Karner, Gunther

The Schrödinger equation on non-stationary domains

Electron. J. Differ. Equ. 1998, Paper No. 20, 20 p. (1998)


Summary: We investigate the dynamical effects of non-stationary boundaries on the stability of a quantum Hamiltonian system described by a periodic family $$\Omega (a) = \left\{(t,x)\in{\Bbb R}^2 : x\in{(a(t),\infty)}, a\in{\cal C}^3({\Bbb R}),a(t)=a(t+k\Gamma), k\in{\Bbb Z}\right\}\,,$$ 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.

Mathematics Subject Classification

35P05, 81Q10


stability of dense point spectra, boundary induced perturbations, Krein's resolvent formula