arXiv subjects

Dmitri Yafaev

Publications and source records attributed to Dmitri Yafaev.

15 recordsLinked to original sources

arXiv2022

Scattering Theory in Quantum Mechanical Problems

The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schr\"odinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.

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arXiv2019

Asymptotic behavior of orthogonal polynomials without the Carleman condition

Our goal is to find an asymptotic behavior as $n\to\infty$ of orthogonal polynomials $P_{n}(z)$ defined by the Jacobi recurrence coefficients $a_{n}, b_{n}$. We suppose that the off-diagonal coefficients $a_{n}$ grow so rapidly that the series $\sum a_{n}^{-1}$ converges, that is, the Carleman condition is violated. With respect to diagonal coefficients $b_{n}$ we assume that $-b_{n} (a_{n}a_{n-1})^{-1/2}\to 2β_{\infty}$ for some $β_{\infty}\neq \pm 1$. The asymptotic formulas obtained for $P_{n}(z)$ are quite different from the case $\sum a_{n}^{-1}=\infty$ when the Carleman condition is satisfied. In particular, if $\sum a_{n}^{-1}<\infty$, then the phase factors in these formulas do not depend on the spectral parameter $z\in{\Bbb C}$. The asymptotic formulas obtained in the cases $|β_{\infty}|<1 $ and $|β_{\infty}|>1 $ are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.

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arXiv2019

Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators $T$ with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator $T$, that is, by the behavior of $\exp(-iTt) f$ for $t\to\pm\infty$. It turns out that the spectrum is naturally partitioned into three disjoint subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation properties are modeled by the continuous part of the symbol, whereas on the thin spectrum, the model operator is determined by the jumps of the symbol. On the mixed spectrum, these two types of the asymptotic evolution of $\exp(-iTt) f$ coexist. This classification is justified in the framework of scattering theory. We prove the existence of wave operators that relate the model operators with the Toeplitz operator $T$. The ranges of these wave operators are pairwise orthogonal, and their orthogonal sum exhausts the whole space, i.e., the set of these wave operators is asymptotically complete.

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arXiv2019

On spectral analysis of self-adjoint Toeplitz operators

The paper pursues three objectives. Firstly, we provide an expanded version of spectral analysis of self-adjoint Toeplitz operators, initially built by M. Rosenblum in the 1960's. We offer some improvements to Rosenblum's approach: for instance, our proof of the absolute continuity, relying on a weak version of the limiting absorption principle, is more direct. Secondly, we study in detail Toeplitz operators with finite spectral multiplicity. In particular, we introduce generalized eigenfunctions and investigate their properties. Thirdly, we develop a more detailed spectral analysis for piecewise continuous symbols. This is necessary for construction of scattering theory for Toeplitz operators with such symbols.

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arXiv2016

Spectral asymptotics for compact self-adjoint Hankel operators

We describe large classes of compact self-adjoint Hankel operators whose eigenvalues have power asymptotics and obtain explicit expressions for the coefficient in front of the leading term. The results are stated both in the discrete and continuous representations for Hankel operators. We also elucidate two key principles underpinning the proof of such asymptotic relations. We call them {\it the localization principle} and {\it the symmetry principle}. The localization principle says that disjoint components of the singular support of the symbol of a Hankel operator make independent contributions into the asymptotics of eigenvalues. The symmetry principle says that if the singular support of a symbol does not contain the points $1$ and $-1$ in the discrete case (or the points $0$ and $\infty$ in the continuous case), then the spectrum of the corresponding Hankel operator is asymptotically symmetric with respect to the reflection around zero.

math.SP↗

arXiv2016

Best rational approximation of functions with logarithmic singularities

We consider functions $ω$ on the unit circle $\mathbb T$ with a finite number of logarithmic singularities. We study the approximation of $ω$ by rational functions and find an asymptotic formula for the distance in the BMO-norm between $ω$ and the set of rational functions of degree $n$ as $n\to\infty$. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.

math.CA↗

arXiv2015

Spectral and scattering theory for differential and Hankel operators

We consider a class of Hankel operators $H$ realized in the space $L^2 ({\Bbb R}_{+}) $ as integral operators with kernels $h(t+s)$ where $h(t)=P (\ln t) t ^{-1}$ and $P(X)= X^n+p_{n-1} X^{n-1}+\cdots$ is an arbitrary real polynomial of degree $n$. This class contains the classical Carleman operator when $n =0$. We show that a Hankel operator $H$ in this class can be reduced by an {\it explicit} unitary transformation (essentially by the Mellin transform) to a differential operator $A = v Q(D) v$ in the space $L^2 ({\Bbb R}) $. Here $Q(X)= X^n+ q_{n-1} X^{n-1}+\cdots$ is a polynomial determined by $P(X)$ and $v(ξ)=π^{1/2} (\cosh(πξ))^{-1/2} $ is the universal function. Then the operator $A = v Q(D) v$ reduces by the generalized Liouville transform to the standard differential operator $B = D^n+ b_{n-1} (x)D^{n-1}+\cdots+ b_{0} (x)$ with the coefficients $b_{m}(x)$, $m=0,\ldots, n-1$, decaying sufficiently rapidly as $|x|\to \infty$. 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 $H$ is simple and coincides with $\Bbb R$ if $n$ is odd, and it has multiplicity $2$ and coincides with $[0,\infty)$ if $n\geq 2$ is even. The singular continuous spectrum of $H$ is empty, and its eigenvalues may accumulate to the point $0$ only. As a by-product of our considerations, we develop spectral theory of a new class of {\it degenerate} differential operators $A = v Q(D) v$ where $Q(X)$ is an arbitrary real polynomial and $v(ξ)$ is a sufficiently arbitrary real function decaying at infinity.

