Green function for Schrödinger operator and conditioned Feynman-Kac gauge

✍ Scribed by Zhongxin Zhao

• Publisher: Elsevier Science
• Year: 1986
• Tongue: English
• Weight: 704 KB
• Volume: 116
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1016/s0022-247x(86)80001-4

✦ Synopsis

Consider the Schr6dinger operator H= (A/2)+ q in a bounded C 1,~ domain D with q e K~ °c. (D, q) satisfies the condition: sup[spec((A/2) + q)lD] < 0. Let V and G denote the Green functions in D for H and L1/2, respectively. We prove that

where (xr) is the conditioned Brownian motion starting at x and ending at y with the life time T, and V/G can be extended to a continuous and positive function on D x D. Specifically, we give the comparison theorem: There exists a constant C = C(D, q) > 0 such that 1G(x,y)<~V(x,y)<~CG(x,y), x, yED.

Another result is that (8 V/Sn:)(x, z), x ~ D, z ~ OD exists and represents the Poisson kernel for the Dirichlet problem corresponding to Schr6dinger operator H.