Rafael del Rio

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. We study when masses and springs can be recovered from the natural frequencies of the original and the perturbed systems. This is a problem about rank 2 or rank 3 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 systems, respectively.

Point spectrum and mixed spectral types for rank one perturbations

We consider examples A_ λ = A + λ(ϕ, •)ϕ of rank one perturbations with ϕ a cyclic vector for A. We prove that for any bounded measurable set B ⊂ I, an interval, there exist A, ϕ so that {E ∈ I | some A_ λ has E as an eigenvalue} agrees with B up to sets of Lebesgue measure zero. We also show that there exist examples where A_ λ has a.c. spectrum [0,1] for all λ, and for sets of λs of positive Lebesgue measure, A_ λ also has singular continuous spectrum in [0,1].

Spectral averaging techniques for Jacobi matrices

Spectral averaging techniques for one-dimensional discrete Schrodinger 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.

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 integrated density of states 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.

Random Sturm–Liouville operators

Selfadjoint Sturm-Liouville operators

Sturm–Liouville operators in the half axis with shifted potentials

H_\omega

Resonances under rank-one perturbations

on

Boundary Conditions and Spectra of Sturm-Liouville Operators

L_2(a,b)

Operators with singular continuous spectrum

with random potentials are considered and it is proven, using positivity conditions, that for almost every

Corrections and addendum to “Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions”

\omega