Related papers: Discrete spectral symmetries of low-dimensional di…

We establish inequalities for the eigenvalues of Schr\"{o}dinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related…

Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy generalized $m$-th order Gaussian…

We study Sturm-Liouville operators on closed sets of a special structure, which are sometimes referred as time scales and often appear in modelling various real processes. Depending on the set structure, such operators unify both…

Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral…

We construct a singular differential operator attached to a class of singular metrics on the line bundles over the complex projective space, $\mathbb{P}^1$. This operator extends the classical notion of the generalized Laplacian. We prove…

One of the difficulties encountered when studying physical theories in discrete space-time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to…

The generic simplicity of the spectrum of a Schr\"odinger-type operator on the n-dimensional torus is studied using the Rayleigh-Schr\"odinger perturbation theory. The existence of a perturbation potential of the Laplacian is proved and…

The Schroedinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schroedinger operators act not on functions on the space-time but on sections of…

We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the…

Spectral representations of the dilation and translation operators on $L^2({\mathbb R})$ are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions…

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent)…

With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux…

We study the spectral properties of discrete Schr\"odinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction…

Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension.…

The work is devoted to the study of Laplace operator when the potential is a singular generalized function and plays the role of a singular perturbation of a Laplace operator. Abstract theorem obtained earlier by the authors B.N. Biyarov…

We develop the method of similar operators to study the spectral properties of unbounded perturbed linear operators that can be represented by matrices of various kinds. The class of operators under consideration includes various…

The spectrum of discrete Schr\"odinger operator $L+V$ on the $d$-dimensional lattice is considered, where $L$ denotes the discrete Laplacian and $V$ a delta function with mass at a single point. Eigenvalues of $L+V$ are specified and the…

We introduce and study {\it new} relative spectral invariants of {\it two} elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depend on both the eigenvalues and the…

We discuss the relationship between target space modular invariance and discrete gauge symmetries in four-dimensional orbifold-like strings. First we derive the modular transformation properties of various string vertex operators of the…

One dimensional Dirac operators $$ L_{bc}(v) \, y = i \begin{pmatrix} 1 & 0 0 & -1 \end{pmatrix} \frac{dy}{dx} + v(x) y, \quad y = \begin{pmatrix} y_1 y_2 \end{pmatrix}, \quad x\in[0,\pi],$$ considered with $L^2$-potentials $ v(x) =…