formulation and removal of singularities[END_REF][START_REF] Lauvergnat | Tnum-Tana: a Fortran code to deal with curvilinear coordinates and kinetic energy operator[END_REF] developed the Fortran program TANA which provides an automatic computation of the kinetic energy operator in analytical form for polyspherical coordinates for any number of atoms. In this approach, the expression of the metric tensor is obtained analytically and it can be used in Eq. ( 1) or in Eq. ( 2). Furthermore, these expressions can be exported in several formats (MCTDH, [START_REF] Richter | Anharmonic Vibrational Treatment Exclusively in Curvilinear Valence Coordinates: The Case of Formamide[END_REF][START_REF] Schröder | The coupling of the hydrated proton to its first solvation shell[END_REF] MidasCpp, [START_REF] Klinting | Vibrational Coupled Cluster Computations in Polyspherical Coordinates with the Exact Analytical Kinetic Energy Operator[END_REF] Fortran). The 3D-vibrational motion of the chloronium ion can be described in terms of the three valence coordinates q = {q 1 , q 2 , q 3 } ≡ {R 1 , R 2 , θ}, where R 1 and R 2 are the distances between the chlorine atom and the two hydrogen atoms and θ is the angle between the two Cl -H bonds. The application of the general KEO formula Eqs. [START_REF] Pilbratt | Herschel Space Observatory-An ESA facility for far-infrared and submillimetre astronomy[END_REF][START_REF] Salez | Hydrogen Chloride Detection in Orion A and Monoceros R2 and Derivation of the H 35 Cl/H 37 Cl Isotopic Ratio[END_REF][START_REF] Neufeld | Herschel observations of interstellar chloronium[END_REF] with the Wilson normalization convention, ρ(q) = 1, (dν 1 = dR 1 • dR 2 • dθ) yields the following KEOs of the H 2 Cl + and its isotopologues:

T(R 1 , R 2 , θ) =F 11 2 ∂ 2 ∂R 2 1 + F 22 2 ∂ 2 ∂R 2 2 + F 33 2 ∂ 2 ∂θ 2 + F 12 2 ∂ 2 ∂R 1 ∂R 2 + F 13 2 ∂ 2 ∂R 1 ∂θ + F 23 2 ∂ 2 ∂R 2 ∂θ + F 1 1 ∂ ∂R 1 + F 2 1 ∂ ∂R 2 + F 3 1 ∂ ∂θ + V ep , (7)
where the functions, F ij 2 , F i 1 and V ep , are given below:

F 11 2 = - h2 2µ R 1 , F 22 2 = - h2 2µ R 2 , F 12 2 = -h 2 cos(θ) m Cl , F 33 2 = - h2 2 1 µ R 1 • R 1 2 + 1 µ R 2 • R 2 2 - 2 cos(θ) m Cl • R 1 • R 2 , F 13 2 = h2 2 1 m Cl • R 2 sin(θ) + 1 sin(θ) - cos 2 (θ) sin(θ) , F 23 2 = h2 2 1 m Cl • R 1 sin(θ) + 1 sin(θ) - cos 2 (θ) sin(θ) , F 1 1 = h2 4 1 m Cl • R 2 3 cos(θ) - cos(θ) sin 2 (θ) - cos 3 (θ) sin 2 (θ) , F 2 1 = h2 4 1 m Cl • R 1 3 cos(θ) - cos(θ) sin 2 (θ) - cos 3 (θ) sin 2 (θ) , F 3 1 =h 2 1 m Cl • R 1 • R 2 cos 2 (θ) sin(θ) - 1 sin(θ) , (8)
V e (q) = -h2 4

1 m Cl • R 1 • R 2 2 cos(θ) + cos 3 (θ) sin 2 (θ) -2 cos(θ) sin 2 (θ) - h2 8 1 µ R 1 • R 1 2 2 + cos 2 (θ) sin 2 (θ) - h2 8 1 µ R 2 • R 2 2 2 + cos 2 (θ) sin 2 (θ) , (9)
where µ R 1 and µ R 2 are reduced masses :

1 µ R 1 = 1 m Cl + 1 m H 1 and 1 µ R 2 = 1 m Cl + 1 m H 2 .
Furthermore, m Cl , m H 1 and m H 2 are, respectively, the isotopic masses of the chlorine, the first hydrogen and the second hydrogen atoms. It is worth noting, that the TANA code has been used to obtain the analytical expression and the Fortran subroutines of the KEO. Furthermore, some simplifications are possible in the previous expressions (Eq. (8-9)).

For example, sin(θ) + 1 sin(θ) -cos 2 (θ) sin(θ) can be simplified in (2 sin(θ)). However, we prefer to present the unsimplified expressions, which are used in our quantum dynamics code and come directly from the TANA code [START_REF] Ndong | Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach: General formulation and removal of singularities[END_REF]. The present triatomic KEO has been published in the past by several group [START_REF] Carney | Variational approaches to vibration-rotation spectroscopy for polyatomic molecules[END_REF][START_REF] Carter | A variational method for the calculation of vibrational levels of any triatomic molecule[END_REF][START_REF] Sutcliffe | A comment on a recent proposal for the calculation of vibrational energies in the general triatomic molecule[END_REF]. Isotope masses (g.mol -1 ) masses (au) 35

Electronic structure and potential energy surface

The electronic structure has been carried out with the explicitly correlated, F12, coupled cluster method, CCSD(T)-F12b [START_REF] Adler | A simple and efficient CCSD(T)-F12 approximation[END_REF], with the F12 adapted triple zeta basis set [START_REF] Peterson | Systematically convergent basis sets for explicitly correlated wavefunctions: The atoms H, He, B-Ne, and Al-Ar[END_REF], cc-pVTZ-F12 (CCSD(T)-F12b/VTZ-F12) using molpro2010 code [START_REF] Werner | A package of ab initio programs[END_REF]. As noted in several studies, this level of computation provides a good compromise between quality and computational efficiency [START_REF] Martin | Assessment of CCSD(T)-F12 Approximations and Basis Sets for Harmonic Vibrational Frequencies[END_REF]. Cl + isotopologues) [START_REF] Lee | Difference-frequency laser spectroscopy of the ν 1 and ν 3 fundamental bands of H 2 Cl + : Determination of the equilibrium molecular structure[END_REF].

The H 2 Cl + potential energy surface (PES), V(Q), was obtained by fitting 145 ab initio points at CCSD(T)-F12b/VTZ-F12 level using symmetrized coordinate displacements (θθ eq , 1/2(R 1 + R 2 ) -R eq and 1/2(R 1 -R 2 )). Along these displacements, the grid ranges in bohr are [-0.90 : 1.03], [-0.41 : 0.65] and [0.00 : 0.48], respectively. The full grid covers energies up to about 10000 cm -1 and it has been fitted as linear combinations of coordinate displacement products. The fitted potential reproduces almost perfectly the ab initio points. Indeed, the root-mean-square error and the maximum error are about 0.1 cm -1 and 0.5 cm -1 , respectively. Furthermore, Fig. 1 shows the ab initio energy points (red crosses) and the fitted potential (solid blue line) along the three symmetrized coordinates (θ, R + and R -), which clearly shows the almost perfect agreement.

The grid generation and the fitting procedure are explained in detail in the Appendix. This PES has been implemented in the Quantum Model Library, freely available on github [START_REF] Lauvergnat | Quantum Model Library: a library of potentials[END_REF].

Vibrational computational approach

The computation of the vibrational levels is relatively standard and it is related on early works on variational methods [START_REF] Carney | Variational approaches to vibration-rotation spectroscopy for polyatomic molecules[END_REF][START_REF] Whitehead | Variational calculation of vibration-rotation energy levels for triatomic molecules[END_REF][START_REF] Botschwina | Vibrational frequencies from anharmonic ab initio/empirical potential energy functions. I. Method and application to H 2 O, HNO, HOF and HOCl[END_REF][START_REF] Tennyson | The ab initio calculation of the vibrational-rotational spectrum of triatomic systems in the close-coupling approach, with KCN and H 2 Ne as examples[END_REF]. For the present study, it has been performed with the help of our home-made pseudo-spectral Fortran code, TI_Schrod [START_REF] Afansounoudji | TI_Schrod: a Fortran code to solve the Schrödinger equation[END_REF]. It allows us to solve the time-independent Schrödinger equation with a Hamiltonian expressed in terms of the three internal coordinates, q = [R 1 , R 2 , θ]. The 3D-quantum vibrational bound states are expanded on a direct product of one-dimensional harmonic oscillator basis sets, and thus, ρ(q) associated with the non-Euclidean volume element is equal to one (ρ(q) = 1). Furthermore, the integration procedure is performed using the direct product of Gauss-Hermite quadrature grids.

