Brownian motion and harnack inequality for Schrödinger operators

1982 ◽  
Vol 35 (2) ◽  
pp. 209-273 ◽  
Author(s):  
M. Aizenman ◽  
B. Simon
Keyword(s):  
Brownian Motion ◽  
Harnack Inequality ◽  
Schrödinger Operators ◽  
Schrodinger Operators

scholarly journals Brownian motion and finite approximations of quantum systems over local fields

2017 ◽  
Vol 29 (05) ◽  
pp. 1750016 ◽  
Author(s):  
Erik Makino Bakken ◽  
Trond Digernes ◽  
David Weisbart
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Local Field ◽  
Schrödinger Operators ◽  
Quantum Systems ◽  
Local Fields ◽  
Schrodinger Operators ◽  
Finite Systems ◽  
Finite Approximations ◽  
Finite Approximability

We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman–Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

scholarly journals Harnack Inequality for Non-Local Schrödinger Operators

2017 ◽  
Vol 48 (4) ◽  
pp. 515-551
Author(s):  
Siva Athreya ◽  
Koushik Ramachandran
Keyword(s):  
Harnack Inequality ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Non Local

scholarly journals PATH INTEGRAL REPRESENTATION FOR SCHRÖDINGER OPERATORS WITH BERNSTEIN FUNCTIONS OF THE LAPLACIAN

2012 ◽  
Vol 24 (06) ◽  
pp. 1250013 ◽  
Author(s):  
FUMIO HIROSHIMA ◽  
TAKASHI ICHINOSE ◽  
JÓZSEF LŐRINCZI
Keyword(s):  
Brownian Motion ◽  
Path Integral ◽  
Schrödinger Operators ◽  
Integral Representations ◽  
Schrodinger Operators ◽  
Kato Class ◽  
Path Integral Representation ◽  
Bernstein Functions ◽  
Subordinate Brownian Motion ◽  
The One

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

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