Semiclassical resolvent estimates for Schrödinger matrix operators with eigenvalues crossing

✍ Scribed by Thierry Jecko

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 294 KB
• Volume: 257
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310076

✦ Synopsis

Abstract

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.