Solutions of abstract wave equations, eigenvalues and resonances

Yuri Latushkin; Alin Pogan

Home/Archive/arXiv

arXiv · 2510.03637

Solutions of abstract wave equations, eigenvalues and resonances

Yuri Latushkin·Alin Pogan

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove general representation formulas for strongly continuous cosine and sine operator families in terms of scattering resonances of their generators. This generalizes known results related to decay, growth and oscillatory behavior of solutions of abstract wave equations to a wide class of non-self-adjoint operators in Banach spaces. Inspired by the classical results on scattering resonances for Schrödinger operators with compactly supported potentials, we develop quite general abstract scheme of resonances that involves extensions of the resolvent operators from Banach to Frechet spaces. We split the solutions of the wave equations in two parts: The first part is related to finite rank operators induced by the eigenvalues and resonances while the second part involves a partial inversion of the Laplace transform whose exponential behavior is effectively controlled. Illustrations and applications cover a wide class of generators including the Schrödinger operators with non-symmetric complex matrix potentials, linearizations of nonlinear wave equations, Aharonov-Bohm and block-box Hamiltonians, etc.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yuri Latushkin, Alin Pogan. 2025-10-04. Solutions of abstract wave equations, eigenvalues and resonances. https://arxiv.org/abs/2510.03637

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Complex Geometric Approach to the Discrete Gabor Transform and Localization Operators on the Flat Torus

In a recent paper, the discrete Gabor transform was connected to a Gabor transform with a time frequency domain given by the flat torus. We show that the corresponding Bargmann-Fock spaces can be expressed as theta functions (or equivalently line bundles on Abelian varieties). We give applications of this viewpoint to frame results for the discrete Gabor transform. In particular, we get necessary conditions which hold in higher dimensions and can expand the known results in the one dimensional case, the primary tool being the theorem of the square. We also give an application to asymptotics of restriction operators which arises via the asymptotic behavior of Bergman kernels and Toeplitz operators for high tensor powers of line bundles and find that time frequency restriction operators on the flat torus will exhibit "plunge" behaviors similar to those of time frequency restriction operators in other contexts.

math.FA↗

arXiv2026

On a minimal Andô dilation for a pair of strict contractions

The isometric dilation of a pair of commuting contractions due to Andô is not minimal. We modify Andô's dilation and construct a minimal isometric dilation on $\mathcal H \oplus_2 \ell_2(\mathcal H \oplus_2 \mathcal H)$ for a commuting pair of strict contractions on a Hilbert space $\mathcal H$. In the same spirit, we construct under certain conditions a minimal Andô dilation for a commuting pair of strict Banach space contractions. Further, we show that an Andô dilation is possible even for a more general pair of commuting contractions $(T_1,T_2)$ on a normed space $\mathbb X$ provided that the function $A_{T_i}: \mathbb X \rightarrow \mathbb R$ given by $A_{T_i}(x)=(\|x\|^2-\|T_ix\|^2)^{\frac{1}{2}}$ defines a norm on $\mathbb X$ for $i=1,2$.

math.FA↗

arXiv2026

Some properties of Fourier quasicrystals and measures on a strip

We extend certain results of the theory of Fourier quasicrystals on the real line to the case of a horizontal strip of finite width. For measures in a strip we use a natural generalization of the usual Fourier transform for measures on the line. We consider positive or translation bounded measures $μ$ on a strip whose Fourier transform is a pure point measure $\hatμ=\sum_{γ\inΓ}b_γδ_γ$ (as usual, $δ_γ$ is the unit mass at the point $γ$). We prove that the measure $ν=\sum_{γ\inΓ}|b_γ|^2δ_γ$ has the exponential growth. Moreover, if for some $η>0$ the points of $Γ$ in every interval of length $η$ are linearly independent over integers, then the measure $\hatμ$ also has the exponential growth.

math.FA↗