Linear adiabatic dynamics generated by operators with continuous spectrum. I. (English) Zbl 1152.35336

Summary: 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 \({\mathcal M}= M(t,i\varepsilon\partial_t)\) 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 Schrödinger equation on the axis with the \(\delta\)-function potential: \(A(t)=-\partial_{xx}+ \alpha(t)\delta(x)\).


35G10 Initial value problems for linear higher-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B25 Singular perturbations in context of PDEs