Related papers: Spectral Analysis on Damek-Ricci Space

The spectral properties of the singular Schr\"odinger operator with complex-valued potential which takes values in a wider region than the half-plane, have been little studied. In general case, the operator is non-sectorial, and the…

We show that Arthur's Paley-Wiener theorem for K-finite compactly supported smooth functions on a real reductive Lie group G of the Harish-Chandra class can be deduced from the Paley-Wiener theorem we established in the more general setting…

We consider the wave equation with a distributional Dirac damping and Dirichlet boundary conditions on a compact interval. It is shown that the spectrum of the corresponding wave operator is fully determined by zeroes of an entire function.…

In this article we discuss a few spectral properties of a paranormal closed operator (not necessarily bounded) defined in a Hilbert space. This class contains closed symmetric operators. First we show that the spectrum of such an operator…

The resolvent of an operator in a Banach space is defined on an open subset of the complex plane and is holomorphic. It obeys the resolvent equation. A generalization of this equation to Schwartz distributions is defined and a Schwartz…

In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on…

The $p$-adic unitary operator $U$ is defined as an invertible operator on $p$-adic ultrametric Banach space such that $\left |U\right |=\left |U^{-1}\right |=1$. We point out $U$ has a spectral measure valued in $\textbf{projection…

The Dirac operators $$ Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi],$$ with $L^2$-potentials $$ v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,\pi]), $$ considered on $[0,\pi]$ with periodic,…

For a second order operator on a compact manifold satisfying the strong H\"ormander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class…

For a broad class of polynomial potentials $V$, with an important and instructive representative being $V(x) = x^{2a} + i x^b$, $x \in \mathbb R$, $a, b \in \mathbb N$, we show that the system of spectral projections $\{P_n\}_n$ of an…

This article delves into the analysis of various spectral properties pertaining to totally paranormal closed operators, extending beyond the confines of boundedness and encompassing operators defined in a Hilbert space. Within this class,…

A description of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a reflexive complex Banach space is shown to remain true without the reflexivity requirement. A similar nature…

In this article we give an overview on some recent development of Littlewood-Paley theory for Schr\"odinger operators. We extend the Littlewood-Paley theory for special potentials considered in the authors' previous work. We elaborate our…

This paper gives a survey of methods for the construction of space-frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general…

Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence of topological obstructions, they are at least exponentially localized. In some cases such as flat-band…

Polynomial invariants are fundamental objects in analysis on Lie groups and symmetric spaces. Invariant differential operators on symmetric spaces are described by Weyl group invariant polynomial. In this article we give a simple criterion…

Given a quadratic CR manifold $\mathcal{M}$ embedded in a complex space, we study Paley-Wiener-Schwartz theorems for spaces of Schwartz functions and tempered distributions on $\mathcal{M}$.

Let S(X) be the Schwartz space of compactly supported smooth functions on the p-adic points of a spherical variety X, and let C(X) be the space of Harish-Chandra Schwartz functions. Under assumptions on the spherical variety, which are…

We study the spectral asymptotics of wave equations on certain compact spacetimes where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime $S^1 \times S^2$. For the Laplacian on $S^1 \times S^2$ the Weyl…

Polynomial invariants are fundamental objects in analysis on Lie groups and symmetric spaces. Invariant differential operators on symmetric spaces are described by Weyl group invariant polynomial. In this article we give a simple criterion…