The Hausdorff dimension of spectral measure for the graph Laplacian is shown exactly in terms of an intermittency function. The intermittency function can be estimated by using one-dimensional discrete Schrödinger operator method.
We give a simple proof of the Weyl asymptotic formula for eigenvalues of the Dirichlet Laplacian, the buckling problem, and the Dirichlet bi-Laplacian in Euclidean domains of finite volume, with no assumptions about the boundary.
In this paper, the regularized trace formulas for a diffusion operator which include conformable fractional derivatives of order α (0<{α\leq 1}) is obtained.
We prove that on the spectrum the integrated density of states (IDS for short) of periodic Jacobi matrices is related to the discriminant. The method is to count the number of generalized zeros of Bloch wave solutions.
We extend the classical Ambarzumyan's theorem to the quasi-periodic boundary value problems by using only a part knowledge of one spectrum. We also weaken slightly the Yurko's conditions on the first eigenvalue.
In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential spectral singularities and singular quasimomenta.
Abstract We show that some hypergeometric monodromy groups in ${\rm Sp}(4,\mathbf{Z})$ split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank $2$ . In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in $\mathbf{P}^{4}$ splits as $\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$ . As a consequence, for a smooth quintic threefold $X$ we show that the group of autoequivalences $D^{b}(X)$ generated by the spherical twist along ${\mathcal{O}}_{X}$ and by tensoring with ${\mathcal{O}}_{X}(1)$ is an Artin group of dihedral type.
A new technique for approximating eigenvalues and eigenvectors of a self-adjoint operator is presented. The method does not incur spectral pollution, uses trial spaces from the form domain, has a self-adjoint algorithm, and exhibits superconvergence.
It is shown that a series of solvable polynomials is attached to the series of zero modes constructed by Adam, Muratori and Nash \cite{AdamMuratoriNash1
We show the completeness of the system of generalized eigenfunctions of closed extensions of elliptic cone operators under suitable conditions on the symbols.
We consider Schroedinger operator $-Δ+V$ in $R^d$ ($d\ge 2$) with smooth periodic potential $V$ and prove that there are only finitely many gaps in its spectrum.
In the scattering theory framework, we point out a connection between the spectrum of the scattering matrix of two operators and the spectrum of the difference of spectral projections of these operators.
We consider OPRL and OPUC with measures regular in the sense of Ullman-Stahl-Totik and prove consequences on the Jacobi parameters or Verblunsky coefficients. For example, regularity on $[-2,2]$ implies $\lim_{N\to\infty} N^{-1} [\sum_{n=1}^N (a_n-1)^2 + b_n^2] =0$.