The Schrodinger equation on non-stationary domains

We investigate the dynamical effects of non-stationary boundaries on the stability of a quantum Hamiltonian system described by a periodic family ${H(gamma,t), tin[0,Gamma],Gamma>0}$ of Sturm-Liouville operators, a Schr"odinger equation $ipartial_{t}psi = H(gamma,t)psi$ defined on $$Omega (a) = left{(t,x)in{Bbb R}^2 : xin{(a(t),infty)}, ain{cal C}^3({Bbb R}),a(t)=a(t+kGamma), kin{Bbb Z} ight},,$$ 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 ${H(gamma,t)}$ under perturbations induced by variations of boundary oscillations, respectively conditions, are derived.