Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π ? 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π ?2β) for these eigenvalues.     

The essential spectrum of the three-particle discrete operator corresponding to a system of three fermions on a lattice

We consider a family of three-particle discrete Shrödinger operators H _μ (K). These operators are associated with the Hamiltonian for a system of three identical particles (fermions) with pairwise two-particle interactions on neighboring junctions of the d-dimensional lattice Z ^d . We describe the location and the structure of the essential spectrum of the operator H _μ (K) for all values of the three-particle quasi-momentum K ∈ T ^d and the interaction energy μ > 0.     

Semiclassical resonances associated with a periodic orbit

We consider resonances for a h-pseudo-differential operator H(x, hD _x; h) induced by a periodic orbit of hyperbolic type. We generalize the framework of Gérard and Sjöstrand, in the sense that we allow hyperbolic and elliptic eigenvalues of the Poincarémap, and look for so-called semi-excited resonances with imaginary part of magnitude ?h log h, or h ^δ, with 0 < δ < 1.     

Stability of nonlocal difference schemes in a subspace

We consider a family of two-layer difference schemes for the heat equation with nonlocal boundary conditions containing the parameter γ. In some interval γ ∈ (1, γ ₊), the spectrum of the main difference operator contains a unique eigenvalue λ ₀ in the left complex half-plane, while the remaining eigenvalues λ ₁, λ ₂, …, λ _N?1 lie in the right half-plane. The corresponding grid space H _N is represented as the direct sum H _N = H ₀⊕H _N?1 of a one-dimensional subspace and the subspace H _N?1 that is the linear span of eigenvectors µ⁽¹⁾, µ⁽²⁾, …, µ^(N?1). We introduce the notion of stability in the subspace H _N?1 and derive a stability criterion.     

Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction

We consider the discrete spectrum of the two-dimensional Hamiltonian H = H ₀ + V, where H ₀ is a Schrödinger operator with a non-constant magnetic field B that depends only on one of the spatial variables, and V is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under certain general conditions on B and V, we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of H.     

Some results for operators on a model space

We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space K _θ = H²ΘθH² is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H²-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.     

Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion

We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L ₂(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X _H with Hurst parameter H∈(0,1), indexed by a self–similar set T?? ^N of Hausdorff dimension D. This rate turns out to be of order n ^?H/D (log?n)^1/2. In the case T=[0,1] ^N we present a concrete wavelet representation of X _H leading to an approximation of X _H with the optimal rate n ^?H/N (log?n)^1/2.     

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L~p(R)and the Hardy space H~1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H~p(R) with 0 p 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H~p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H~1(R).     

Traces of Higher Negative Orders for a String with a Singular Weight

We study the linear operator pencil A(λ) = L?λV, λ ∈ ?, where L is the Sturm–Liouville operator with potential q(x) and V is the operator of multiplication by the weight ρ(x). The potential and the weight are assumed to belong to the space W ₂ ^?1 [0, π]. For the pencil A(λ), we seek formulas for the traces of higher negative orders, i.e., for the sums \(\sum\nolimits_{n = 1}^\infty {\lambda _n^{ - p}} \), p ≥ 2, where λ_n, n ∈ ?, is the sequence of eigenvalues of the pencil numbered in nondescending order of absolute values. Trace formulas in terms of the weight ρ(x) and the integral kernel of the operator L^?1 are obtained, and the relationship between these formulas and the classical results about traces of integral operators is described. The theoretical results are illustrated by examples.     

On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder

We consider an operator A^ε on L₂(\({\mathbb{R}^{{d_1}}} \times {T^{{d_2}}}\)) (d₁ is positive, while d₂ can be zero) given by A^ε = ?div A(ε^?1x₁,x₂)?, where A is periodic in the first variable and smooth in a sense in the second. We present approximations for (A^ε ? μ)^?1 and ?(A^ε ? μ)^?1 (with appropriate μ) in the operator norm when ε is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.     

Regularization of a three-element functional equation

In this paper we study the three-element functional equation

$(V\Phi )(z) \equiv \Phi (iz) + \Phi ( - iz) + G(z)\Phi \left( {\frac{1}{z}} \right) = g(z), z \in R,$

, subject to

$R: = \{ z:\left| z \right| < 1, \left| {\arg z} \right| < \frac{\pi }{4}\} .$

We assume that the coefficients G(z) and g(z) are holomorphic in R and their boundary values G ⁺(t) and g ⁺(t) belong to H(Γ), G(t)G(t ^?1) = 1. We seek for solutions Φ(z) in the class of functions holomorphic outside of \(\bar R\) such that they vanish at infinity and their boundary values Φ^?(t) also belong to H(Γ). Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.

