The Schrodinger Equation on Non-Stationary Domains




Karner, Gunther

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Southwest Texas State University, Department of Mathematics


We investigate the dynamical effects of non-stationary boundaries on the stability of a quantum Hamiltonian system described by a periodic family {H(γ, t), t ∈ [0, Γ], Γ > 0} of Sturm-Liouville operators, a Schrödinger equation i∂tψ = H(γ, t)ψ defined on Ω(α) = {(t, x) ∈ ℝ2 : x ∈ (α(t), ∞), α ∈ C3(ℝ), α(t) = α(t + kΓ), k ∈ ℤ}, as well as boundary conditions at x = α(t) modeled by the Γ-periodic function γ. Employing extended Hilbert space methods, stability conditions for the spectra of the evolution operators U(α, γ, Γ, 0) to the families {H(γ, 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 U(a, γˆ, Γ, 0)) implies pure point U(a, γ, Γ, 0) for all γ ∈ C1(ℝ), α ∈ C3(ℝ), whereas in case of infinitely degenerate σpp (U(α, γˆ, Γ, 0)) the existence of σac (U(α, γ, Γ, 0)) ≠ 0, respectively σsc(U (a, γ, Γ, 0)) ≠ ∅, is possible.



Stability of dense point spectra, Boundary induced perturbations, Krein's resolvent formula


Karner, G. (1998). The Schrodinger equation on non-stationary domains. <i>Electronic Journal of Differential Equations, 1998</i>(20), pp. 1-20.


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