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.