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Rafael del Rio

Publications and source records attributed to Rafael del Rio.

17 recordsLinked to original sources
arXiv2025

On Spectral Stability for Rank One Perturbations

Embedded point spectra of rank one singular perturbations of an arbitrary self-adjoint operator A on a Hilbert space H is studied. It is shown that these perturbations can be regarded as self-adjoint extensions of a densely defined closed symmetric operator B with deficiency indices (1; 1). Assuming the deficiency vector of B is cyclic for its self-adjoint extensions, we prove that the spectrum of A contains a dense Gδ subset where it is not possible to have eigenvalues for any rank one singular perturbation. Moreover, for a dense Gδ set of rank one singular perturbations of A their eigenvalues are isolated. The approach presented here unifies points of view taken by different authors.

math.SP↗
arXiv2020

Point mass perturbations of spectral measures

Using a generalization of the moment problem and the extremal properties of spectral measures corresponding to the selfadjoint extensions of a regular symmetric operator, we study point mass perturbations of spectral measures. We obtain general results for a wide class of operators and apply them to the analysis of point mass perturbations of spectral measures pertaining to Bessel and generalized Schrödinger operators.

math-ph↗
arXiv2019

Random Sturm-Liouville Operators with Generalized Point Interactions

In this work we study the point spectra of selfadjoint Sturm-Liouville operators with generalized point interactions, where the two one-sided limits of the solution data are related via a general $\mathrm{SL}(2,\mathbb{R})$ matrix. We are particularly interested in the stability of eigenvalues with respect to the variation of the parameters of the interaction matrix. As a particular application to the case of random generalized point interactions we establish a version of Pastur's theorem, stating that except for degenerate cases, any given energy is an eigenvalue only with probability zero. For this result, independence is important but identical distribution is not required, and hence our result extends Pastur's theorem from the ergodic setting to the non-ergodic setting.

math.SP↗
arXiv2017

Stability of determinacy and inverse spectral problems for Jacobi operators

This work studies the interplay between Green functions, the index of determinacy of spectral measures and interior finite rank perturbations of Jacobi operators. The index of determinacy quantifies the stability of uniqueness of solutions of the moment problem. We give results on the constancy of this index in terms of perturbations of the corresponding Jacobi operators. The permanence of the $N$-extremality of a measure is also studied. A measure $μ$ is $N$-extremal when the polynomials are dense in $L_2(\mathbb{R},μ)$. As a by-product, we give a characterization of the index in terms of cyclic vectors. We consider a new inverse problem for Jacobi operators in which information on the place where the interior perturbation occurs is obtained from the index of determinacy.

math-ph↗
arXiv2017

A direct proof of F. Riesz representation Theorem

A direct proof of the Riesz representation theorem is provided. This theorem characterizes the linear functionals acting on the vector space $C(K)$ of continuous functions defined on a compact subset $K$ of the real numbers $\mathbb{R}$. This proof avoids complicated arguments commonly used in generalizations of Riesz original theorem.

math.FA↗
arXiv2017

Resonances under Rank One Perturbations

We study resonances generated by rank one perturbations of selfadjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant states is exhibited. We show how these results can be applied to Sturm-Liouville operators. Main tools are the Aronszajn-Donoghue theory for rank one perturbations, a reduction process of the resolvent based on Feshbach-Livsic formula, the Fermi golden rule and a careful analysis of the Fourier transform of quasi-Lorentzian functions. We relate these results to sojourn time estimates and spectral concentration phenomena

math.SP↗
arXiv2016

Inverse problems for Jacobi operators IV: Interior mass-spring perturbations of semi-infinite systems

This work gives results on the interplay of the spectra of two Jacobi operators corresponding to an infinite mass-spring system and a modification of it obtained by changing one mass and one spring of the system. It is shown that the system can be recovered from these two spectra. Necessary and sufficient conditions for two sequences to be the spectra of the mass-spring system and the perturbed one are provided.

math-ph↗
arXiv2014

Spectral analysis for semi-infinite mass-spring systems

We study how the spectrum of a Jacobi operator changes when this operator is modified by a certain finite rank perturbation. The operator corresponds to an infinite mass-spring system and the perturbation is obtained by modifying one interior mass and one spring of this system. In particular, there are detailed results of what happens in the spectral gaps and which eigenvalues do not move under the modifications considered. These results were obtained by a new tecnique of comparative spectral analysis and they generalize and include previous results for finite and infinite Jacobi matrices.

math-ph↗
arXiv2013

Spectra of Random Operators with absolutely continuous Integrated Density of States

The structure of the spectrum of random operators is studied. It is shown that if the density of states measure of some subsets of the spectrum is zero, then these subsets are empty. In particular follows that absolute continuity of the IDS implies singular spectra of ergodic operators is either empty or of positive measure. Our results apply to Anderson and alloy type models, perturbed Landau Hamiltonians, almost periodic potentials and models which are not ergodic.

math.SP↗
arXiv2013

Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems

Consider an infinite linear mass-spring system and a modification of it obtained by changing the first mass and spring of the system. We give results on the interplay of the spectra of such systems and on the reconstruction of the system from its spectrum and the one of the modified system. Furthermore, we provide necessary and sufficient conditions for two sequences to be the spectra of the mass-spring system and the perturbed one.

math-ph↗
arXiv2011

Inverse problems for Jacobi operators I: Interior mass-spring perturbations in finite systems

We consider a linear finite spring mass system which is perturbed by modifying one mass and adding one spring. From knowledge of the natural frequencies of the original and the perturbed systems we study when masses and springs can be reconstructed. This is a problem about rank two or rank three type perturbations of finite Jacobi matrices where we are able to describe quite explicitly the associated Green's functions. We give necessary and sufficient conditions for two given sets of points to be eigenvalues of the original and modified system respectively.

math.SP↗
arXiv2010

Spectral measures of Jacobi operators with random potentials

Let $H_ω$ be a self-adjoint Jacobi operator with a potential sequence $\{ω(n)\}_n$ of independently distributed random variables with continuous probability distributions and let $μ_ϕ^ω$ be the corresponding spectral measure generated by $H_ω$ and the vector $ϕ$. We consider sets $A(ω)$ which depend on $ω$ in a particular way and prove that $μ_ϕ^ω(A(ω))=0$ for almost every $ω$. This is applied to show equivalence relations between spectral measures for random Jacobi matrices and to study the interplay of the eigenvalues of these matrices and their submatrices.

math-ph↗
arXiv2009

Random Sturm Liouville Operators

Selfadjoint Sturm-Liouville operators $H_ω$ on $L_2(a,b)$ with random potentials are considered and it is proven, using positivity conditions, that for almost every $ω$ the operator $H_ω$ does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as $H_ω$ but in $L_2(\tilde a,\tilde b)$ where $(\tilde a,\tilde b)\subset(a,b)$.

math.SP↗
arXiv2008

Spectral averaging techniques for Jacobi matrices

Spectral averaging techniques for one-dimensional discrete Schroedinger operators are revisited and extended. In particular, simultaneous averaging over several parameters is discussed. Special focus is put on proving lower bounds on the density of the averaged spectral measures. These Wegner type estimates are used to analyze stability properties for the spectral types of Jacobi matrices under local perturbations.

math-ph↗
arXiv1994

Singular continuous spectrum is generic

In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense $G_δ$.

math.SP↗