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Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum. (English) Zbl 0776.35056

Summary: The Schrödinger equation in the adiabatic limit when the Hamiltonian depends analytically on time and possesses for any fixed time two nondegenerate eigenvalues \(e_ 1(t)\) and \(e_ 2(t)\) bounded away from the rest of the spectrum is considered herein. An approximation of the evolution called superadiabatic evolution is constructed and studied. Then a solution of the equation which is asymptotically an eigenfunction of energy \(e_ 1(t)\) when \(t\to-\infty\) is considered. Using superadiabatic evolution, an explicit formula for the transition probability to the eigenstate of energy \(e_ 2(t)\) when \(t\to+\infty\), provided the two eigenvalues are sufficiently isolated in the spectrum, is derived. The end result is a decreasing exponential in the adiabaticity parameter times a geometrical prefactor.

MSC:

35Q40 PDEs in connection with quantum mechanics
35P15 Estimates of eigenvalues in context of PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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