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Schrodinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a one-size-fits-all algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schrodinger operators with periodic potentials that are sufficiently smooth. In both cases implementable algorithms are provided, along with examples. For certain Schrodinger operators whose potentials may diverge at a single point (but are otherwise well-behaved) it is shown that there does not exist such an algorithm, though it is shown that the computation is possible if one allows for two successive limits.
For any multi-graph $G$ with edge weights and vertex potential, and its universal covering tree $mathcal{T}$, we completely characterize the point spectrum of operators $A_{mathcal{T}}$ on $mathcal{T}$ arising as pull-backs of local, self-adjoint ope
We study perturbations of the self-adjoint periodic Sturm--Liouville operator [ A_0 = frac{1}{r_0}left(-frac{mathrm d}{mathrm dx} p_0 frac{mathrm d}{mathrm dx} + q_0right) ] and conclude under $L^1$-assumptions on the differences of the coefficient
We introduce the concept of essential numerical range $W_{!e}(T)$ for unbounded Hilbert space operators $T$ and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known fo
Smoothing (and decay) spacetime estimates are discussed for evolution groups of self-adjoint operators in an abstract setting. The basic assumption is the existence (and weak continuity) of the spectral density in a functional setting. Spectral ident
The compression of the resolvent of a non-self-adjoint Schrodinger operator $-Delta+V$ onto a subdomain $Omegasubsetmathbb R^n$ is expressed in a Krein-Naimark type formula, where the Dirichlet realization on $Omega$, the Dirichlet-to-Neumann maps, a