07-113 Vladimir S. Buslaev and Catherine Sulem

Linear adiabatic dynamics generated by operators with continuous spectrum.I (297K, pdf) May 4, 07

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Abstract. We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation $$ i\varepsilon \partial_t \psi = A(t) \psi, \quad A(t) =A_0 +V(t),\quad \psi(0) = \psi_0, $$ in the limit $\varepsilon \to 0$. A case of special interest is when the operator $A(t)$ has continuous spectrum and the initial data $\psi_0$ is, in particular, an improper eigenfunction of the continuous spectrum of $A(0)$. Under suitable assumptions on $A(t)$, we derive a formal asymptotic solution of the problem whose leading order has an explicit representation. A key ingredient is a reduction of the original Cauchy problem to the study of the semiclassical pseudo-differential operator ${\lM}= M(t, i\varepsilon\partial_t)$ with compact operator-valued symbol $M(t,E) = V_1(t)(A_0-EI)^{-1} V_2(t)$ , $V(t) =V_2(t)V_1(t),$ and an asymptotic analysis of its spectral properties. We illustrate our approach with a detailed presentation of the example of the Schr\"odinger equation on the axis with the $\delta$-function potential: $A(t) =-\partial_{xx} + \alpha(t) \delta(x).$

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