Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1809.03079

Resolvent of the generator of the $C_0$-group with non-basis family of eigenvectors and sharpness of the XYZ theorem

Grigory M. Sklyar·Vitalii Marchenko
View original source ↗Read source document ↗Export metadata ↓

Abstract

The paper presents an explicit form of the resolvent for the class of generators of $C_0$-groups with purely imaginary eigenvalues, clustering at $i\infty$, and complete minimal non-basis family of eigenvectors, constructed recently by the authors in~\cite{Sklyar3}. The growth properties of the resolvent are described. The discrete Hardy inequality serves as the cornerstone for the proofs of the corresponding results. Moreover, it is shown that the main result on the Riesz basis property for invariant subspaces of the generator of the $C_0$-group, obtained a decade ago by G.Q.~Xu, S.P.~Yung and H.~Zwart in~\cite{Xu},~\cite{Zwart}, is sharp.

Explore related subjects

math.SPmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Grigory M. Sklyar, Vitalii Marchenko. 2018-09-10. Resolvent of the generator of the $C_0$-group with non-basis family of eigenvectors and sharpness of the XYZ theorem. https://arxiv.org/abs/1809.03079

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

KEEP EXPLORING

Related papers

arXiv2026

Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes

This article is concerned with mixed Steklov--Robin eigenvalues on bounded domains in $\mathbb{R}^{n}, n \geq 2$, with Lipschitz boundary. Specifically, we consider domains with symmetry of order $4$ containing a spherical hole. We obtain isoperimetric inequalities for the $k$-th Steklov-Robin eigenvalues for each $k \in \{2, 3, \dots, n+1\}$. We provide examples to emphasize the fact that the symmetry assumptions, on the family of domains considered, are crucial.

math.SP↗
arXiv2026

Steklov isospectrality of conformal metrics

The Steklov spectrum of a smooth compact Riemannian manifold $(M,g)$ with boundary is the set of eigenvalues counted with multiplicities of its Dirichlet-to-Neumann map. (DN map) This article is devoted to the Steklov spectral inverse problem of recovering the metric $g$, up to natural gauge invariance, from its Steklov spectrum. Positive results are established in dimension $n\geq 3$ for conformal metrics under the assumption that the geodesic flow on the boundary is Anosov with simple length spectrum. The paper combines wave trace formula techniques with the injectivity of the geodesic X-ray transform for functions on closed Anosov manifolds. It is shown that knowledge of the Steklov spectrum determines the jet at the boundary of the underlying Riemannian metric within its conformal class. In this particular context, this parallels the well-known results of the Calderón problem, where we are given the entire Dirichlet-to-Neumann map instead. As a simple corollary, assuming real-analyticity of the conformal factor, Steklov isospectral metrics must coincide. Using similar arguments, we are also able to prove under the same assumption of hyperbolicity of the geodesic flow on the boundary, that generically any smooth potential $q$ can be recovered from the Steklov spectrum, in the sense that its jet at the boundary is determined by the spectrum of the DN map for the Schrödinger operator with potential $q$. Consequently, in this case, two analytic Steklov isospectral potentials must be equal.

math.SP↗
arXiv2026

Revisiting the Weak Coupling Phenomenon for Two-Dimensional Schrödinger Operators

We study the existence of negative eigenvalues for two-dimensional Schrödinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a century ago, Simon was the first to describe the unique negative eigenvalue emerging from the threshold of the essential spectrum of one- and two-dimensional Schrödinger operators. The aim of this paper is to extend Simon's results in two dimensions to a broader class of potentials, allowing for both stronger singularities and slower decay at infinity, at the cost of losing uniqueness of weakly coupled eigenvalues.

math.SP↗