In this paper we consider eigenvalues asymptotics of the energy operator in the one of the most interesting models of quantum physics, describing an interaction between two-level system and harmonic oscillator. The energy operator of this model can be reduced to some class of infinite Jacobi matrices. Discrete spectrum of this class of operators represents the perturbed spectrum of harmonic oscillator. The perturbation is an unbounded operator compact with respect to unperturbed one. We use slightly modified Janas-Naboko successive diagonalization approach and some new compactness criteria for infinite matrices. Two first terms of eigenvalues asymptotics and the estimation of remainder are found.
We fnd the asymptotics of eigenvalues of polynomially compact zero order pseudodiferential operators, the motivating example being the Neumann- Poincare operator in linear elasticity.
Let $H_0$ be a periodic operator on $\R^+$(or periodic Jacobi operator on $\N$). It is known that the absolutely continuous spectrum of $H_0$ is consisted of spectral bands $\cup[α_l,β_l]$. Under the assumption that $\limsup_{x\to \infty} x|V(x)|<\infty$ ($\limsup_{n\to \infty} n|V(n)|<\infty$), the asymptotical behaviour of embedded eigenvalues approaching to the spectral boundary is established.
The one-particle density matrix $γ(x, y)$ for a bound state of an atom or molecule is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula $λ_k \sim (Ak)^{-8/3}$, $A \ge 0$, as $k\to\infty$, for the eigenvalues $λ_k$ of the self-adjoint operator $\boldsymbolΓ\ge 0$ with kernel $γ(x, y)$.
The kinetic energy of a multi-particle system is described by the one-particle kinetic energy density matrix $τ(x, y)$. Alongside the one-particle density matrix $γ(x, y)$, it is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula $λ_k \sim (Bk)^{-2}$, $B \ge 0$, as $k\to\infty$, for the eigenvalues $λ_k$ of the self-adjoint operator $\boldsymbol{\sf T}\ge 0$ with kernel $τ(x, y)$.
We consider the Bochner Laplacian on high tensor powers of a positive line bundle on a closed symplectic manifold (or, equivalently, the semiclassical magnetic Schrödinger operator with the non-degenerate magnetic field). We assume that the operator has discrete wells. The main result of the paper states asymptotic expansions for its low-lying eigenvalues.
This paper is devoted to the asymptotic behavior of all eigenvalues of Symmetric (in general non Hermitian) Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the complex plane line as the dimension of the matrices increases to infinity. The main result describes the asymptotic structure of all eigenvalues. The constructed expansion is uniform with respect to the number of eigenvalues. Keywords: Toeplitz matrices, eigenvalues, asymptotic expansions
In this paper, we consider the asymptotic behavior for the principal eigenvalue of an elliptic operator with piecewise constant coefficients. This problem was first studied by Friedman in 1980. We show how the geometric shape of the interface affects the asymptotic behavior for the principal eigenvalue. This is a refinement of the result by Friedman.
In this work we obtain the integrated density of states for the Schrödinger operators with decaying random potentials acting on $\ell^2(\mathbb{Z}^d)$. We also study the asymptotic of the largest and smallest eigenvalues of its finite volume approximation
In the present paper, the Karhunen-Lo{è}ve eigenvalues for a sub-fractional Brownian motion are considered in the case of $H>\frac12$. Rigorous large $n$ asymptotics for those eigenvalues are shown, based on functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotically estimates for the closely related problem of small $L^2$-ball probabilities for a sub-fractional Brownian motion are derived. By the way, asymptotic analysis on the Karhunen-Lo{è}ve eigenvalues for the corresponding "derivative" process is also established.
In this work, we find the asymptotic formulas for the sum of the negative eigenvalues smaller than $-\varepsilon$ $(\varepsilon >0)$ of a self-adjoint operator $L$ which is defined by the following differential expression $$\ell(y)=-(p(x)y'(x))'-Q(x)y(x)$$ with the boundary condition $y(0) = 0$ in the space in the space $L_{2}(0,\infty ;H)$.
Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials are analyzed and their asymptotic formulas are obtained. Our approach also yields asymptotic formulas for diverging eigenvalues in the strong coupling regime for the imaginary part of the potential.
We find uniform asymptotic formulas for all the eigenvalues of certain 7-diagonal symmetric Toeplitz matrices of large dimension. The entries of the matrices are real and we consider the case where the real-valued generating function such that its first five derivatives at the one endpoint of interval are equal zero. This is not the simple-loop case considered earlier. We obtain nonlinear equations for the eigenvalues. It should be noted that our equations have a more complicated structure than the equations for the simple loop case.
We study the 3D Neuman magnetic Laplacian in the presence of a semi-classical parameter and a non-uniform magnetic field with constant intensity. We determine a sharp two term asymptotics for the lowest eigenvalue, where the second term involves a quantity related to the magnetic field and the geometry of the domain. In the special case of the unit ball and a helical magnetic field, the concentration takes place on two symmetric points of the unit sphere.
We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.