More precisely, each eigenstate is developed as follows:

Ψ(R 1 , R 2 , θ) = ∑ i 1 ,i 2 ,i 3 C i 1 ,i 2 ,i 3 HO i 1 (R 1 )HO i 2 (R 2 )HO i 3 (θ), (10)
where, for the coordinate, q j , the HO i (q j ) are the normalized Harmonic oscillator basis functions, HO i (q j ) = N sc i .H i-1 (x j (q j )) • exp[-1 2 x j (q j ) 2 ]. x j (q j ) is the scaled and shifted coordinate, x j (q j ) = sc j • (q jq j re f ), N sc j i is a normalization factor and H i-1 (x) are the Hermite polynomials. re f is the value of q j at the equilibrium geometry and the expression of sc j is given by 4 k j /G jj where k j and G jj are, respectively, the curvature and the value of contravariant component of the metric tensor at the q re f geometry [START_REF] Marsili | Quantum and Quantum-Classical Studies of the Photoisomerization of a Retinal Chromophore Model[END_REF]. The values of the q j re f , sc j and G jj are given in Table [START_REF] Pilbratt | Herschel Space Observatory-An ESA facility for far-infrared and submillimetre astronomy[END_REF] for all studied isotopologues.

Before discussing in more detail the vibrational spectra of H 2 Cl + and its isotopologues, we have checked the convergence of the energy levels with respect to the basis set and the grid sizes. In our tests, the three 1D-basis sets have the same number of basis functions, nb which increases from 9 to 12 and the grid point number, nq, is fixed to 12. These tests have shown, that, the largest error between the energy levels up to 4000-5000 cm -1 calculated with nb = 11 and nb = 12 is smaller than 10 -2 cm -1 . Therefore, we have chosen 12 basis functions for each Harmonic oscillator basis and for all isotopologues.

Discussion and conclusion

The table 3 lists the computed and experimental anharmonic vibrational frequencies, expressed in cm -1 , for the H 2 35 Cl + and H 2 37 Cl + isotopologues. The differences between the experimental and the computed values are 0.5 cm -1 for the bending mode (ν 2 ) and about 5 cm -1 for the two stretching modes (ν 1 and ν 3 ). This shows a good agreement between our calculated values and the experimental ones. Furthermore, these values show a clear improvement with respect to those computed by Botschwina with his unscaled potential [START_REF] Botschwina | Anharmonic potential-energy surfaces, vibrational frequencies and infrared intensities calculated from highly correlated wavefunctions[END_REF], in particular, for the bending mode. Indeed, Botschwina developed two potentials: (i) the first one (unscaled) is obtained after the fitting of the ab-initio calculations; (ii) the second one is derived from the first one, but some parameters are scaled to reproduce the experimental fundamental transitions. Therefore, we compare our full ab-initio potential with the first Botschwina potential, and our results clearly show the quality of our potential. Table 4 gives the vibrational frequencies for the other isotopologues of this ion, namely HD 35 Cl + , HD 37 Cl + , D 2 35 Cl + , D 2 37 Cl + . The equilibrium molecular structure was therefore determined by taking the expectation values of each coordinate (⟨R 1 ⟩, ⟨R 2 ⟩ and ⟨Θ⟩) for the vibrational ground-state. The values resulting from these calculations are reported in the table 5. These values obtained for the isotopologues, H 2 35 Cl + , HD 35 Cl + and H 2 37 Cl + agree very well with the experimental values (the distances and the angle are extracted from the experimental rotational constants) [START_REF] Araki | Submillimeter-wave spectra of H 2 Cl + and its isotopic species: Molecular structure[END_REF][START_REF] Kawaguchi | Magnetic field modulated infrared laser spectroscopy of the chloronium ClH 2 + ion ν 2 band[END_REF]. Indeed, for the three isotopologues, the angle values are very close to the experimental ones and the errors are less than 0.1 • [START_REF] Araki | Submillimeter-wave spectra of H 2 Cl + and its isotopic species: Molecular structure[END_REF][START_REF] Kawaguchi | Magnetic field modulated infrared laser spectroscopy of the chloronium ClH 2 + ion ν 2 band[END_REF]. With respect to the Cl-H distance of H 2 35 Cl + obtained from ref [START_REF] Kawaguchi | Magnetic field modulated infrared laser spectroscopy of the chloronium ClH 2 + ion ν 2 band[END_REF], the difference with our calculation is about 0.1 Å, which is surprisingly large. However, we are confident with our values. Indeed, our equilibrium values agree very well with Botschwina ones [START_REF] Botschwina | Anharmonic potential-energy surfaces, vibrational frequencies and infrared intensities calculated from highly correlated wavefunctions[END_REF] and also from the experimental ones [START_REF] Araki | Submillimeter-wave spectra of H 2 Cl + and its isotopic species: Molecular structure[END_REF][START_REF] Kawaguchi | Magnetic field modulated infrared laser spectroscopy of the chloronium ClH 2 + ion ν 2 band[END_REF]. Furthermore, with respect to a more recent study of Araki et al. [START_REF] Araki | Submillimeter-wave spectra of H 2 Cl + and its isotopic species: Molecular structure[END_REF], the experimental Cl-H and Cl-D distances (associated to the zero point average structure [START_REF] Araki | Submillimeter-wave spectra of H 2 Cl + and its isotopic species: Molecular structure[END_REF]) are in good agreement with our calculated values since the largest difference with respect to the experimental value is only 0.005 Å with respect to ref [START_REF] Araki | Submillimeter-wave spectra of H 2 Cl + and its isotopic species: Molecular structure[END_REF]. Furthermore, our calculations nicely reproduce the shortening of the Cl-D bond length with respect to the Cl-H one for the HD 35 Cl + ion [START_REF] Araki | Submillimeter-wave spectra of H 2 Cl + and its isotopic species: Molecular structure[END_REF]. In the Table 5, we also give the expectation values for other isotopologues, D 2 35 Cl + , HD 37 Cl + and D 2 37 Cl + . In addition, we were able to compute the three rotational constants associated with the vibrational ground state. However, this has to be done with care. Indeed, since we did not perform a full ro-vibrational calculation, the rotational Eckart conditions [START_REF] Eckart | Some Studies Concerning Rotating Axes and Polyatomic Molecules[END_REF] must be applied to minimize the Coriolis coupling. The Cartesian equilibrium geometry (obtained from R 1 = R 2 = 2.46729 bohr and θ = 1.64588 rad) is used as the reference geometry to apply the Eckart conditions. They have been implemented in Tnum [START_REF] Lauvergnat | Tnum-Tana: a Fortran code to deal with curvilinear coordinates and kinetic energy operator[END_REF] as described in ref [START_REF] Lauvergnat | Numerical and exact kinetic energy operator using Eckart conditions with one or several reference geometries: Application to HONO[END_REF]. The rotational constants have been computed with the Fortran codes ElVibRot [START_REF] Lauvergnat | ElVibRot: a Fortran Quantum Dynamics code[END_REF] coupled with Tnum using the following procedure: (i) Calculate the average values (⟨µ α,β ⟩, with α and β being x, y or z) of the contravariant rotational components of the metric tensor. This is done with the numerical conditions (basis set and grid parameters) used to compute the energy levels. (ii) Diagonalize the previous 3 × 3 matrix to get the rotational constants as its eigenvalues. The comparison between the computed and the experimental values given in Table 5 shows a nice agreement. Indeed, the largest error is about 1. GHz. Furthermore, the rotational constant shifts due to the isotopic substitution (from H 2 35 Cl + to H 2 37 Cl + ) are well reproduced. In this computational work, we have generated the 3D-potential energy surface of the ground state of the chloronium ion using a state-of-the-art electronic structure level, namely the F12-explicitly correlated coupled cluster method with a F12-triple zeta atomic basis set. This fitted polynomial expansion was incorporated into a home-made variational Fortran code to compute the energy levels, the zero-point average coordinate values and the rotational constants of several isotopologues, H

A Potential energy surface fitting procedure

The potential, V(Q), has been expanded in terms of displacements,

∆Q i = Q i -Q re f i , (i = 1, 2, 3)) of symmetrized coordinates, Q ([θ, R + , R -]) with Q re f = [1.
645916 rad, 2.467341 bohr, 0. bohr] and expressed as a sum of four n-mode contributions, V 0 (Q), ∆V 1 (Q), ∆V 2 (Q) and ∆V 3 (Q), which depend on the coupling between modes. The first contribution, V 0 (Q), is just the reference energy, i.e. the energy of the minimum. The other contributions are fitted from the energies generated similarly to the n-mode expansion:

∆V K (Q) = ∑ R V K (i 1 ,i 2 ,i 3 ) C i 1 ,i 2 ,i 3 • ∆Q 1 i 1 ∆Q 2 i 2 ∆Q 3 i 3 . (11)
The C i 1 ,i 2 ,i 3 are the coefficients obtained by a least square procedure of the multimode expansion ab initio values,

∆V ab K (Q). Each R V K (i 1 , i 2 , i 3 ) defines a constraint on the [i 1 , i 2 , i 3 ] and its general expression is (i 1 + i 2 + i 3 ) ≤ L V
K and with i 1 ≥ 0, i 2 ≥ 0 and i 3 ≥ 0. Furthermore, Since the potential is symmetric with respect to ∆Q 3 , only the even values of i 3 are possible. L V K is a parameter which controls the number of terms or functions in the expansion. More precisely, for each value of K, R K is defined as follows:

(i) One mode expansion (K=1), R V 1 :
The possible values of the indices are [i 1 , 0, 0] or [0, i 2 , 0] or [0, 0, i 3 ] (with i 1 ≥ 1, i 2 ≥ 1 and i 3 ≥ 2) and L V 1 = 9, the number of functions is 22 (9 functions for [i 1 , 0, 0] and [0, i 2 , 0] and only 4 functions for [0,

0, i 3 ] since i 3 is even. (ii) Two mode expansion (K=2), R V 2 : The possible values of the indices are [i 1 , i 2 , 0] or [i 1 , 0, i 3 ] or [0, i 2 , i 3 ] (with i 1 ≥ 1, i 2 ≥ 1 and i 3 ≥ 2). With L V 2 =
8, the number of functions is 52 (28 functions for [i 1 , i 2 , 0] and 12 functions for [i 1 , 0, i 3 ] or [0, i 2 , i 3 ]. (iii) Three mode expansion (K=3), R V 3 : Here, all indices are larger than 0 and with L V 3 = 6, the number of functions is 7.

