Linear adiabatic dynamics generated by operators with continuous spectrum. I
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Bibliographic record
Abstract
We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation iε ∂ t ψ=A(t)ψ, A(t)=A 0 +V(t), ψ(0)=ψ 0 , in the limit ε→0. A case of special interest is when the operator A(t) has continuous spectrum and the initial data ψ 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 ℳ=M(t, iε ∂ 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 δ-function potential: A(t)=−∂ xx +α(t)δ(x).
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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