Scattering by magnetic fields

Consider the scattering amplitude $s(\omega,\omega^\prime;\lambda)$, $\omega,\omega^\prime\in{\Bbb S}^{d-1}$, $\lambda > 0$, corresponding to an arbitrary short-range magnetic field $B(x)$, $x\in{\Bbb R}^d$. This is a smooth function of $\omega$ and $\omega^\prime$ away from the diagonal $\omega=\omega^\prime$ but it may be singular on the diagonal. If $d=2$, then the singular part of the scattering amplitude (for example, in the transversal gauge) is a linear combination of the Dirac function and of a singular denominator. Such structure is typical for long-range scattering. We refer to this phenomenon as to the long-range Aharonov-Bohm effect. On the contrary, for $d=3$ scattering is essentially of short-range nature although, for example, the magnetic potential $A^{(tr)}(x)$ such that ${\rm curl} A^{(tr)}(x)=B(x)$ and $=0$ decays at infinity as $|x|^{-1}$ only. To be more precise, we show that, up to the diagonal Dirac function (times an explicit function of $\omega$), the scattering amplitude has only a weak singularity in the forward direction $\omega = \omega^\prime$. Our approach relies on a construction in the dimension $d=3$ of a short-range magnetic potential $A (x)$ corresponding to a given short-range magnetic field $B(x)$

The semiclassical limit of eigenfunctions of the Schr\"odinger equation and the Bohr-Sommerfeld quantization condition, revisited

Consider the semiclassical limit, as the Planck constant \hbar\ri 0, of bound states of a quantum particle in a one-dimensional potential well. We justify the semiclassical asymptotics of eigenfunctions and recover the Bohr-Sommerfeld quantization condition

Spectral and scattering theory for perturbations of the Carleman operator

We study spectral properties of the Carleman operator (the Hankel operator with kernel $h_{0}(t)=t^{-1}$) and, in particular, find an explicit formula for its resolvent. Then we consider perturbations of the Carleman operator $H_{0}$ by Hankel operators $V$ with kernels $v(t)$ decaying sufficiently rapidly as $t\to\infty$ and not too singular at t=0. Our goal is to develop scattering theory for the pair $H_{0}$, $H=H_{0} +V$ and to construct an expansion in eigenfunctions of the continuous spectrum of the Hankel operator $H$. We also prove that under general assumptions the singular continuous spectrum of the operator $H$ is empty and that its eigenvalues may accumulate only to the edge points 0 and $\pi$ in the spectrum of $H_{0}$. We find simple conditions for the finiteness of the total number of eigenvalues of the operator $H$ lying above the (continuous) spectrum of the Carleman operator $H_{0}$ and obtain an explicit estimate of this number. The theory constructed is somewhat analogous to the theory of one-dimensional differential operators.Comment: dedicated to the memory of Vladimir Savel'evich Buslaev there are some minor and editorial changes compared to 1210.5709 of 21 oc

On semibounded Toeplitz operators

We show that a semibounded Toeplitz quadratic form is closable in the space $\ell^2({\Bbb Z}_{+})$ 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.Comment: This 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 theor

Quasi-Carleman operators and their spectral properties

The Carleman operator is defined as integral operator with kernel $(t+s)^{-1}$ in the space $L^2 ({\Bbb R}_{+})$. This is the simplest example of a Hankel operator which can be explicitly diagonalized. Here we study a class of self-adjoint Hankel operators (we call them quasi-Carleman operators) generalizing the Carleman operator in various directions. We find explicit formulas for the total number of negative eigenvalues of quasi-Carleman operators and, in particular, necessary and sufficient conditions for their positivity. Our approach relies on the concepts of the sigma-function and of the quasi-diagonalization of Hankel operators introduced in the preceding paper of the author