Those three contributions are fitted independently of each other using well-defined grids generated in terms of displacement with respect to a reference geometry, Q re f . A least-square procedure is used with the ∆V ab K (Q) values on the grid points. Those values are related to the ab initio energy values, V ab (Q). The full 3D-grid is split in four parts, G 0 , G 1 , G 2 and G 3 depending on the coupling between modes. The first grid, G 0 , contains a single point, the reference geometry and the corresponding ab initio energy, E 0 . The other points are generated with constraints, R G K , on the indices [u 1 , u 2 , u 3 ]: u 1 + u 2 + u 3 ≤ L G K , where L G K control the size of the grid. The constraints for K = 1, 2, 3 are the following: (i) One mode expansion (K=1), R G 1 : The possible values of the indices are [u 1 , 0, 0] or [0, u 2 , 0] or [0, 0, u 3 ] (with u 1 ≥ 1, u 2 ≥ 1 and u 3 ≥ 1). With L G 1 = 5, the number of grid points is 25 (10 for [u 1 , 0, 0] or [0, u 2 , 0] and only 5 for [0, 0, u 3 ]. Then, the ∆V ab 1 (Q) values are obtained as follows: ∆V ab 1 (Q) = V ab (Q) -E 0 . (ii) Two mode expansion (K=2), R G 2 : The possible values of the indices are [u 1 , u 2 , 0] or [u 1 , 0, u 3 ] or [0, u 2 , u 3 ] (with u 1 ≥ 1, u 2 ≥ 1 and u 3 ≥ 2). With L G 2 = 5, the number of grid points is 80 (40 for [u 1 , y 2 , 0] and 20 [u 1 , 0, u 3 ] or [0, u 2 , u 3 ]. The ∆V ab 2 (Q) is given by:

∆V ab 2 (Q) = V ab (Q) -E 0 -∆V ab 1 (Q 1 , Q re f 2 , Q re f 3 ) -∆V ab 1 (Q re f 1 , Q 2 , Q re f 3 ) -∆V ab 1 (Q re f 1 , Q re f 2 , Q 3 ). (12)
(iii) Three mode expansion (K=3), R G 3 : Here, all index absolute values are larger than 0 and with L G 3 = 5, the number of grid points is 40. The ∆V ab 3 (Q) is given by:

∆V ab 3 (Q) = V ab (Q) -E 0 -∆V ab 1 (Q 1 , Q re f 2 , Q re f 3 ) -∆V ab 1 (Q re f 1 , Q 2 , Q re f 3 ) -∆V ab 1 (Q re f 1 , Q re f 2 , Q 3 ) -∆V ab 2 (Q 1 , Q 2 , Q re f 3 ) -∆V ab 2 (Q 1 , Q re f 2 , Q 3 ) -∆V ab 2 (Q re f 1 , Q 2 , Q 3 ) (13)
In those three grids (G K ), for each point the k th -coordinate reads:

Q k = Q re f k + ∆Q k (u k ) (14)
where the ∆Q k (u k ) (k = 1, 2, 3) are given in table 6.

Figure 1 :

 1 Figure 1: Ab initio energy point (red cross) and fitted potential (full blue line) along the three symmetrized coordinates. The energy is in Hartree, θ in radian and R + and R -are in bohr.

Fig. 2

 2 Fig.2shows the two iso-contours of the PES cuts. The left panel displays the contour as function of the two HCl distances with the angle fixed around its equilibrium value (1.64 radian), while the right panel displays the contour as function of the angle and one HCl distance with the other distance fixed around its equilibrium value (2.46 bohr).

Figure 2 :

 2 Figure 2: 2D-contour plots of H 2 Cl + potential energy surface at CCSD(T)F12b/cc-pVTZ-F12. Left panel: R 1 -R 2 contour. Right panel: R 1θ contour.

Table 1 :

 1 Isotopic masses used in this study and obtained from the 2018 NIST table.

  The geometry was optimized at CCSD(T)-F12b/VTZ-F12 and the optimized H-Cl distance and the HClH angle are 1.306 Å and 94.3°, respectively. These values are close to the Botschwina CEPA-1 values (1.302 Å and 94.2°)[START_REF] Botschwina | Anharmonic potential-energy surfaces, vibrational frequencies and infrared intensities calculated from highly correlated wavefunctions[END_REF] and the experimental ones (1.304 Å and 94.24°f or both H 2 35 Cl + or H 2 37

Table 2 :

 2 Basis set parameters (in atomic unit) for the isotopologues.

	q j			1 (R 1 )	2 (R 2 )	3 (θ)
	q	j re f			2.467	2.467	1.646
	k j			0.282	0.282	0.168
			35 Cl + HD 35 Cl + 1785.69 3471.53 7158.55 1785.69 1785.69 5424.03 H 2
	35 Cl + 37 Cl + HD 37 Cl + 1788.39 3481.78 7173.92 3471.53 3471.53 10523.94 1/G jj D 2 1788.39 1788.39 5432.85 H 2
			D 2	37 Cl +	3481.78 3481.78 10557.19
			35 Cl + HD 35 Cl + H 2	4.74 4.74	4.74 5.59	5.49 5.89
	sc j	35 Cl + 37 Cl + HD 37 Cl + D 2 H 2	5.59 4.74 4.74	5.59 4.74 5.60	6.48 5.49 5.89
			D 2	37 Cl +	5.60	5.60	6.49

Table 3 :

 3 computed and experimental anharmonic vibrational transitions (in cm -1 ) for H 235 Cl + and H 237 Cl + isotopologues.

		H 2	35 Cl +		H 2	37 Cl +
	Band	This work Botschwina[13]	Exp.	This work	Exp.
	ν 2	1184.6	1206.8	1184.1 a	1183.7	1183.2 a
	2ν 2	2356.6	2398.1		2354.8
	ν 3	2634.9	2637.4	2630.1 b	2632.8	2628.1 b
	ν 1	2648.2	2650.0	2643.2 b	2646.5	2641.5 b
	3ν 2	3515.3	3565.7		3512.6
	ν 2 + ν 3	3794.0	3816.5		3791.1
	ν 1 + ν 2	3813.1	3838.3		3810.5
	4ν 2	4660.2	4690.5		4656.6
	2ν 2 + ν 3	4940.7	4979.8		4937.0
	2ν 2 + ν 1	4965.5	5011.1		4962.1
	ZPE	3320.9	3335.1		3318.4

a: Ref

[START_REF] Kawaguchi | Magnetic field modulated infrared laser spectroscopy of the chloronium ClH 2 + ion ν 2 band[END_REF]

b: Ref

[START_REF] Lee | Difference-frequency laser spectroscopy of the ν 1 and ν 3 fundamental bands of H 2 Cl + : Determination of the equilibrium molecular structure[END_REF]

Table 4 :

 4 Computed anharmonic vibrational frequencies (in cm -1 ) for HD35 Cl + , HD 37 Cl + , D 235 Cl + and D 237 Cl + isotopologues.BandHD35 Cl + HD 37 Cl + D 235 Cl + D 237 Cl +

	ν 2	1034.6	1033.5	856.9	855.7
	2ν 2	2059.4	2057.4	1707.4	1705.0
	ν 3	2641.3	2639.4	1915.3	1912.4
	ν 1	1917.5	1914.8	1920.0	1917.5
	3ν 2	3074.2	3071.2	2551.4	2547.8
	ν 2 + ν 3	3653.5	3650.6	2759.0	2754.9
	ν 1 + ν 2	2939.9	2936.2	2766.7	2763.0
	4ν 2	4078.5	4074.4	3388.5	3383.8
	2ν 2 + ν 3	4656.2	4652.4	3596.5	3591.3
	2ν 2 + ν 1	3952.7	3948.0	3607.0	3602.2
	ZPE	2862.9	2860.0	2391.0	2387.6

Table 5 :

 5 Experimental and computed expectation values of coordinates and rotationnal constants for several isotoplogues with the distances in Å, the angles in °and the rotationnal constants in GHz.

