1,720,998 research outputs found
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
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
A new representation of Hankel operators and its spectral consequences
No full textDedicated to the memory of Mikhail Zakharovich SolomyakInternational audienceWe describe a new representation of Hankel operators as pseudo-differential operators in the space of functions defined on the whole axis. The amplitudes of such operators have a very special structure: they are products of functions of a one variable only. This representation has numerous spectral consequences both for compact Hankel operators and for operators with the continuous spectrum
A trace formula for the Dirac operator
No full text10 pagesInternational audienceOur goal is to extend the theory of the spectral shift function to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, we consider a couple of Dirac operators
A particle in the Bio-Savart-Laplace magnetic field: explicit solutions
No full text6 pagesWe consider the Schr\\ödinger operator in the space with a magnetic potential created by an infinite straight current. We perform a spectral analysis of the operator almost explicitly. In particular, we show that the operator is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions of the time dependent Schr\\ödinger equation. Equations of classical mechanics are also integrated. Our main observation is that both quantum and classical particles have always a preferable (depending on its charge) direction of propagation along the current and both of them are confined in the plane orthogonal to the current
On semibounded Toeplitz operators
No full textThis is a slightly revised version of the article, arXiv:1603.06229v1, with the same tittle. Some misprints has been removed and some arguments has been made more clear. The results are unchaged. To appear in J. Operator theoryInternational audienceWe show that a semibounded Toeplitz quadratic form is closable in the space if and only if its matrix elemens are Fourier coefficients of an absolutely continuous measure. We also describe the domain of the corresponding closed form. This allows us to define semibounded Toeplitz operators under minimal assumptions on their matrix elements
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