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Semiclassical resolvent estimates for Schrödinger matrix operators with eigenvalues crossing
Author(s) -
Jecko Thierry
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310076
Subject(s) - semiclassical physics , resolvent , mathematics , eigenvalues and eigenvectors , bounded function , operator (biology) , codimension , matrix (chemical analysis) , mathematical analysis , mathematical physics , pure mathematics , quantum mechanics , physics , biochemistry , materials science , repressor , transcription factor , composite material , quantum , gene , chemistry
For semiclassical Schrödinger 2×2–matrix operators, the symbol of which has crossing eigenvalues, we investigate the semiclassical Mourre theory to derive bounds O ( h −1 ) ( h being the semiclassical parameter) for the boundary values of the resolvent, viewed as bounded operator on weighted spaces. Under the non–trapping condition on the eigenvalues of the symbol and under a condition on its matricial structure, we obtain the desired bounds for codimension one crossings. For codimension two crossings, we show that a geometrical condition at the crossing must hold to get the existence of a global escape function, required by the usual semiclassical Mourre theory.