Joye, Alain; Pfister, Charles-Edouard Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum. (English) Zbl 0776.35056 J. Math. Phys. 34, No. 2, 454-479 (1993). 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. Cited in 16 Documents 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 Keywords:evolution of the Schrödinger equation; adiabatic regime of quantum dynamics PDFBibTeX XMLCite \textit{A. Joye} and \textit{C.-E. Pfister}, J. Math. Phys. 34, No. 2, 454--479 (1993; Zbl 0776.35056) Full Text: DOI References: [1] DOI: 10.1016/0003-4916(91)90297-L · Zbl 0875.60022 · doi:10.1016/0003-4916(91)90297-L [2] DOI: 10.1088/0305-4470/24/4/012 · Zbl 0722.60086 · doi:10.1088/0305-4470/24/4/012 [3] DOI: 10.1103/PhysRevA.44.4280 · doi:10.1103/PhysRevA.44.4280 [4] DOI: 10.1007/BF02099288 · Zbl 0755.35104 · doi:10.1007/BF02099288 [5] Garrido L. M., J. Math. Phys. 6 pp 335– (1964) [6] DOI: 10.1098/rspa.1987.0131 · doi:10.1098/rspa.1987.0131 [7] DOI: 10.1098/rspa.1990.0051 · doi:10.1098/rspa.1990.0051 [8] DOI: 10.1098/rspa.1990.0096 · doi:10.1098/rspa.1990.0096 [9] DOI: 10.1103/PhysRevA.43.3232 · doi:10.1103/PhysRevA.43.3232 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.