1202.4120 (Palle Jorgensen et al.)

Spectral Theory of Multiple Intervals    [PDF]

Palle Jorgensen, Steen Pedersen, Feng Tian

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.

View original: http://arxiv.org/abs/1202.4120