     

Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface

Let(Σ, g) be a compact Riemannian surface without boundary and λ_1(Σ) be the first eigenvalue of the Laplace-Beltrami operator ?_g. Let h be a positive smooth function on Σ. Define a functional J_(α,β)(u) =1/2∫Σ(|?_gu|~2-αu~2)dv_g-β log∫Σhe~udv_g on a function space H = {u ∈ W~(1,2)(Σ) :∫Σudvg = 0}. If α λ_1(Σ) and J_(α,8π) has no minimizer on H,then we calculate the infimum of Jα,8π on H by using the method of blow-up analysis. As a consequence,we give a sufficient condition under which a Kazdan-Warner equation has a solution. If αλ_1(Σ), then infu∈HJ_(α,8π)(u) =-∞. If β 8π, then for any α∈ R, there holds infu∈H Jα,β(u) =-∞. Moreover, we consider the same problem in the case that α is large, where higher order eigenvalues are involved.     

Schrödinger operators with δ′-interactions and the Krein-Stieltjes string

We investigate one dimensional symmetric Schrödinger operator H _{X, β} with δ′-interactions of strength β = “β _n ” _{n = 1} ^∞ ? ? on a discrete set X = “x _n ” _{n = 1} ^∞ ? [0, b), b ≤ +∞ (x _n ↑ b). We consider H _{X, β} as an extension of the minimal operator H _min:= ?d ²/dx ²?W ₀ ^2.2 (?\X) and study its spectral properties in the frame-work of the extension theory by using the technique of boundary triplets and the corresponding Weyl functions. The construction of a boundary triplet for H _min ^* is given in the case d _*:= inf_{n ∈ ?}\x _n ? x _{n ? 1}\ = 0. We show that spectral properties like self-adjointness, lower semiboundedness, nonnegativity, and discreteness of the spectrum of the operator H _{X, β} correlate with the corresponding properties of a certain Jacobi matrix. In the case β _n > 0, n ∈ ?, these matrices form a subclass of Jacobi matrices generated by the Krein-Stieltjes strings. The connection discovered enables us to obtain simple conditions for the operator H _{X, β} to be self-adjoint, lower semibounded and discrete. These conditions depend significantly not only on β but also on X. Moreover, as distinct from the case d _* > 0, the spectral properties of Hamiltonians with δ- and δ′-interactions in the case d _* = 0 substantially differ.     

Asymptotic <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">u</Emphasis></Subscript>-Toeplitzness of weighted composition operators on <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* ⁿ TS ⁿ } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T _u -Toeplitzness of weighted composition operators on the Hardy space H², where u is a nonconstant inner function.     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

Spectral Properties of Discontinuous Sturm–Liouville Problems with a Finite Number of Transmission Conditions

In this paper we investigate discontinuous two-point boundary value problems with eigenparameter in the boundary conditions and with transmission conditions at the finitely many points of discontinuity. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of the considered problem coincide with those of A. We obtain asymptotic formulas for the eigenvalues and eigenfunctions. Also we show that the eigenelements of A are complete in H.     

Transmission of Waves Through a Small Aperture in the Cross-Wall in an Acoustic Waveguide

We study wave diffraction at near-threshold frequencies in an acoustic waveguide with a cross-wall that has a small aperture of diameter ε > 0. We describe the effects of almost complete reflection or transmission of waves related to the classical Vainstein anomaly and the presence of almost standing waves for the threshold value Λ _k of the spectral parameter λ in continuous spectrum. The greatest attention is paid to analyzing the range λ ^ε = Λ _k + ε²μ² of the spectral parameter with μ ≤ μ₀, which generates scattering coefficients depending on μ > 0 and presents the greatest difficulties in constructing and justifying the asymptotics. Almost complete reflection and transmission correspond to the cases of going away from the threshold (as μ → +∞) and approaching it (as μ → +0) characterized by simpler asymptotics.     

Two classes of <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated with a von Neumann algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P ₁ and P ₂ of τ-measurable operators and investigate their properties. The class P ₂ contains P ₁. If a τ-measurable operator T is hyponormal, then T lies in P ₁; if an operator T lies in P _k , then UTU* belongs to P _k for all isometries U from M and k = 1, 2; if an operator T from P ₁ admits the bounded inverse T ^?1, then T ^?1 lies in P ₁. We establish some new inequalities for rearrangements of operators from P ₁. If a τ-measurable operator T is hyponormal and T ⁿ is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P ₁ coincides with the set of all paranormal operators on H.     

Homogenization of a nonstationary model equation of electrodynamics

In L ₂(?₃;?₃), we consider a self-adjoint operator ? _ε , ε > 0, generated by the differential expression curl η(x/ε)^?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? _ε ^1/2 ) and ? _ε ^?1/2 sin(τ? _ε ^1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?⁰)^1/2) and (?₀)^?1/2 sin(τ(?₀)^1/2), respectively, in the norm of the operators acting from the Sobolev space H ^s (with a suitable s) to ?₂. Here ?⁰ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? _τ ² v _ε = ?? _ε v _ε , div v _ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).     

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator A _{ε, ε} > 0, on L ₂(R ^d ;C ⁿ ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA _ε ^1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA _ε ^1/2 ) converges to cos(τ(A ⁰)^1/2) in the norm of operators acting from the Sobolev space H ^s (R ^d ;C ⁿ ) (with a suitable s) to L ₂(R ^d ;C ⁿ ). Here A ₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? _τ ² u _ε (x, τ) = ?A _ε u _ε (x, τ).