Electronic Journal of Differential Equations,
Vol. 1998(1998), No. 20, pp. 1-20.
Title: The Schrodinger equation on non-stationary domains
Author: Gunther Karner (Univ. Karlsruhe, Germany)
Abstract:
We investigate the dynamical effects of non-stationary boundaries on the
stability of a quantum Hamiltonian system described by a periodic family
$\{H(\gamma,t), t\in[0,\Gamma],\Gamma>0\}$
of Sturm-Liouville operators, a Schr\"odinger equation
$i\partial_{t}\psi = H(\gamma,t)\psi$
defined on
$$\Omega (a) = \left\{(t,x)\in{\Bbb R}^2 : x\in{(a(t),\infty)},
a\in{\cal C}^3({\Bbb R}),a(t)=a(t+k\Gamma), k\in{\Bbb Z}\right\}\,,$$
as well as boundary conditions at $x=a(t)$ modeled by the
$\Gamma$-periodic function $\gamma$.
Employing extended Hilbert space methods, stability
conditions for the spectra of the evolution operators
${\cal U}(a,\gamma,\Gamma,0)$
to the families
$\bigl\{H(\gamma,t)\}$
under perturbations induced by variations of boundary oscillations,
respectively conditions, are derived.
In particular, it is shown that the existence of a pure point finitely
degenerate realization
${\cal U}(a,\hat{\gamma},\Gamma,0))$
implies pure point
${\cal U}(a,\gamma,\Gamma,0)$
for all
$\gamma\in{\cal C}^1({\Bbb R}), a\in{\cal C}^3({\Bbb R})$,
whereas in case of infinitely degenerate
$\sigma_{pp}\bigl({\cal U}(a,\hat{\gamma},\Gamma,0)\bigr)$
the existence of
$\sigma_{{\rm ac}}\bigl({\cal U}(a,\gamma,\Gamma,0)\bigr)\neq\emptyset$,
respectively
$\sigma_{sc}\bigl({\cal U}(a,\gamma,\Gamma,0)\bigr)\neq\emptyset$,
is possible.
Submitted December 18, 1997. Published July 17, 1998.
Math Subject Classification: 35P05, 81Q10.
Key Words: Stability of dense point spectra; boundary induced perturbations;
Krein's resolvent formula.