		H 2	35 Cl +		HD 35 Cl +		H 2	37 Cl +
		This work Ref [12] This work Ref[12] This work Ref[12]
	⟨R 1 ⟩	1.325	1.3206	1.325	1.3207	1.325	1.3206
	⟨R 2 ⟩	1.325	1.3206	1.319	1.3165	1.325	1.3206
	⟨Θ⟩	94.33	94.24	94.32		94.33	94.24
	A	336.9	337.352	308.8	309.476	336.0	336.360
	B	272.5	273.587	152.7	153.168	272.5	273.587
	C	147.9	148.100	100.7	100.837	147.7	147.907
		HD 37 Cl +	D 2	35 Cl +		D 2	37 Cl +
		This work	This work		This work
	⟨R 1 ⟩	1.325		1.320		1.319
	⟨R 2 ⟩	1.319		1.320		1.319
	⟨Θ⟩	94.32		94.31		94.31
	A	308.2		177.6		176.7
	B	152.4		136.5		136.5
	C	100.5		76.2		76.0

  2 35 Cl + , H 237 Cl + , HD35 Cl + , HD 37 Cl + , D 2 35 Cl + and D 2 37 Cl + . Our results are in good agreement with the available experimental data. DL gratefully acknowledges the hospitality of the Laboratoire de Physique des Matériaux et des Composants à Semi-conducteurs, Département de Physique, Université de Lomé, where part of this work was carried out. RA gratefully thanks the ASESMANET network for giving me the opportunity to spend a month in the Institut de Chimie Physique of the Université Paris-Saclay.

	4 Acknowledgement

Table 6 :

 6 Values of the ∆Q k (u k ) as function of u k (k = 1, 2, 3) to define the grid points. ∆Q 1 0.00 0.21 -0.18 0.41 -0.36 0.62 -0.54 0.83 -0.72 1.03 -0.90 ∆Q 2 0.00 0.13 -0.08 0.26 -0.17 0.39 -0.25 0.52 -0.33 0.65 -0.41

	u k	0	1	-1	2	-2	3	-3	4	-4	5	-5
	∆Q 3 0.00 0.10		0.19		0.29		0.38		0.48


Nassim Athmouni
email: athmouninassim@yahoo.fr

Mondher Damak
email: mondher_damak@yahoo.com

Brice Franke
email: brice.franke@univ-brest.fr

Nejib Yaakoubi
email: nejibyaakoubi@gmail.com

ON SHIFTING THE PRINCIPAL EIGENVALUE OF DIRICHLET PROBLEM TO INFINITY WITH NON-TRANSVERSAL INCOMPRESSIBLE DRIFT

Keywords:

We prove that it is always possible to add some divergence free drift vector-eld to some two dimensional spherical Dirichlet problem, such that the resulting principal eigenvalue lies above a prescribed bound. By construction those drift vector-elds vanish on the boundary and their ow lines individually stay away from the boundary. The capacity of those drift vectorelds to accelerate diusivity originates from high frequency oscillation of the associated ow lines. The lower bounds for the spectrum are obtained through isoperimetric inequalities for ow invariant functions.

Introduction

In this paper, we consider, in a bounded domain Ω ⊂ R d , the following eigenvalue problem with Dirichlet boundary condition:

∆ϕ + V • ∇ϕ = λϕ in Ω, ϕ = 0 on ∂Ω,
where ∆ is the Laplace operator on R d and V is a divergence free C 1 vector-eld on Ω. We are interested in the inuences of the drift vector-eld V on the principal eigenvalue λ V := -sup Re(z); z ∈ spec(A V ) of the elliptic dierential operator

A V = ∆ + V • ∇
under the above boundary condition. It is well known that in this situation the principal eigenvalue is real and the corresponding eigen-function takes only real values. The principal value is an important indicator for the time the diusion generated by A V needs to reach the absorbing boundary ∂Ω. It is interesting to understand which geometric features of the ow generated by the vector-eld V inuence the principal eigenvalue. In [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF] the authors Berestycki, Hamel and Nadirashvili studied the asymptotic behavior of the principal eigenvalue λ cV as some real parameter c grows to innity. More precisely, they proved that [START_REF] Adams | Sobolev Spaces[END_REF]

lim |c|→∞ λ cV = inf Ω |∇ϕ| 2 dx; ϕ ∈ ker(V • ∇) ∩ H 1 0 , ϕ L 2 = 1 ,
where H 1 0 denotes the space of rst order Sobolev functions satisfying Dirichlet boundary conditions on Ω. An analogous result for the spectral gap

ρ V := -sup Re(z); z ∈ spec(A V ) \ {0}
of some diusion generator A V on a compact Riemannian manifold M without boundary was proved in [START_REF] Franke | The behavior of the spectral gap under growing drift[END_REF]. The expression in [START_REF] Adams | Sobolev Spaces[END_REF] shows, that in a situation, where almost all trajectories of the ow generated by the vector-eld V come arbitrarily close to the boundary ∂Ω of Ω, the limit of the principal eigenvalue becomes innite. Vector-elds with this property can easily be constructed; i.e.: the vector-eld V (x) = (1, 0, ..., 0); x ∈ Ω generates a ow following straight parallel lines which leaves the bounded domain Ω in nite time. Those type of examples are constructed via vector-elds which are transversal to the boundary on a portion of ∂Ω. One might ask, whether there exist examples, where the vector-eld is tangential to ∂Ω or even vanishing on ∂Ω. It can be seen from ( 1) that in order to have a diverging principal value it is sucient that the ow generated by V has a dense set of trajectories coming arbitrarily close to the boundary ∂Ω, since this then implies that the intersection of the kernel of the operator V • ∇ with H 1 0 only contains the zero function. This means that in this situation there exists a sequence of divergence free vector-elds of the form c n V such that the associated principal value diverges to innity. In dimension two, due to Jordan's curve theorem, it is not possible to construct measure preserving ows having a dense set of trajectories coming arbitrarily close to the boundary without ever reaching it. This also rules out the existence of ergodic ows in this situation. One can then relax the question and ask, whether one can nd a sequence of vector-elds V n with ow lines keeping some distance to the boundary ∂Ω such that the associated principal eigenvalues diverge to innity.

In the rst part of the paper, we construct a sequence of divergence free vectorelds satisfying this property on the domain Ω = B 1 (0) in R 2 . The result can be generalized to more general bounded domains in R d by using more advanced techniques. In order to reduce the complexity of our arguments, we choose to state and prove the special case of the open ball B 1 (0) extensively and to add a discussion on how to improve our result to more general bounded domains in a nal section at the end of the paper. Theorem 1. On the open ball B 1 (0) in R 2 there exists a sequence of continuous and piece-wise

C 1 vector-elds V n ; n ∈ N with the properties div(V n ) = 0 and V n | ∂B1(0) = 0 generating a ow Φ (n) t ; t ∈ R with dist {Φ (n) t (x); t ∈ R}, ∂B 1 (0) > 0, for all x ∈ B 1 (0)
such that the sequence of associated principal eigenvalues satises

lim n→∞ λ Vn = ∞.
The analogous problem on various compact Riemannian surfaces without boundary and on d-torus was studied in [START_REF] Franke | On how to push the spectral gap of a diusion on S 2 to innity[END_REF], [START_REF] Franke | Accelerating diusion on compact surfaces by drift[END_REF] and [START_REF] Hwang | Accelerating Brownian motion on N-torus[END_REF]. There the authors prove that the spectral gap ρ V can become arbitrarily large after a suitable choice of V .

The prove of Theorem 1, which will be prepared in the following sections and completed in Section 4, is based on the construction of explicit divergence free vectorelds V . Each of those vector-elds is constructed in Section 2 through its ow, which will be composed of smooth closed curves which do not leave the domain B 1 (0). Those curves are constructed to strongly oscillate in the radial direction as they orbit around the origin in Section 1. This feature of the trajectories, will ensure that ow-invariant functions have relatively large energy integrals compared to their L 2 -norms, where the ratio can be increased through augmentation of the number of oscillations. The resulting vector-eld V can then be multiplied with a large constant c and the asymptotic behavior of the resulting principal eigenvalue λ cV can be studied as c grows to innity. Note that the limit of the principal eigenvalue given in ( 1) is expressed through variation of energy integrals over the space of ow invariant functions. As noted above, this expression can be made arbitrarily large by increasing the ows oscillation and thus in order to obtain a vector-eld with a large principal eigenvalue we rst construct a ow with a suciently high number of oscillations and then multiply a suciently large constant to the underlying vector-eld. In order to prove Theorem 1 it is then necessary to show that the inmum of the variation problem can be made arbitrarily large by pushing up the number of oscillations. For this, in Section 3, we will introduce some suitable rotationally symmetric comparison problem, which takes into account the oscillation, and, in Subsection 3.1, use some Faber Krahn type argument in order to obtain lower bounds for the above variation problem in terms of the symmetrized variation problem. In Subsection 3.2 we then use an argument based on some kind of Cheeger constant in order to obtain lower bounds for the symmetrized problem which grows to innity as the number of oscillations increases. Those arguments are based on the careful analysis of the ratio between the length of the ows trajectories and the area that they encircle which is presented in Section 1.

One might ask, whether it is possible to give some estimate on the amplitude of the vector-eld which is necessary to push the principal eigenvalue over a prescribed value. However, this is not possible with the approach presented in this paper, since the method is based on applying the asymptotic result from [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF], which does not provide such information.

construction of the flow lines

In this section, our goal is to construct ows of closed curves which oscillate in the radial direction as they revolve around the origin. In order to t into the unit ball B 1 (0) the amplitude of the oscillation has to be reduced to zero as the curves approach the unit circle ∂B 1 (0). Also the oscillation has to fade down to zero as the curves come closer to the origin. This will be incorporated into the construction through some suitable amplitude function η, which will be constructed below. In Proposition 1.2 we will show that the ratio between the length of those closed curves and the area that they encircle goes to innity uniformly over compact sets of radii as the number of oscillations increases. This will be essential for achieving the divergence of the Cheeger constants in Subsection 3.2.

