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 the 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 the spectrum includes both periodic and non-periodic distributions.