Subelliptic resolvent estimates for non-self-adjoint semiclassical Schrödinger operators

Abstract

We examine semiclassical magnetic Schrödinger operators with complex electric potentials. Under suitable conditions on the magnetic and electric potentials, we prove a resolvent estimate for spectral parameters in an unbounded parabolic neighborhood of the imaginary axis.

Semigroup expansions for non-selfadjoint Schrödinger operators

“…See [5], [22], [23], and also [11]. The recent results of [3], [4] establish polynomial resolvent estimates for a broad class of non-selfadjoint semiclassical magnetic Schrödinger operators in an unbounded parabolic region near the imaginary axis, and these works have been the starting point for this note. Let us now proceed to describe the assumptions and state the main results.…”

“…The purpose of this note is to apply the spectral results of [13], [14] together with the resolvent bounds obtained in [3], [4], to establish an expansion for the evolution semigroup associated to a class of semiclassical non-selfadjoint magnetic Schrödinger operators, in the limit of large times. It is well known that the exponential decay of a contraction semigroup on a Hilbert space is closely connected to the resolvent estimates for the corresponding semigroup generator, see [9], [19], and the references given there.…”

“…To this end, let us now proceed to introduce some additional assumptions, which will allow us to obtain a precise description of the spectrum of P in a disc around the origin of radius O(h), and an estimate for the resolvent of P in the disc, as well as further away from this set, as a consequence of the results of [13], [14], [4]. See also [11].…”

“…Indeed, one of the assumptions made in [11,Theorem 1.4], namely assumption (H3) there, demands that the principal symbol p = p 1 +ip 2 of the operator in question should be such that ∂ α p 1 = O(1), |α| ≥ 2. This assumption is not satisfied in the magnetic Schrödinger case, for p given by (1.3) and indeed, a crucial role in our proof of Theorem 1.2 is played by the recent work [4] of the first-named author, where resolvent estimates of subelliptic type are established for the operator P in (1.9), in an unbounded parabolic neighborhood of the imaginary axis, away from a bounded region near the origin.…”

“…While the techniques of [13], [14], [3], [4], on which this note is based, are somewhat different, here too we rely essentially on the quantitative version of the Gearhart-Prüss theorem, due to [9].…”