Inverse Scattering at a Fixed Energy for Long-Range Potentials

In this paper we consider the inverse scattering problem at a fixed energy for the Schr\"odinger equation with a long-range potential in \ere^d, d\geq 3. We prove that the long-range part can be uniquely reconstructed from the leading forward singularity of the scattering amplitude at some positive energy

On Inverse Scattering at a Fixed Energy for Potentials with a Regular Behaviour at Infinity

We study the inverse scattering problem for electric potentials and magnetic fields in \ere^d, d\geq 3, that are asymptotic sums of homogeneous terms at infinity. The main result is that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at some positive energy.Comment: This is a slightly edited version of the previous pape

A point interaction for the discrete Schr\"odinger operator and generalized Chebyshev polynomials

article number: 063511International audienceWe consider semi-infinite Jacobi matrices corresponding to a point interaction for the discrete Schr\"odinger operator. Our goal is to find explicit expressions for the spectral measure, the resolvent and other spectral characteristics of such Jacobi matrices. It turns out that their spectral analysis leads to a new class of orthogonal polynomials generalizing the classical Chebyshev polynomials

Quasi-diagonalization of Hankel operators

42 pagesInternational audienceWe show that all Hankel operators $H$ realized as integral operators with kernels $h(t+s)$ in $L^2 ({\Bbb R}_{+})$ can be quasi-diagonalized as $H= {\sf L}^* \Sigma {\sf L}$. Here ${\sf L}$ is the Laplace transform, $\Sigma$ is the operator of multiplication by a function (distribution) $\sigma(\lambda)$, $\lambda\in {\Bbb R}$. We find a scale of spaces of test functions where ${\sf L}$ acts as an isomorphism. Then ${\sf L}^*$ is an isomorphism of the corresponding spaces of distributions. We show that $h= {\sf L}^* \sigma$ which yields a one-to-one correspondence between kernels $h(t)$ and sigma-functions $\sigma(\lambda)$ of Hankel operators. The sigma-function of a self-adjoint Hankel operator $H$ contains substantial information about its spectral properties. Thus we show that the operators $H$ and $\Sigma$ have the same numbers of positive and negatives eigenvalues. In particular, we find necessary and sufficient conditions for sign-definiteness of Hankel operators. These results are illustrated at examples of quasi-Carleman operators generalizing the classical Carleman operator with kernel $h(t)=t^{-1}$ in various directions. The concept of the sigma-function directly leads to a criterion (equivalent of course to the classical Nehari theorem) for boundedness of Hankel operators. Our construction also shows that every Hankel operator is unitarily equivalent by the Mellin transform to a pseudo-differential operator with amplitude which is a product of functions of one variable only (of $x\in{\Bbb R}$ and of its dual variable)

The Schrödinger operator: Perturbation determinants, the spectral shift function, trace identities, and all that

Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 41, No. 3, pp. 60-83, 2007 Original Russian Text Copyright © by D. R. Yafaev Dedicated to the 100th anniversary of the birth of Mark Grigor'evich KreinInternational audienceWe discuss applications of the M. G. Krein theory of the spectral shift function to the multidimensional Schrödinger operator. Specific properties of this function, for example, its high-energy asymptotics are studied. Trace identities are derive

Passage through a potential barrier and multiple wells

International audienceConsider the semiclassical limit, as the Planck constant \hbar\ri 0, of bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. We show that, for each eigenvalue of the Schr\"odinger operator, the Bohr-Sommerfeld quantization condition is satisfied at least for one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. We show that, at least from one side, the barrier fixes the phase of wave functions in the same way as a potential barrier of infinite width. On the other hand, it turns out that for each well there exists an eigenvalue in a small neighborhood of every point satisfying the Bohr-Sommerfeld condition

Unbounded Hankel operators and moment problems

A correction to this article is available online at https://doi.org/10.1007/s00020-019-2543-1, hal-02379070v1International audienceWe find necessary and sufficient conditions for a non-negative Hankel quadratic form to admit the closure. We also describe the domain of the corresponding closed form. This allows us to define unbounded non-negative Hankel operators under optimal assumptions on their matrix elements. The results obtained supplement the classical Widom condition for a Hankel operator to be bounded