286 research outputs found
The Schrödinger operator: Perturbation determinants, the spectral shift function, trace identities, and all that
No full textTranslated 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
Exponential decay of eigenfunctions of first order systems
No full textPapers from the International Conference on Transport and Spectral Problems in Quantum Mechanics held in honor of Jean-Michel Combes at the Université de Cergy-Pontoise, Cergy-Pontoise, September 4--6, 2006The author studies exponential decay of the eigenfunctions of first-order (matrix) differential operators of the form It is shown that under certain assumptions, the eigenfunctions obey estimates of the type The author emphasizes that these estimates are valid everywhere off the essential spectrum , not just below the minimum of
On semibounded Wiener-Hopf operators
No full textInternational audienceWe show that a semibounded Wiener-Hopf quadratic form is closable in the space if and only if its integral kernel is the Fourier transform of an absolutely continuous measure. This allows us to define semibounded Wiener-Hopf operators and their symbols under minimal assumptions on their integral kernels. Our proof relies on a continuous analogue of the Riesz Brothers theorem obtained in the paper
Quasi-diagonalization of Hankel operators
No full text42 pagesInternational audienceWe show that all Hankel operators realized as integral operators with kernels in can be quasi-diagonalized as . Here is the Laplace transform, is the operator of multiplication by a function (distribution) , . We find a scale of spaces of test functions where acts as an isomorphism. Then is an isomorphism of the corresponding spaces of distributions. We show that which yields a one-to-one correspondence between kernels and sigma-functions of Hankel operators. The sigma-function of a self-adjoint Hankel operator contains substantial information about its spectral properties. Thus we show that the operators and 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 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 and of its dual variable)
Hankel and Toeplitz operators: continuous and discrete representations
No full textin pressInternational audienceWe find a relation guaranteeing that Hankel operators realized in the space of sequences and in the space of functions are unitarily equivalent. This allows us to obtain exhaustive spectral results for two classes of unbounded Hankel operators in the space generalizing in different directions the classical Hilbert matrix. We also discuss a link between representations of Toeplitz operators in the spaces and
Lectures on scattering theory
No full textThe first two lectures are devoted to describing the basic concepts of scattering theory in a very compressed way. A detailed presentation of the abstract part can be found in \\cite{I} and numerous applications in \\cite{RS} and \\cite{Y2}. The last two lectures are based on the recent research of the author
Correction to: Unbounded Hankel Operators and Moment Problems
No full textCorrection to: Unbounded Hankel Operators and Moment Problems (Integral Equations and Operator Theory, (2016), 85, 2, (289-300), 10.1007/s00020-016-2289-y)International audienc
Unbounded Hankel operators and moment problems
No full textA 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
Spectral and scattering theory for differential and Hankel operators
No full textInternational audienceWe consider a class of Hankel operators realized in the space as integral operators with kernels where and is an arbitrary real polynomial of degree . This class contains the classical Carleman operator when . We show that a Hankel operator in this class can be reduced by an {\it explicit} unitary transformation (essentially by the Mellin transform) to a differential operator in the space . Here is a polynomial determined by and is the universal function. Then the operator reduces by the generalized Liouville transform to the standard differential operator with the coefficients , , decaying sufficiently rapidly as . This allows us to use the results of spectral theory of differential operators for the study of spectral properties of generalized Carleman operators. In particular, we show that the absolutely continuous spectrum of is simple and coincides with if is odd, and it has multiplicity and coincides with if is even. The singular continuous spectrum of is empty, and its eigenvalues may accumulate to the point only. As a by-product of our considerations, we develop spectral theory of a new class of {\it degenerate} differential operators where is an arbitrary real polynomial and is a sufficiently arbitrary real function decaying at infinity
Radiation conditions and scattering theory for -particle hamiltonians (main ideas of the approach)
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