Resolvent Expansion and Time Decay of the Wave Functions for Two-Dimensional Magnetic Schrödinger Operators

Abstract

Abstract. We consider two-dimensional Schrödinger operators H(B, V ) given by equation (1.1) below. We prove that, under certain regularity and decay assumptions on B and V , the character of the expansion for the resolvent (H(B, V )−λ) −1 as λ → 0 is determined by the flux of the magnetic field B through R 2 . Subsequently, we derive the leading term of the asymptotic expansion of the unitary group e −itH(B,V ) as t → ∞ and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e −itH(0,V ) .

Improved time-decay for a class of scaling critical electromagnetic Schrödinger flows

“…On the other hand, by combining Lemma 2.1 and equation (1.9) with (2.4) and (2.15) we obtain 20) where the convergence of the series is guaranteed by (2.6). This together with (2.19) implies that sup (x,y)∈R 2n…”

“…, being µ 1 the first eigenvalue of the angular operator L as above. The results of [14] where then extended to a wide class of regular magnetic fields with finite flux in [20].…”

The Hardy inequality and the heat equation with magnetic field in any dimension

“…An alternative approach would be to establish a Laurent expansion of the resolvent at zero energy and apply it to the heat semigroup with help of functional calculus. This approach has been recently undertaken by Kovařík [33] to study a large-time behaviour of the Schrödinger equation in the present magnetic setting for d = 2 (the established Laurent expansion can be used for the heat semigroup as well). More generally, the low-energy properties of the resolvent are subject of an intensive study in geometric scattering theory; see [50], [26], [27] and [28] for recent developments in a much more general geometric setting.…”

Spectral properties and time decay of the wave functions of Pauli and Dirac operators in dimension two

“…This vector potential A 0 was used already in [31] in the study of magnetic Schrödinger operators. If we now make a suitable choice of the gauge in the Pauli operator, see the remark below, then the coefficients of…”