A Semi-classical Trace Formula for Schrödinger Operators in the Case of a Critical Energy Level

✍ Scribed by David Khuat-Duy

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 602 KB
• Volume: 146
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3035

✦ Synopsis

Let H =&( 2 Â2) 2+V(x) be a Schro dinger operator on R n , with smooth potential V(x) Ä + as |x| Ä + . The spectrum of H is discrete, and one can study the asymptotic of the smoothed spectral density

We shall investigate the case where E is a critical value of the symbol H of H and, extending the work of Brummelhuis, Paul and Uribe in [3], we will prove the existence of a full asymptotic expansion for ( in terms ofand ln and compute the leading coefficient. We will consider new Weyl-type estimates for the counting function:

1997 Academic Press CONTENTS 1. Introduction and main statements. 2. A result concerning the classical dynamic. 3. Microlocal decomposition and Proof of the main statements. 4. Proof of Theorem 3.2. 4.1. The phase function. 4.2. The asymptotic expansion. 4.3. Calculation of the leading term. 5. The case with an observable. 6. An example for the case &=0. Appendix A. Determination of the density defined on 3. Appendix B. A theorem for the singularity in t=0.