Let B r (0) denote the ball of radius r centered at zero in R 2 . For a dierentiable curve γ : [0, T ] → R 2 we denote by (γ) its curve length in R 2 and for an measurable r .

and C 1 on ]0, 1[×[0, 2π[. Remark 1. For every r > 0, the set Γ (n) r is a closed curve in B 1 (0).
Lemma 1.1. For all n ∈ N and one has as r → 0 that (Γ (n) r ) ∼ (∂B r (0)) and also that a( Γ (n) r ) ∼ a(B r (0)).

Proof. At rst, note that, for r < 1 3 , the curve Γ (n) r oscillates n times between the circle of radius (r -r 2

2 ) and the circle of radius (r + r 2 2 ). Thus,

2π r - r 2 2 ≤ (Γ (n) r ) ≤ 2π r + r 2 2 + 2nr 2 .
After dividing the 3 sides of those inequalities by the arc length of ∂B r (0), we obtain

2π(r -r 2 2 ) 2πr ≤ (Γ (n) r ) (∂B r (0)) ≤ (2π(r + r 2 2 ) + 2nr 2 2πr .
Then, some computation yields

1 - r 2 ≤ (Γ (n) r ) (∂B r (0)) ≤ 1 + r 2 + nr π .
We obtain the rst assertion when taking the limit r → 0. The second assertion follows form the rst by integration over r.

a( Γ (n) r ) = a( Γ (1) r ).
Remark 3. For c ∈]0, π[ let r c denote the unique radius such that

a( Γ (n) rc ) = c.
It follows from the previous remark that r c does not depend on n. It thus follows from Lemma 1.1 that for any choice of > 0 there exists a δ > 0 such that for c < δ one has that

c = a( Γ (n) rc ) ≤ (1 + )πr 2 c . Proposition 1.2. We have, for all c 0 < π that (2) inf 0<c<c0 (Γ (n) rc ) c -→ n→∞ ∞.
Proof. At rst, for a xed r, by computing the innitesimal arc length of the curve

Γ (n) r
and integrating over [0, π] with respect to the θ-variable, one has that

(Γ (n) r ) ≥ π 4 r 1 + n 2 η 2 (r) 8 .
For proving the assertion (2), we will identify 3 cases

inf 0<c<c0 (Γ (n) rc ) c ≥ min inf 0<c<c0 rc < 1 3 (Γ (n) rc ) c ; inf 0<c<c0 1 3 ≤rc ≤ 2 3 (Γ (n) rc ) c ; inf 0<c<c0 2 3 <rc (Γ (n) rc ) c .
For the rst case, i.e. r c < 1 3 , let us consider a sequence α n which converges to zero as n → ∞. One has

inf 0<c<c0 rc < 1 3 (Γ (n) rc ) c ≥ min inf 0<c<c0 0<rc <αn (Γ (n) rc ) c ; inf 0<c<c0 αn <rc < 1 3 (Γ (n) rc ) c .
For 0 < r c < α n , we can use Lemma 1.1 and Remark 3 to see that

c = γ( Γ (n) rc ) ∼ πr 2 c . It follows then that inf 0<c<c0 rc < 1 3 (Γ (n) rc ) c ≥ inf 0<c<c0 0<rc <αn π 4 r c 1 + n 2 η 2 (rc) 8 c ≥ inf 0<c<c0 rc <αn π 4 r c 1 + n 2 r 2 c 8 (1 + )πr 2 c . ≥ 1 4(1 + ) 1 α 2 n + n 2 8 -→ n→∞ ∞.
We now choose α n = 1 4 √ n which clearly satises α n → 0 as n → ∞. For the case when α n < r c < 1/3, one has

inf 0<c<c0 αn<rc < 1 3 (Γ (n) rc ) c ≥ inf 0<c<c0 rc <αn π 4 r c 1 + n 2 η 2 (rc) 8 c ≥ inf 0<c<c0 rc <αn π 2 r c 1 + n 2 r 2 c 8 c 0 ≥ π 4 α n 1 + n 2 α 2 n 8 c 0 ≥ π 4c 0 1 4 √ n + n 8 -→ n→∞ ∞.
For the second case, when

1 3 ≤ r c ≤ 2 3 , holds inf 0<c<c0 1 3 ≤rc ≤ 2 3 (Γ (n) rc ) c ≥ inf 0<c<c0 1 3 ≤rc ≤ 2 3 π 4 r c 1 + n 2 η 2 (rc) 8 c ≥ inf 0<c<c0 1 3 ≤rc ≤ 2 3 π 4 1 3 1 + n 2 1 3 + 1 3π sin(π(3rc-1)) 2 8 c 0 ≥ π 4 1 3 1 + n 2 ( 1 π -1 3π ) 2 8 c 0 ≥ π 4c 0 1 + n 2 ( 1 π -1 3π ) 2 8 -→ n→∞ ∞.
For the third case, one can see that

inf 0<c<c0 2 3 <rc <1 (Γ (n) rc ) c ≥ inf 0<c<c0 2 3 <rc <1 π 4 r c 1 + n 2 η 2 (rc) 8 c ≥ inf 0<c<c0 2 3 <rc <1 π 4 r c 1 + n 2 (1-rc) 2 8 c ≥ π 4 2 3 1 + n 2 K 8 c 0 -→ n→∞ ∞,
where K is a suitable positive constant which does not depend on n.

Construction of the vector-field

In this section, we use the previously constructed curves to construct their underlying vector-elds. In order to do so we rst introduce some suitable height function α n whose level sets are exactly the closed curves generated by the ow.

Taking the orthogonal gradient ∇ ⊥ α n of those functions will generate divergence free vector-elds V 0 n , whose ow lines correspond to the level sets of the function α n . Finally, we multiply the vector-eld V 0 n with a suitable ow-invariant function S which vanishes on the boundary, in order to guarantee that the resulting vectoreld V n = SV 0 n vanishes on the boundary.

We denote by (r, θ) ∈ R + × [0, 2π[ the polar coordinates on R 2 \{0}. With those new variables we have the following expression for the gradient operator of any C 1 -function F : R 2 → R:

∇F = ∂F ∂r , 1 r ∂F ∂θ .
We dene the orthogonal gradient of F denoted ∇ ⊥ F as follows:

∇ ⊥ F = - 1 r ∂F ∂θ , ∂F ∂r .
Remark 4. The orthogonal gradient ∇ ⊥ F of a function F satises the following two properties:

• (∇F ) t (∇ ⊥ F ) = 0; (v t denotes the transpose of a column vector v ∈ R 2 ); • div(∇ ⊥ F) = 0.
Remark 5. We can use the implicit function theorem to dene a C 1 -function

α n :]0, 1[×[0, 2π[ → R + which satises for all (r, θ) ∈ ]0, 1[×[0, 2π[ that h n (α n (r, θ), θ) = (r, θ).

Note that the function

α n is C ∞ on the open set O n := B 1 (0)\({0} ∪ Γ (n) 1 3 ∪ Γ (n) 2 3
).

From this family of functions, we get a sequence of vector-elds V 0 n dened as

V 0 n := ∇ ⊥ α n (r, θ).
The vector-eld

V 0 n is continuous on B 1 (0)\{0} and C ∞ on the open set O n ⊂ B 1 (0). Remark 6. By construction V 0
n is a divergence free vector-eld which generates a ow following the curves Γ (n) r . For our purpose, we use the function η and r x , which where dened in the previous section, to introduce a strictly positive C 1 function S : B 1 (0) → R; x → η(r x ). The function S vanishes at the boundary ∂B 1 (0) and can be extended to a continuous function vanishing at the origin. We nally dene V n := SV 0 n . Remark 7. By construction the vector-eld V n is continuous on B 1 (0) and C ∞ on O n with zero divergence. Also V n vanishes on ∂B 1 (0) and in the origin. Its ow

Φ (n) t ; t ∈ R
preserves the measure a and follows the curves Γ (n) r ; r ∈]0, 1[, which do not come arbitrarily close to ∂B 1 (0); i.e.:

dist {Φ (n) t (x); t ∈ R}, ∂B 1 (0) > 0, for all x ∈ B 1 (0).
Thus the sequence of vector-elds V n ; n ∈ N is a good candidate for proving Theorem 1. In the upcoming sections, we will analyze the energy of functions

ϕ ∈ H 1 0 ∩ ker(V n • ∇).

Comparison arguments

In this section we show that the minimal energy over the space of ow invariant functions diverges to innity as the number of oscillations is increased. In order to do so, we introduce some suitable comparison problem, which is based on some change of the underlying area measure a to a rotationally symmetric measure a * , which incorporates the ratio between the ow lines length and the area that they encircle (see Proposition 3.3). This relation between the length of the boundary and the area of ow-invariant sets is then used in Subsection 3.1 to obtain some lower bound for the energy of ow invariant functions in terms of the energy of some rearranged rotationally invariant function with respect to the measure a * (see Proposition 3.4). Finally, in Subsection 3.2, we introduce some Cheeger type constant in order to show that the minimal energy of the symmetrized variation problems diverges to innity as the number of oscillations increases (see Proposition 3.5).

