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Let $A$ be a densely defined symmetric operator with equal deficiency indices in a Hilbert space. We introduce the notion of a Weyl function $M(z)$ of $A$ corresponding to an ordinary boundary triplet of the operator $A^*$ and then…

Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach uses some elements of the spectral shift…

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $N\times N$ truncated Hilbert matrix for large values of $N$. In this paper, we extend this formula to Hankel matrices with symbols in the class of…

In this article we consider asymptotics for the spectral function of Schr\"odinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order…

Let $\Lambda$ be a uniformly discrete set and $S$ be a compact set in $R$. We prove that if there exists a bounded sequence of functions in Paley--Wiener space $PW_S$, which approximates $\delta-$functions on $\Lambda$ with $l^2-$error $d$,…

Based on a canonical approach and functional-integration techniques, a series expansion of Green's function of a scalar field, in the presence of a medium, is obtained. A series expansion for Lifshitz-energy, in finite-temperature, in terms…

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'(t)-B(t)y(t)=\D(t) f(t)$ on an interval $[a,b> $ with the regular endpoint $a$. The deficiency indices $n_\pm$ of the corresponding minimal relation $\Tmi$…

The properties of the high energy behavior of the scattering amplitude of massive, neutral and spinless particles in higher dimensional field theories are investigated. The axiomatic formulation of Lehmann, Symanzik and Zimmermann is…

The main goal of this thesis is to show the crucial role that plays the symbol in analysing the spectrum the sequence of matrices resulting from PDE approximation and in designing a fast method to solve the associated linear problem. In the…

We extend the Euler-Bernoulli beam problem, formulated as a matrix string equation with a matrix-valued density, to a setting where the density takes values in a Clifford algebra, and we analyze its isospectral deformations. For discrete…

We extend generalized functional linear models under independence to a situation in which a functional covariate is related to a scalar response variable that exhibits spatial dependence-a complex yet prevalent phenomenon. For estimation,…

We consider discrete Schroedinger operator J with Wigner-von Neumann potential not belonging to l^2. We find asymptotics of orthonormal polynomials associated to J. We prove the Weyl-Titchmarsh type formula, which relates the spectral…

We consider the joint distribution of eigenvalue clusters of the Wigner ensemble separated by macroscopic distances (i.e., on the same scale as the difference between the edges of the semicircle law). We prove that under an averaging…

The scope of this text is to study a process that induces another proof of the Spectral Embedding Theorem: that any densely defined symmetric operator can be extended by a multiplication operator through an embedding of the Hilbert space…

Semibounded symmetric operators have a distinguished self-adjoint extension, the Friedrichs extension. The eigenvalues of the Friedrichs extension are given by a variational principle that involves only the domain of the symmetric operator.…

We prove an explicit formula for the spectral expansions in $L^2(\R)$ generated by selfadjoint differential operators $$ (-1)^n\frac{d^{2n}}{dx^{2n}}+\sum\limits_{j=0}^{n-1}\frac{d^{j}}{dx^{j}} p_j(x)\frac{d^{j}}{dx^{j}},\quad…

We show that the Herglotz functions that arise as Weyl-Titchmarsh $m$ functions of one-dimensional Schr\"odinger operators are dense in the space of all Herglotz functions with respect to uniform convergence on compact subsets of the upper…

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the…

For any real limit-$n$ $2n$th-order selfadjoint linear differential expression on $[0,\infty)$, Titchmarsh- Weyl matrices $M(\lambda)$ can be defined. Two matrices of particu lar interest are the matrices $M_D(\lambda)$ and $M_N(\lambda)$…

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the $A-$amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl $m-$function associated with the…