Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the origin

Publication date: 30 January 2023

Abstract: We consider, for

h, E > 0

, the semiclassical Schr"odinger operator

- h^{2} D e l t a + V - E

in dimension two and higher. The potential

V

, and its radial derivative

d e l l_{r} V

are bounded away from the origin, have long-range decay and

V

is bounded by

r^{- d e l t a}

near the origin while

d e l l_{r} V

is bounded by

r^{- 1 - d e l t a}

, where

0 l e q d e l t a l e q 4 (s q r t 2 - 1)

. In this setting, we show that the resolvent bound is exponential in

h^{- 1}

, while the exterior resolvent bound is linear in

h^{- 1}

.