Let Φ (n) t ; t ∈ R be the ow generated by the vector-eld

V n in B 1 (0). By construc- tion the ow is C 1 on the open set of full measure O n := B 1 (0)\({0} ∪ Γ (n) 1 3 ∪ Γ (n) 2 3
) and continuous on B 1 (0). In the following for a bounded domain Ω we denote by H 1 0 (Ω) the closure of the set of smooth functions with compact support in Ω with respect to the rst order Sobolev norm. In case of the domain B 1 (0) we abbreviate the expression to

H 1 0 . Lemma 3.1. For all n ∈ N the space C 1 (O n ) ∩ C(B 1 (0)) ∩ ker(V n • ∇) is dense in the space H 1 0 ∩ker(V n •∇)
with respect to L 2 -norm and also with respect to H 1 -norm.

Proof. In the same way as in Proposition 2.2 in [START_REF] Franke | On how to push the spectral gap of a diusion on S 2 to innity[END_REF] we can nd for a given > 0 and a given

v ∈ H 1 0 ∩ker(V n •∇) a v ∈ H 1 0 satisfying for a-almost all x ∈ B 1 (0) that v • φ t (x) = v (x) for all t ∈ R and v -v H 1 <
. By denition of H 1 0 , there exists some smooth function u with compact support in B 1 (0) such that u -v H 1 < . However, the function u is not in ker(V n • ∇). We thus average out the function u along the trajectories Γ (n) r ; 0 < r < 1 as follows

ū = 1 0 u • Φ (n) t/tn(x) dt,
where t n (x) is the rst return time of the ow

Φ (n) t ; t ∈ R in location x. Note that the map x → t n (x) is dierentiable on O n . Thus the transformation x → Φ t/tn(x) (x) is dierentiable on O n ⊂ B 1 (0). It follows from the construction that ū is in C 1 (O n )∩C(B 1 (0))∩ker(V n •∇)
. Also, by Theorem 3.35 in [START_REF] Adams | Sobolev Spaces[END_REF] there exist a constant K > 0 such that for all t ∈ [0, 1]

u -u • Φ (n) t/tn H 1 ≤ u -v H 1 + v -v • Φ (n) t/tn H 1 + (v -u ) • Φ (n) t/tn H 1 ≤ + 0 + K v -u H 1 ≤ (1 + K) .
Thus it follows that

u -ū H 1 ≤ 1 0 u -u • Φ (n) t/tn H 1 dt ≤ (1 + K) .
This implies

v -ū H 1 ≤ v -v H 1 + v -ū H 1 ≤ + v -u H 1 + u -ū H 1 ≤ (3 + K) .
The proof is complete, since can be chosen arbitrarily small and K is a constant not depending on .

Lemma 3.2. The minimizers of the function Q :

H 1 0 ∩ ker(V n • ∇) → R dened by Q(u) = B1(0) |∇u| 2 dx B1(0) |u| 2 dx
are non-negative functions u which satisfy the following monotonicity property:

r 1 < r 2 ⇒ u(x) ≥ u(y), ∀x ∈ Γ (n) r1 , y ∈ Γ (n) r2 .
Proof. By Lemma 3.1 we can assume that the minimizer u is in

C 1 (O n ) ∩ C(O n ) ∩ ker(V n • ∇).
This implies that u is constant along the ow lines

Γ (n) r ; r ∈ [0, 1].
Suppose u does not satisfy the above property. Let r crit be a critical radius such that there exist at least a radius r 0 < r crit with u(x) > u(y) for all

x ∈ Γ (n) rcrit , y ∈ Γ (n) r0 . One can construct a function u ∈ H 1 0 ∩ ker(V n • ∇) as follows: u(x) := u(x) for x ∈ B 1 (0(\ Γ rcrit ; 2 t crit -u(x) + + u(x) for x ∈ Γ rcrit ;
where t crit is the level of the function u on the set Γ rcrit and where x + denotes the positive part of a real x. Recall that for u ∈ H 1 0 then t crit -u(x)

+ ∈ H 1 0 (see [START_REF] Gilbarg | Elliptic Partial Dierential Equations of Second Order[END_REF] p.145). Note at rst that by simple calculus one has that |∇u(x)| = |∇ u(x)| for almost all x ∈ Ω. It follows that

B1(0) |∇u| 2 dx = B1(0) |∇ u| 2 dx.
The construction of u gives that

B1(0) | u| 2 dx > B1(0) |u| 2 dx.

As a consequence

B1(0) |∇u| 2 dx B1(0) |u| 2 dx > B1(0) |∇ u| 2 dx B1(0) | u| 2 dx .
This proves the claim.

Proposition 3.3. For every n ∈ N, there exists a measure a * n on B 1 (0) which is absolutely continuous with respect to the Lebesgue measure a, invariant with respect to rotations around zero and which satises for all c ∈ [0, π] the property

a * n (B r (0)) = c implies * n (∂B r (0)) := ∂ ∂r a * n B r (0) = (Γ (n) r(c) ).
Proof. The following proof is motivated from the analogous construction of comparison manifolds in [START_REF] Bérard | Spectral Geometry: Direct and Inverse Problems[END_REF] and [START_REF] Franke | Integral inequalities for the fundamental solutions of diusions on manifolds with divergence-free drift[END_REF] and the construction of comparison measures in [START_REF] Damak | Accelerating planar Ornstein-Uhlenbeck diusion with suitable drift[END_REF].

To ease notation, we drop the reference to n in the notation of the comparison measure a * . Suppose that a * is invariant by rotations around the origin and absolutely continuous with respect to a, then it must be of the form a * (dx) = ψ 0 (|x|)dx. In the following we want to determine the function ψ 0 . Let R(c) : [0, π] → [0, 1] be implicitly dened through

c = a * (B R(c) (0)) = 2π R(c) 0 sψ 0 (s)ds.
Dierentiating both sides of this equality with respect to c yields:

(3)

1 = 2πR(c)ψ 0 (R(c)) d dc R(c).
Recall now that, by dierentiating with respect to the r variable, one has that * (∂B r (0

)) = ∂ ∂r a * B r (0) = ∂ ∂r r 0 2πψ 0 (s)sds = 2πrψ 0 (r).
Due to (3), we obtain that

d dc R(c) = 1 2πR(c)ψ 0 (R(c)) = 1 * (∂B R(c) (0)) = 1 (Γ (n) rc )
.

Thus we obtain the following explicit formula for the function R(c):

R(c) = c 0 1 (Γ (n) ru )
du.

As a conclusion, the explicit formula for ψ 0 is given by:

ψ 0 :]0, 1[-→ R; s → (Γ (n) r R -1 (s) ) 2πs .
This ends the proof.

Remark 8. Note that it follows from the explicit expression for ψ 0 given at the end of the proof of Proposition 3.3, that ψ 0 (r) → 1 as r → 1.

3.1.

The rearrangement method and Faber-Krahn type inequalities. Let A be a measurable set in B 1 (0). Its symmetric rearrangement A * is the open ball centered around zero in B 1 (0) satisfying

a * n (A * ) = a(A).
For a non negative measurable function f , the following representation holds almost everywhere with respect to a

f (x) = ∞ 0 χ y|f (y)>t (x)dt.
Here we used the notation χ A for the indicator function over the set A. We dene its rearrangement function f * by:

f * (x) = ∞ 0 χ y|f (y)>t * (x)dt.
Then f * is lower semi-continuous (since its level sets are open), and is uniquely determined by the distribution function a {x; f (x) > s} of the function f .

Remark 9. Note that as the measure a * n depends on n the rearrangement function f * also depends on n. It then follows that f L 2 (B1(0),a) = f * L 2 (B1(0),a * n ) . By using standard arguments and Proposition 3.3 we can prove the following Faber-Krahn type inequality in this context.

Proposition 3.4. For all f ∈ C 1 (O n ) ∩ C(O n ) ∩ ker(V n • ∇) we have that B1(0) |∇f | 2 da ≥ B1(0) |∇f * | 2 da * n .
Remark 11. It follows from the construction and from the Faber-Krahn inequality that for f ∈ H 1 0 we have

f * ∈ H 1 0 (a * ) = g ∈ L 2 (B 1 (0), a * n ); B1(0) |∇g| 2 da * n < ∞ .
In the next paragraph, we will use a modied Cheeger type argument to nd a lower bound for the Rayleigh quotient of functions which are in ker(V n • ∇).

Cheeger type arguments.

