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Alexander Pushnitski

Publications and source records attributed to Alexander Pushnitski.

At least 55 records · Page 3Linked to original sources
arXiv2015

The spectral density of a product of spectral projections

We consider the product of spectral projections $$ Π_ε(λ) = 1_{(-\infty,λ-ε)}(H_0) 1_{(λ+ε,\infty)}(H) 1_{(-\infty,λ-ε)}(H_0) $$ where $H_0$ and $H$ are the free and the perturbed Schrödinger operators with a short range potential, $λ>0$ is fixed and $ε\to0$. We compute the leading term of the asymptotics of $\mathrm{Tr}\ f(Π_ε(λ))$ as $ε\to0$ for continuous functions $f$ vanishing sufficiently fast near zero. Our construction elucidates calculations that appeared earlier in the theory of "Anderson's orthogonality catastrophe" and emphasizes the role of Hankel operators in this phenomenon.

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arXiv2015

The spectral density of a difference of spectral projections

Let $H_0$ and $H$ be a pair of self-adjoint operators satisfying some standard assumptions of scattering theory. It is known from previous work that if $λ$ belongs to the absolutely continuous spectrum of $H_0$ and $H$, then the difference of spectral projections $$D(λ)=1_{(-\infty,0)}(H-λ)-1_{(-\infty,0)}(H_0-λ)$$ in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations $D_\varepsilon(λ)$ of $D(λ)$, given by $$D_\varepsilon(λ)=ψ_\varepsilon(H-λ)-ψ_\varepsilon(H_0-λ),$$ where $ψ_\varepsilon(x)=ψ(x/\varepsilon)$ and $ψ(x)$ is a smooth real-valued function which tends to $\mp1/2$ as $x\to\pm\infty$. We prove that the eigenvalues of $D_\varepsilon(λ)$ concentrate to the absolutely continuous spectrum of $D(λ)$ as $\varepsilon\to+0$. We show that the rate of concentration is proportional to $|\log\varepsilon|$ and give an explicit formula for the asymptotic density of these eigenvalues. It turns out that this density is independent of $ψ$. The proof relies on the analysis of Hankel operators.

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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 $t\to \infty$.

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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 $t\to\infty$.

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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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arXiv2014

An inverse problem for self-adjoint positive Hankel operators

For a sequence $\{α_n\}_{n=0}^\infty$, we consider the Hankel operator $Γ_α$, realised as the infinite matrix in $\ell^2$ with the entries $α_{n+m}$. We consider the subclass of such Hankel operators defined by the "double positivity" condition $Γ_α\geq0$, $Γ_{S^*α}\geq0$; here $S^*α$ is the shifted sequence $\{α_{n+1}\}_{n=0}^\infty$. We prove that in this class, the sequence $α$ is uniquely determined by the spectral shift function $ξ_α$ for the pair $Γ_α^2$, $Γ_{S^*α}^2$. We also describe the class of all functions $ξ_α$ arising in this way and prove that the map $α\mapstoξ_α$ is a homeomorphism in appropriate topologies.

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arXiv2014

Trace class conditions for functions of Schrödinger operators

We consider the difference $f(-Δ+V)-f(-Δ)$ of functions of Schrödinger operators in $L^2(\mathbb R^d)$ and provide conditions under which this difference is trace class. We are particularly interested in non-smooth functions $f$ and in $V$ belonging only to some $L^p$ space. This is motivated by applications in mathematical physics related to Lieb--Thirring inequalities. We show that in the particular case of Schrödinger operators the well-known sufficient conditions on $f$, based on a general operator theoretic result due to V. Peller, can be considerably relaxed. We prove similar theorems for $f(-Δ+V)-f(-Δ)-\frac{d}{dα} f(-Δ+αV)|_{α=0}$. Our key idea is the use of the limiting absorption principle.

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arXiv2013

Spectral perturbation theory and the two weights problem

The famous two weights problem consists in characterising all possible pairs of weights such that the Hardy projection is bounded between the corresponding weighted $L^2$ spaces. Koosis' theorem of 1980 gives a way to construct a certain class of pairs of weights. We show that Koosis' theorem is closely related to (in fact, is a direct consequence of) a spectral perturbation model suggested by de Branges in 1962. Further, we show that de Branges' model provides an operator-valued version of Koosis' theorem.

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arXiv2012

Scattering theory and Banach space valued singular integrals

We give a new sufficient condition for existence and completeness of wave operators in abstract scattering theory. This condition generalises both trace class and smooth approaches to scattering theory. Our construction is based on estimates for the Cauchy transforms of operator valued measures.

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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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arXiv2012

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.

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arXiv2012

The spectrum of the scattering matrix near resonant energies in the semiclassical limit

The object of study in this paper is the on-shell scattering matrix $S(E)$ of the Schrödinger operator with the potential satisfying assumptions typical in the theory of shape resonances. We study the spectrum of $S(E)$ in the semiclassical limit when the energy parameter $E$ varies from $E_\text{res}-\varepsilon$ to $E_\text{res}+\varepsilon$, where $E_\text{res}$ is a real part of a resonance, and $\varepsilon$ is sufficiently small. The main result of our work describes the spectral flow of the scattering matrix through a given point on the unit circle. This result is closely related to the Breit-Wigner effect.

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arXiv2011

The spectral density of the scattering matrix for high energies

We determine the density of eigenvalues of the scattering matrix of the Schrodinger operator with a short range potential in the high energy asymptotic regime. We give an explicit formula for this density in terms of the X-ray transform of the potential.

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arXiv2010

Scattering matrix and functions of self-adjoint operators

In the scattering theory framework, we consider a pair of operators $H_0$, $H$. For a continuous function $ϕ$ vanishing at infinity, we set $ϕ_δ(\cdot)=ϕ(\cdot/δ)$ and study the spectrum of the difference $ϕ_δ(H-λ)-ϕ_δ(H_0-λ)$ for $δ\to0$. We prove that if $λ$ is in the absolutely continuous spectrum of $H_0$ and $H$, then the spectrum of this difference converges to a set that can be explicitly described in terms of (i) the eigenvalues of the scattering matrix $S(λ)$ for the pair $H_0$, $H$ and (ii) the singular values of the Hankel operator $H_ϕ$ with the symbol $ϕ$.

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arXiv2009

On the spectrum of Bargmann-Toeplitz operators with symbols of a variable sign

The paper discusses the spectrum of Toeplitz operators in Bargmann spaces. Our Toeplitz operators have real symbols with a variable sign and a compact support. A class of examples is considered where the asymptotics of the eigenvalues of such operators can be computed. These examples show that this asymptotics depends on the geometry of the supports of the positive and negative parts of the symbol. Applications to the perturbed Landau Hamiltonian are given.

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