Sharp Inequalities for Heat Kernels of Schrödinger Operators and Applications to Spectral Gaps

✍ Scribed by Rodrigo Bañuelos; Pedro J. Méndez-Hernández

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 208 KB
• Volume: 176
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3611

✦ Synopsis

This paper derives inequalities for multiple integrals from which sharp inequalities for ratios of heat kernels and integrals of heat kernels of certain Schro dinger operators follow. Such ratio inequalities imply sharp inequalities for spectral gaps. The multiple integral inequalities, although very different, are motivated by the now classical Brascamp Lieb Luttinger rearrangement inequalities.

2000 Academic Press

Contents. 1. Introduction and statement of results.

2. An inequality for multiple integrals. 3. Proof of Theorem 1: Inequalities for integrals of heat kernels. 4. Proofs of Corollaries 1 and 2 and consequences for nodal lines. 5. A Schro dinger operator with a closed nodal line. 6. Pointwise inequalities for heat kennels. * V 3, D ... . When the potential is identically zero we will just write * i, D for these eigenvalues. The quantity