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D. R. Yafaev

Publications and source records attributed to D. R. Yafaev.

34 records · Page 2Linked to original sources
arXiv2014

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.

math.FA↗
arXiv2014

Quasi-diagonalization of Hankel operators

We 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}^* Σ{\sf L} $. Here ${\sf L}$ is the Laplace transform, $Σ$ is the operator of multiplication by a function (distribution) $σ(λ)$, $λ\in {\Bbb R}$. 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 $h= {\sf L}^* σ$ which yields a one-to-one correspondence between kernels $h(t)$ and sigma-functions $σ(λ)$ 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 $Σ$ 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).

math.FA↗
arXiv2013

Diagonalizations of two classes of unbounded Hankel operators

We show that every Hankel operator $H$ is unitarily equivalent to a pseudo-differential operator $A$ of a special structure acting in the space $L^2 ({\Bbb R}) $. As an example, we consider integral operators $H$ in the space $L^2 ({\Bbb R}_{+}) $ with kernels $P (\ln (t+s)) (t+s)^{-1}$ where $P(x)$ is an arbitrary real polynomial of degree $K$. In this case, $A$ is a differential operator of the same order $K$. This allows us to study spectral properties of Hankel operators $H$ with such kernels. In particular, we show that the essential spectrum of $H$ coincides with the whole axis for $K$ odd, and it coincides with the positive half-axis for $K$ even. In the latter case we additionally find necessary and sufficient conditions for the positivity of $H$. We also consider Hankel operators whose kernels have a strong singularity at some positive point. We show that spectra of such operators consist of the zero eigenvalue of infinite multiplicity and eigenvalues accumulating to $+\infty$ and $-\infty$. We find the asymptotics of these eigenvalues.

math.FA↗
arXiv2013

On finite rank Hankel operators

For self-adjoint Hankel operators of finite rank, we find an explicit formula for the total multiplicity of their negative and positive spectra. We also show that very strong perturbations, for example, a perturbation by the Carleman operator, do not change the total number of negative eigenvalues of finite rank Hankel operators.

math.FA↗
arXiv2013

Criteria for Hankel operators to be sign-definite

We show that total multiplicities of negative and positive spectra of a self-adjoint Hankel operator $H$ with kernel $h(t)$ and of an operator of multiplication by some real function $s(x)$ coincide. In particular, $\pm H\geq 0$ if and only if $\pm s(x)\geq 0$. The kernel $h(t)$ and its "sign-function" $s(x)$ are related by an explicit formula. An expression of $h(t)$ in terms of $s(x)$ leads to an exponential representation of $h(t)$. Our approach directly applies to various classes of Hankel operators. In particular, for Hankel operators of finite rank, we find an explicit formula for the total multiplicity of their negative and positive spectra.

math.FA↗
arXiv2012

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 $π$ 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.

math.SP↗
arXiv2011

A trace formula for differential operators of arbitrary order

An operator $H=H_{0}+V$ where $H_{0}=i^{-N} \partial^{N}$ ($N$ is arbitrary) and $V$ is a differential operator of order $N-1$ with coefficients decaying sufficiently rapidly at infinity is considered in the space $L^2(\Bbb R)$. The goal of the paper is to find an expression for the trace of the difference of the resolvents $(H-z)^{-1}$ and $(H_{0}-z)^{-1}$ in terms of the Wronskian of appropriate solutions to the differential equation $Hu=zu$. This also leads to a representation for the perturbation determinant of the pair $H_{0}, H$.

math.SP↗
arXiv2010

A commutator method for the diagonalization of Hankel operators

We present a method for the explicit diagonalization of some Hankel operators. This method allows us to recover classical results on the diagonalization of Hankel operators with the absolutely continuous spectrum. It leads also to new results. Our approach relies on the commutation of a Hankel operator with some differential operator of second order.

math.SP↗
arXiv2008

Spectral and scattering theory of fourth order differential operators

An ordinary differential operator of the fourth order with coefficients converging at infinity sufficiently rapidly to constant limits is considered. Scattering theory for this operator is developed in terms of special solutions of the corresponding differential equation. In contrast to equations of second order "scattering" solutions contain exponentially decaying terms. A relation between the scattering matrix and a matrix of coefficients at exponentially decaying modes is found. In the second part of the paper the operator $D^4$ on the half-axis with different boundary conditions at the point zero is studied. Explicit formulas for basic objects of the scattering theory are found. In particular, a classification of different types of zero-energy resonances is given.

math.SP↗
arXiv2007

Exponential decay of eigenfunctions of first order systems

For first order systems, we obtain an efficient bound on the exponential decay of an eigenfunction in terms of the distance between the corresponding eigenvalue and the essential spectrum. As an example, the Dirac operator is considered.

math.SP↗
arXiv2005

Scattering by magnetic fields

Consider the scattering amplitude $s(ω,ω^\prime;λ)$, $ω,ω^\prime\in{\Bbb S}^{d-1}$, $λ> 0$, corresponding to an arbitrary short-range magnetic field $B(x)$, $x\in{\Bbb R}^d$. This is a smooth function of $ω$ and $ω^\prime$ away from the diagonal $ω=ω^\prime$ 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 $ω$), the scattering amplitude has only a weak singularity in the forward direction $ω= ω^\prime$. 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 $B(x)$.

math.SP↗
arXiv2004

A trace formula for the Dirac operator

Our 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.

math.SP↗
arXiv2004

Trace-class approach in scattering problems for perturbations of media

We consider the operators $H_0=M_0^{-1}(x) P(D)$ and $H =M^{-1} (x) P(D)$ where $M_0 (x)$ and $M (x)$ are positively definite bounded matrix-valued functions and $P(D)$ is an elliptic differential operator. Our main result is that the wave operators for the pair $H_0$, $H$ exist and are complete if the difference $ M(x)-M_0(x)=O(|x|^{- rho})$, $ rho>d$, as $|x| to infty$. Our point is that no special assumptions on $M_0(x)$ are required. Similar results are obtained in scattering theory for the wave equation.

math.SP↗