For a xed c 0 ∈]0, π[, dene a Cheeger type constant by:

C (n) c0 := inf 0<c<c0 (Γ (n) rc ) c .
In the following we denote by R 2 + the set of positive H 1 0 functions on B 1 (0) ⊂ R 2 which are invariant with respect to rotations around the origin of R 2 . Proposition 3.5. Let

u n ∈ H 1 0 (a * n ) ∩ R 2
+ be a minimizer of the quotient

Q * n (u) = B1(0) |∇u| 2 da * n B1(0) u 2 da * n .
We have the following asymptotic behavior

Q * n (u n ) -→ n→∞ ∞.
Proof. Let u be a minimizer of Q * and consider

Ω t = {x ∈ B 1 (0) : u 2 (x) > t}.
The fact that ∇u 2 = 2u∇u and Cauchy Schwarz inequality gives that (4)

1 4 B1(0) |∇u 2 |da * n B1(0) u 2 da * n 2 ≤ B1(0) |∇u| 2 da * n B1(0) u 2 da * n .
Let us, for some t 0 > 0, put c 0 := a * (Ω t0 ). By the coarea formula (see p.88 in [START_REF] Chavel | Eigen-Values in Riemannian Geometry[END_REF]) this yields,

B1(0) |∇u 2 |da * n = ∞ 0 * n (∂Ω t )dt (5) = t0 0 * n (∂Ω t )dt + ∞ t0 * n (∂Ω t )dt ≥ C c0 ∞ t0 a * n (Ω t )dt.
It is evident that we can choose a suitable t 0 > 0 such that

t0 0 a * n (Ω t )dt = ∞ t0 a * n (Ω t )dt.
This yields ( 6)

B1(0) |∇u 2 |da * n ≥ 1 2 C c0 ∞ 0 a * n (Ω t )dt = 1 2 C c0 B1(0) u 2 da * .
Note that t 0 and c 0 depend on n since a * n depends on n. We have now to understand the behavior of the sequence t (n) 0 and c (n) 0 as n goes to innity. This is done in the following succession of lemmata. We can suppose without loss of generality that

u L 2 (a * ) = 1.
Lemma 3.6. There exist > 0 such that for all n ∈ N one has t (n) 0 > . Proof. Suppose that for a sub-sequence

t (n) 0 → 0 as n → ∞. It then would follow that ∞ t (n) 0 a * n (Ω (n) t )dt = t (n) 0 0 a * n (Ω (n) t )dt = B1(0) 1 {u 2 n ≤t (n) 0 } u 2 n da * n -→ n→∞ 0, which is not possible since ∞ 0 a * n (Ω t )dt = B1(0) u 2 n da * n = 1.
Lemma 3.7. For all r 0 ∈]0, 1[, the minimal value V 0 over all functions γ ∈ C([r 0 , 1]) with γ(1) = 0 and γ(r 0 ) = t 0 of the functional

L(γ) → 1 r0 r|γ (r)| 2 dr
is given by

V 0 = t 2 0 8(ln r 0 ) 2 2 -r 0 (ln r 0 ) 2 + 2r 0 ln r 0 -2r 0 .
Proof. We use calculus of variations. Let ∈ C([r 0 , 1]) with (1) = 0 and (r 0 ) = 0.

If γ realizes the minimum of the functional L, then it must satisfy:

0 = d dt t=0 L(γ + t ) = 2 1 r0 rγ (r) (r)dr = 2 (1)γ (1) + (r 0 )r 0 γ (r 0 ) - 1 r0 (γ (r) + rγ ) (r)dr = -2 1 r0 (γ (r) + rγ (r)) (r)dr.
From this, we obtain, the following dierential equation γ (r) + rγ = 0 with the boundary data γ(r 0 ) = t 0 and γ(1) = 0. It is easy to see that this dierential equation has the following unique solution satisfying the boundary conditions:

γ(r) = t 0 ln r 0 ln r.
If we plug in this solution into the integral, we obtain

L(γ) = 1 r0 r|γ (r)| 2 dr = t 2 0 (ln r 0 ) 2 1 r0 r| ln r| 2 dr = t 2 0 (ln r 0 ) 2 0 ln r0 s 2 e 2s ds = t 2 0 8(ln r 0 ) 2 0 ln r0 s 2 e s ds = t 2 0 8(ln r 0 ) 2 0 -r 0 (ln r 0 ) 2 -2 0 ln r0 se s ds = t 2 0 8(ln r 0 ) 2 0 -r 0 (ln r 0 ) 2 -0 + 2r 0 ln r 0 + 2 0 ln r0 e s ds = t 2 0 8(ln r 0 ) 2 0 -r 0 (ln r 0 ) 2 -0 + 2r 0 ln r 0 + 2 -2r 0 .
This nishes the proof.

Lemma 3.8. If c (n) 0 -→ n→∞ π then it follows that B1(0) |∇u n | 2 da * n -→ n→∞ ∞.
Proof. Let r n be the radius such that a *

n (B rn (0)) = c (n) 0 . Clearly the condition c (n) 0 → π implies r n ↑ 1.
We have that

B1(0) |∇u n | 2 da * n ≥ B1(0)\Br n (0) |∇u n | 2 da * n .
Since u n is rotationally invariant, there exist a function ūn such that ūn (|x|) = u n (x) for all x ∈ B 1 (0). Note that by construction one has that ūn (1) = 0 and ūn (r n ) = t

(n) 0 . It then follows from the remark following Proposition 3.3 that

B1(0)\Br n (0) |∇u n | 2 da * n ≥ 2π 1 rn r|ū n (r)| 2 ψ 0 (r)dr ≥ 2π 1 rn r|ū n (r)| 2 dr inf rn≤s≤1 ψ 0 (s) ∼ 2π 1 rn r|ū n (r)| 2 dr.
Further, from Lemma 3.7 and Lemma 3.6 we have

2π 1 rn r|ū n (r)| 2 dr ≥ 2π(t (n) 0 ) 2 8(ln r n ) 2 2 -r n (ln r n ) 2 + 2r n ln r n -2r n ≥ π 2 4(ln r n ) 2 2r n ln r n -r n (ln r n ) 2 = π 2 r n 2 ln r n - π 2 r n 4 -→ ∞ as r n → 1.
It thus follows that

B1(0)\Br n (0) |∇u n | 2 da * n -→ ∞ as r n → 1.
This nishes the proof of the lemma.

To conclude the proof of Proposition 3.5, we have to analyze two possible situations:

• Case 1: ∃ > 0 such that c (n) 0 < π -for all n ∈ N, then it follows from Proposition 1.2 that C (n)
c0 goes to innity as n goes to innity. Then it follows from inequalities ( 5) and ( 6) that

Q * n (u n ) -→ n→∞ ∞. • Case 2: If c (n) 0 -→ n→∞ π, from Lemma 3.8, we see that Q * n (u n )
goes to innity as n goes to innity.

Proof of Theorem 1.

The main argument follows the arguments given in [START_REF] Franke | On how to push the spectral gap of a diusion on S 2 to innity[END_REF]. Let ω * n be comparison measure associated to the vector-eld V n . For a given K > 0, Proposition 3.5 tells us that there exists N ∈ N suciently large such that inf

B1(0) |∇f | 2 da * n ; f ∈ H 1 (R 2 , a * n ) with f = 1 ≥ 2K, ∀n ≥ N.
By Proposition 3.4 an Remark 10 one has that:

inf B1(0) |∇f | 2 dx; f ∈ H 1 0 ∩ ker(V n • ∇) with f = 1 ≥ 2K.
We now can use the asymptotic behavior of the principal eigenvalue given in [START_REF] Adams | Sobolev Spaces[END_REF] to see that

lim a↑∞ λ aV ≥ 2K.
This implies λ aVn > K for some suitable choice of a > 0.

Generalizations to bounded domains in dimension two and more

In this last section we want to discuss how the material in the previous section can be upgraded do more general domains.

5.1. Simply connected bounded domains in the plane. First we arm that the statement of Theorem 1 also holds for a general bounded simply connected domain Ω.

Theorem 2. Let Ω be a bounded and simply connected domain in R 2 with piece-wise C 2 boundary. There exists a sequence of continuous and piece-wise C 1 vector-elds

V n ; n ∈ N on Ω with the properties div(V n ) = 0 and V n | ∂Ω = 0 generating a ow Φ t ; t ∈ R satisfying dist {Φ (n)
t (x); t ∈ R}, ∂Ω > 0, for all x ∈ Ω such that the sequence of associated principal eigenvalues satises

lim n→∞ λ Vn = ∞.
Proof. The proof is based on the conformal mapping theorem, which states that there exists a bi-holomorphic function ψ : Ω → B 1 (0). We can use this function to transform the function α n , which was dened on a parametrization of B 1 (0) in Section 2 to a function β n := α n • ψ dened on Ω vanishing on ∂Ω. One can then use the orthogonal gradient to generate a divergence free vector-eld V 0 n := ∇ ⊥ β n which generates a ow following the level sets of β n which correspond to regular sets of the form

Λ (n) r := ψ -1 (Γ (n) r ).
As in section 2 one can obtain a vector-eld V n vanishing on ∂Ω by multiplying V 0 n with a suitable function S ∈ ker(V 0 n • ∇). The arguments of Section 3 can all be easily generalized to this new situation once one can show some analogue of Proposition 1.2 for the family of curves Λ (n) r ; 0 < r < 1.

