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We construct concrete examples of time operators for both continuous and discrete-time homogeneous quantum walks, and we determine their deficiency indices and spectra. For a discrete-time quantum walk, the time operator can be self-adjoint…

Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function $P(X,t)$ of packets of spreading random walkers, were observed in numerous situations…

This text contains an alternative presentation, and in certain cases an improvement, of the "hyperbolic dispersive estimate" that was proved by Anantharaman and Nonnenmacher and used to make progress towards the quantum unique ergodicity…

A perturbation decaying to 0 at infinity and not too irregular at 0 introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues…

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by…

Mathematical analysis of the spectral properties of the time evolution operator in quantum walks is essential for understanding key dynamical behaviors such as localization and long-term evolution. The inhomogeneous three-state case, in…

We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kenser's criterion for Jacobi operators follows…

We study planar graphs with large negative curvature outside of a finite set and the spectral theory of Schr{\"o}dinger operators on these graphs. We obtain estimates on the first and second order term of the eigenvalue asymptotics.…

I present an example of a discrete Schr"odinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than…

For a class of zero order pseudodifferential operators we find the asymptotics of eigenvalues converging to a non-isolated tip of the essential spectrum.

We establish exponential decay in H\"older norm of transfer operators applied to smooth observables of uniformly and nonuniformly expanding semiflows with exponential decay of correlations.

By using an exact analytical non-Hermitian approach in terms of resonance (quasinormal) states we express the decaying wave function as the sum of exponential and nonexponential decaying solutions to the time-dependent Schr\"odinger…

In this paper we study spectral properties of Jacobi operators. In particular, we prove two main results: (1) that perturbing the diagonal coefficients of Jacobi operator, in an appropriate sense, results in exponential localization, and…

Statistical properties of critical wave functions at the spin quantum Hall transition are studied both numerically and analytically (via mapping onto the classical percolation). It is shown that the index $\eta$ characterizing the decay of…

The goal of this note is to study the spectrum of a self-adjoint convolution operator in $L^2(\mathbb R^d)$ with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show…

This paper continues the previous work (Quantum Inf. Process (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to…

The eigenvalue equation of a band or a block tridiagonal matrix, the tight binding model for a crystal, a molecule, or a particle in a lattice with random potential or hopping amplitudes: these and other problems lead to three-term…

Explicit representations of the eigenvalues of the peridynamic operator have been recently derived in [5]. These representations are given in terms of generalized hypergeometric functions. Asymptotic analysis of the hypergeometric functions…

Quantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude…

We show that the probability that a finitely supported random walk on a non-elementary subgroup of the the mapping class group gives a non-pseudo-Anosov element decays exponentially in the length of the random walk. More generally, we show…