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arXiv2015

Localization principle for compact Hankel operators

In the power scale, the asymptotic behavior of the singular values of a compact Hankel operator is determined by the behavior of the symbol in a neighborhood of its singular support. In this paper, we discuss the localization principle which says that the contributions of disjoint parts of the singular support of the symbol to the asymptotic behavior of the singular values are independent of each other. We apply this principle to Hankel integral operators and to infinite Hankel matrices. In both cases, we describe a wide class of Hankel operators with power-like asymptotics of singular values. The leading term of this asymptotics is found explicitly.

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arXiv2014

Asymptotic behaviour of eigenvalues of Hankel operators

We consider compact Hankel operators realized in $ \ell^2(\mathbb Z_+)$ as infinite matrices $Γ$ with matrix elements $h(j+k)$. Roughly speaking, we show that if $h(j)\sim (b_{1}+ (-1)^j b_{-1}) j^{-1}(\log j)^{-α}$ as $j\to \infty$ for some $α>0$, then the eigenvalues of $Γ$ satisfy $λ_{n}^{\pm} (Γ)\sim c^{\pm} n^{-α}$ as $n\to \infty$. The asymptotic coefficients $c^{\pm}$ are explicitly expressed in terms of the asymptotic coefficients $b_{1} $ and $b_{-1}$. Similar results are obtained for Hankel operators $\mathbf Γ$ realized in $ L^2(\mathbb R_+)$ as integral operators with kernels $\mathbf h(t+s)$. In this case the asymptotics of eigenvalues $λ_{n}^{\pm} (\mathbf Γ)$ are determined by the behaviour of $\mathbf h(t)$ as $t\to 0$ and as $t\to \infty$.

math.SP↗

arXiv2014

Sharp estimates for singular values of Hankel operators

We consider compact Hankel operators realized in $\ell^2(\mathbb Z_+)$ as infinite matrices $Γ$ with matrix elements $h(j+k)$. Roughly speaking, we show that, for all $α>0$, the singular values $s_{n}$ of $Γ$ satisfy the bound $s_{n}= O(n^{-α})$ as $n\to \infty$ provided $h(j)= O(j^{-1}(\log j)^{-α})$ as $j\to \infty$. These estimates on $s_{n}$ are sharp in the power scale of $α$. Similar results are obtained for Hankel operators $\mathbfΓ$ realized in $L^2(\mathbb R_+)$ as integral operators with kernels $\mathbf h(t+s)$. In this case the estimates of singular values of $\mathbfΓ$ are determined by the behavior of $\mathbf h(t)$ as $t\to 0$ and as $t\to\infty$.

math.SP↗

arXiv2014

Spectral and scattering theory of self-adjoint Hankel operators with piecewise continuous symbols

We develop the spectral and scattering theory for self-adjoint Hankel operators $H$ with piecewise continuous symbols. In this case every jump of the symbol gives rise to a band of the absolutely continuous spectrum of $H$. We construct wave operators relating simple "model" (that is, explicitly diagonalizable) Hankel operators for each jump and the given Hankel operator $H$. We show that the set of all these wave operators is asymptotically complete. This determines the absolutely continuous part of $H$. We also prove that the singular continuous spectrum of $H$ is empty and that its eigenvalues may accumulate only to "thresholds" in the absolutely continuous spectrum. All these results are reformulated in terms of Hankel operators realized as matrix or integral operators.

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arXiv2012

Spectral theory of piecewise continuous functions of self-adjoint operators

Let $H_0$, $H$ be a pair of self-adjoint operators for which the standard assumptions of the smooth version of scattering theory hold true. We give an explicit description of the absolutely continuous spectrum of the operator $\mathcal{D}_θ=θ(H)-θ(H_0)$ for piecewise continuous functions $θ$. This description involves the scattering matrix for the pair $H_0$, $H$, evaluated at the discontinuities of $θ$. We also prove that the singular continuous spectrum of $\mathcal{D}_θ$ is empty and that the eigenvalues of this operator have finite multiplicities and may accumulate only to the "thresholds" of the absolutely continuous spectrum of $\mathcal{D}_θ$. Our approach relies on the construction of "model" operators for each jump of the function $θ$. These model operators are defined as certain symmetrised Hankel operators which admit explicit spectral analysis. We develop the multichannel scattering theory for the set of model operators and the operator $θ(H)-θ(H_0)$. As a by-product of our approach, we also construct the scattering theory for general symmetrised Hankel operators with piecewise continuous symbols.

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arXiv2012

A multichannel scheme in smooth scattering theory

We develop the scattering theory for a pair of self-adjoint operators $A_{0}=A_{1}\oplus...\oplus A_{N}$ and $A=A_{1}+...+A_{N}$ under the assumption that all pair products $A_{j}A_{k}$ with $j\neq k$ satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products $A_{j}A_{k}$, $j\neq k$, can be represented as integral operators with smooth kernels in the spectral representation of the operator $A_{0}$. We show that the absolutely continuous parts of the operators $A_{0}$ and $A$ are unitarily equivalent. This yields a smooth version of Ismagilov's theorem known earlier in the trace class framework. We also prove that the singular continuous spectrum of the operator $A$ is empty and that its eigenvalues may accumulate only to "thresholds" of the absolutely continuous spectra of the operators $A_{j}$. Our approach relies on a system of resolvent equations which can be considered as a generalization of Faddeev's equations for three particle quantum systems.

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arXiv2009

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,λ)$. The purpose of the paper is to study the spectral properties of the difference $D(λ)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(λ)$ in terms of the eigenvalues of the scattering matrix $S(λ)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(λ)$ is empty.

math.SP↗

arXiv2004

Lectures on scattering theory

The 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.

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