Spectral Theory of Multiple Intervals

Abstract

We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space.) The models allow for a scale of complexity, indexed by the natural numbers $\mathbb{N}$. For each $n\in\mathbb{N}$, we get families of selfadjoint operators indexed by: (i) the unitary matrix group U(n), and by (ii) a prescribed set of $n$ non-overlapping intervals. Take $\Omega$ to be the complement in $\mathbb{R}$ of $n$ fixed closed finite and disjoint intervals, and let $L^{2}(\Omega)$ be the corresponding Hilbert space. Moreover, given $B\in U(n)$, then both the lengths of the respective intervals, and the gaps between them, show up as spectral parameters in our corresponding spectral resolutions within $L^{2}(\Omega)$. Our models have two advantages: One, they encompass realistic features from quantum theory, from acoustic wave equations and their obstacle scattering; as well as from harmonic analysis. Secondly, each choice of the parameters in our models, $n\in\mathbb{N}$, $B\in U(n)$, and interval configuration, allows for explicit computations, and even for closed-form formulas: Computation of spectral resolutions, of generalized eigenfunctions in $L^{2}(\Omega)$ for the continuous part of spectrum, and for scattering coefficients. Our models further allow us to identify embedded point-spectrum (in the continuum), corresponding, for example, to bound-states in scattering, to trapped states, and to barriers in quantum scattering. The possibilities for the discrete atomic part of spectrum includes both periodic and non-periodic distributions.