In this paper, we briefly explain the spectral expansion problem for differential operators defined on the entire real line, generated by a differential expression with periodic, complex-valued coefficients.
This paper deals with the concepts of upper and lower almost (Λ, sp)-continuous multifunctions. Moreover, several characterizations of upper and lower almost (Λ, sp)-continuous multifunctions are investigated.
It is known that the spectrum of Schrödinger operators with sparse potentials consists of singular continuous spectrum. We give a sufficient condition so that the edge of the singular continuous spectrum is not an eigenvalue and construct examples with singular continuous spectrum which have no eigenvalues and which have a single negative eigenvalue.
In this paper, we present an approach for explicitly constructing quasi-periodic Schrödinger operators with Cantor spectrum with $C^k$ potential. Additionally, we provide polynomial asymptotics on the size of spectral gaps.
We give a Coulson integral formula and a Coulson-Jacobs formula for the $p$-Schatten energy. We use this formulas to compare the $p$-Schatten energy of different trees by using a quasiorder, and establish the maximality of paths among all trees.
In this paper I give an explicit construction of an analogue of eigenspace for points of the singular spectrum of a self-adjoint operator. This construction is based on an abstract version of homogeneous Lippmann-Schwinger equation.
A generalization of Robin boundary conditions leading to self-adjoint operators is developed for the second derivative operator on metric graphs with compact completion and totally disconnected boundary. Harmonic functions and their properties play an essential role.
Lieb-Thirring type estimates are proved for the sum of powers of negative eigenvalues of a Schrödinger type operator $(-Δ)^l -Vμ$ where $μ$ is a singular measure in $\mathbb{R}^d,$ satisfying a condition on the measure of balls and $V$ is a $μ$-measurable function.
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
We show, in the same vein of Simon's Wonderland Theorem, that, typically in Baire's sense, the rates with whom the solutions of the Schrödinger equation escape, in time average, from every finite-dimensional subspace, depend on subsequences of time going to infinite.
For a compact, connected metric graphs with a boundary that consists of $k$ vertices, we prove that an arbitrary symmetric $k\times k$ matrix with real entries can be realized as the Dirichlet-to-Neumann operator for the Laplacian plus a constant.
Two necessary and sufficient conditions for an operator to be semi-normal are revealed. For a Volterra integration operator the set where the operator and its adjoint are metrically equal is described.
We study spectral properties of energy-dependent Sturm-Liouville equations, introduce the notion of norming constants and establish their interrelation with the spectra. One of the main tools is the linearization of the problem in a suitable Pontryagin space.
We prove that the Folland's fundamental solution for the sub-Laplacian on Heisenberg groups can also be derived form the resolvent kernel of this sub-Laplacian. This provides us with a new integral representation for this fundamental solution.
In this note the following is shown. Consider the quadratic form on (complex) matrices Q(A):=tr(A^2). Let A be such a matrix. Then an ellipse can be found, with the vector from center to focus determined by the value of Q at the traceless part of A, which must be contained in the convex hull of the spectrum of A.
We obtain an asymptotic formula for the counting function of the discrete spectrum for Hankel-type pseudo-differential operators with discontinuous symbols.
In this paper I prove existence of an irreducible pair of operators $H$ and $H+V,$ where $H$ is a self-adjoint operator and $V$ is a self-adjoint trace-class operator, such that the singular spectral shift function of the pair is non-zero on the absolutely continuous spectrum of the operator $H.$
A Szegö-type theorem for Toeplitz operators was proved by Boutet de Monvel and Guillemin for general Toeplitz structures. We give a local version of this result in the setting of positive line bundles on compact symplectic manifolds.
We discuss convergence properties of the spectral shift functions associated with a pair of Schrodinger operators with Dirichlet boundary conditions at the end points of a finite interval (0, r) as the length of interval approaches infinity.