In order to check Proposition 1.2 for this situation, we rst note that Lemma 1.1 still holds if we replace B r (0) in the statement by the lled ellipse E r (a, b) with semi-minor and semi-major axes ra resp. rb, where a and b correspond to the two positive eigenvalues of the matrix Dψ -1 (0). This can be seen through linearization of the map ψ -1 , as for decreasing values of r the family of closed curves Λ (n) r and the sets Λ(n) r that they enclose, concentrate in shrinking neighborhoods of the point ψ -1 (0). This implies that for any sequence of positive numbers n ; n ∈ N with n → 0 one has

inf 0<c≤ n (Λ (n) rc ) a( Λ(n) rc ) ≥ (Λ (n) r n ) a( Λ(n) r n ) ∼ (∂E r n (a, b)) a(E r n (a, b)) -→ n→∞ ∞.
As in Section 2 for a given n ∈ N let r c > 0 be the unique solution of a( Λ(n

) rc ) = c.
To prove a version of Proposition 1.2 for the family of curves Λ (n) r we rst observe that by construction the curves Λ (n) rc oscillate n times between the two sets O c := ψ -1 (∂B rc-η(rc) (0)) and U c := ψ -1 (∂B rc+η(rc) (0)). Moreover, the function c → η(r c ) is bounded below by a positive constant d on any compact interval of the form [ , c 0 ]. Thus it follows that there exists a constant d > 0 such that

inf ≤c≤c0 dist(O c , U c ) > d .
Since the curve Λ (n) rc oscillates n times between O c and U c it follows that

inf ≤c≤c0 (Λ (n) rc ) ≥ nd .
This shows that for all choice of > 0 we have

inf ≤c≤c0 (Λ (n) rc ) a( Λ(n) rc ) ≥ nd c 0 -→ n→∞ ∞.
It is then possible to nd a sequence of positive numbers n ; n ∈ N such that n → 0 and

inf n ≤c≤c0 (Λ (n) rc ) a( Λ(n) rc ) -→ n→∞ ∞.
Alltogether, this shows that a version of Proposition 1.2 holds for the family of curves Λ (n) r ; 0 < r < 1 and thus nishes the proof of the theorem for bounded simply connected domains in R 2 .

Remark 12. For further use we note that the bi-holomorphic function ψ can be constructed to depend continuously on the jordan curve encircling the simply connected domain Ω (see [START_REF] Pommerenke | Boundary behavior of conformal maps[END_REF] Theorem 2.11). This implies by construction that the vector-elds V n in the previous proof also depends continuously on the domain Ω.

5.2.

Bounded domains in the plane. We also emphasize that Theorem 2 can be generalized to general bounded domains when applying some topological surgery method.

Theorem 3. Let Ω be a bounded domain in R 2 with C 2 boundary and suppose that there exists a nite or countable number of pairwise disjoint simply connected domains Ω i ; i ∈ I with piece-wise C 2 boundaries ∂Ω i satisfying (∂Ω i ∩ ∂Ω) > 0 such that i∈I Ω i ⊂ Ω ⊂ i∈I Ω i . There exists a sequence of continuous and piecewise

C 1 vector-elds V n ; n ∈ N on Ω with the properties div(V n ) = 0 and V n | ∂Ω = 0 generating a ow Φ t ; t ∈ R satisfying dist {Φ (n) t (x); t ∈ R}, ∂Ω > 0, for all x ∈ Ω
such that the sequence of associated principal eigenvalues satises

lim n→∞ λ Vn = ∞.
Proof. By using the methods from the proofs of Theorem 1 and Theorem 2, we nd for a given K > 0 and each of the domains Ω i a C 1 vector-eld V i dened on Ω i vanishing on its boundary ∂Ω i and generating a ow composed of closed curves staying away from ∂Ω i and further satisfying

inf Ωi |∇f | 2 dx; f ∈ H 1 0 (Ω i ) ∩ ker(V i • ∇) with f = 1 ≥ 2K.
We can patch together the vector-elds V i to obtain a continuous vector-eld V on Ω. Let f be a function from H 1 0 (Ω) ∩ ker(V • ∇) satisfying f = 1. We want to prove that the energy of the function f over the domain Ω also exceeds 2K. Since the space C 1 (Ω) ∩ ker(V • ∇) is dense in H 1 0 (Ω) ∩ ker(V • ∇) with respect to the H 1 -norm, we can assume without loss of generality that the function f is continuous and constant along the trajectories generated by V i . Since the ow-lines by construction ll the domain Ω i in a concentric way, the trajectories which come close to the boundary of Ω i also come close to some portion of the boundary of Ω. By assumption this portion has positive length and it can be seen that the restriction f i of f to any of the sub-domains Ω i must satisfy a Dirichlet boundary condition on ∂Ω i . This implies f i ∈ H 1 0 (Ω i ) ∩ ker(V i • ∇) for all i ∈ I. Note that since the supports of the functions are disjoint, the products f i f j vanish for i = j and it follows that

f 2 = Ω f 2 dx = Ω i∈I f i 2 dx = i∈I Ω f 2 i dx = i∈I f i 2 .
From this we have

Ω |∇f | 2 dx = i∈I Ωi |∇f i | 2 dx ≥ K i∈I f i 2 = 2K f 2 .
Since, according to the result from [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF], we can push the principal eigenvalue of the Dirichlet problem driven by the vector-eld cV above the value K and since K can be chosen arbitrarily large, the existence of suitable vector-elds V n on Ω is guaranteed. Obviously the resulting ows are composed of ow lines which are closed and stay away from the boundary ∂Ω.

Bounded domains in any dimensions.

In this subsection, we further generalize Theorem 3 to arbitrary dimensions under some slicing condition on the domain.

Theorem 4. Let Ω be a bounded domain in R d with piece-wise C 2 boundary and suppose that the sets Ω x3,...,x d := {(x 1 , x 2 ) ∈ R 2 : (x 1 , x 2 , x 3 , ..., x d ) ∈ Ω} are simply connected for all (x 3 , ..., x d ) ∈ R d-2 (the empty set is considered as simply connected). There exists a sequence of continuous vector-elds V n ; n ∈ N on Ω with V n | ∂Ω = 0 and generating an area-preserving ow Φ t ; t ∈ R satisfying dist {Φ

(n) t (x); t ∈ R}, ∂Ω > 0, for all x ∈ Ω such that the sequence of associated principal eigenvalues satises

lim n→∞ λ Vn = ∞.
Proof. In order to keep notation simple, we consider the situation in three dimensions. The general case follows just the same line of arguments. The construction of the vector-eld is done by slicing the domain Ω along the third axis into two dimensional domains Ω x3 := {(x 1 , x 2 ) ∈ R 2 : (x 1 , x 2 , x 3 ) ∈ Ω}. By Theorem 3 there exists for any K > 0 and each x 3 ∈ R where Ω x3 is not empty a two dimensional vector-eld V x3 (x 1 , x 2 ) = (V (1) x3 (x 1 , x 2 ), V

x3 (x 1 , x 2 )) on Ω x3 such that λ Vx 3 > 2K. From this we can construct a three dimensional vector-eld

V (x 1 , x 2 , x 3 ) := (V (1) x3 (x 1 , x 2 ), V (2)
x3 (x 1 , x 2 ), 0) on Ω. It follows from the continuity of the conformal mapping theorem with respect to the domain that was discussed in Remark 12 that the resulting vector-eld V x3 (x 1 , x 2 ) is continuous as x 3 varies. By construction the vector-eld V has a vanishing third component and is piece-wise C 1 with respect to the rst two components. It follows that div(V ) is well dened and equals zero all over Ω. Obviously, the trajectories generated by V do not leave the slice Ω x3 where they started and the ow lines, which correspond to the ow lines of the V z -vector-elds, stay a positive distance away from the boundary ∂Ω. Suppose that f is a function from H 1 0 (Ω) ∩ ker(V • ∇) with f = 1. It then follows that for almost all x 3 ∈ R one has that (x 1 , x 2 ) → f (x 1 , x 2 , x 3 ) is in H 1 0 (Ω x3 ) ∩ ker(V • ∇). This then implies

Ω |∇f | 2 dx = R Ωx 3 (∂ x1 f ) 2 + (∂ x2 f ) 2 + (∂ x3 f ) 2 dx 1 dx 2 dx 3 ≥ R Ωx 3 (∂ x1 f ) 2 + (∂ x2 f ) 2 dx 1 dx 2 dx 3 ≥ 2K R Ωx 3 f 2 dx 1 dx 2 dx 3 = 2K Ω f 2 dx = 2K.
As in the proofs of Theorem 1, one then concludes by multiplying V with a large constant and using the result from [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF].

Remark 13. The condition requiring simply connected slices Ω x3 in Theorem 4 can certainly be relaxed to some more general condition which ensures that the decomposition of Ω x3 into simply connected sub-domains, that we used in Theorem 3, is suciently regular as the parameter x 3 varies. Remark 14. Also note that in dimension three and more it might be possible to construct vector-elds generating some weakly mixing ow on Ω. It then would be possible to use methods from [START_REF] Feng | Dissipation enhancement by mixing[END_REF] to construct vector-elds pushing the principal eigenvalue to innity through multiplication with increasing constants. However the ow lines generated by those vector-elds would come arbitrarily close to the boundary ∂Ω of the domain.

  Further, since for all θ ∈ [0, 2π[ and r > 0 the derivative ∂ ∂r h n (r, θ) = (0, 0) does not vanish, it follows that the graphs Γ (n) r are disjoint as r varies. Denote by Γ(n) r the open domain enclosed by the curve Γ(n)

Remark 10 .

 10 The function f * constructed above veries, for all h ∈ R a {x; f (x) > h} = a * n {x; f * (x) > h} .