Exponential decay estimates for fundamental solutions of Schrödinger-type operators

Abstract

In the present paper we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators L = −(∇− ia) T A(∇ − ia) + V . The latter class includes, in particular, the magnetic Schrödinger operator − (∇ − ia) 2 +V and the generalized electric Schrödinger operator −divA∇+ V . Our exponential decay bounds rest on a generalization of the Fefferman-Phong uncertainty principle to the present context and are governed by the Agmon distance associated to the corresponding maximal function. In the presence of a scale-invariant Harnack inequality, for instance, for the generalized electric Schrödinger operator with real coefficients, we establish both lower and upper estimates for fundamental solutions, thus demonstrating sharpness of our results. The only previously known estimates of this type pertain to the classical Schrödinger operator −∆ + V [29] .

Second order elliptic partial differential equations driven by Lévy white noise

“…For α > 0 and x, y ∈ R d , x = y it follows from the triangle inequality that (observe that d = 3 and hence the Lebesgue measure of a ball with radius r is 4π 3 ) As (x n ) n≥M is bounded we can find an integrable majorant on R d \ B r 0 (x). We know from [10], chapter 7 that E is continuous and by Lebesgue's dominated convergence theorem we obtain Hence, we obtain the assertion by Theorem 6.…”

“…In the following we define the maximum function m and Agmon distance γ of the potential V to apply the estimates of the fundamental solution of the generalized electric Schrödinger operator shown in [14] and [10]. (R d ) with ω > 0 a.e.…”

“…where k, C > 0, see [10], Corollary 6.16, page 40. From now on the constant k > 0 is fixed and such that (22) is satisfied.…”

“…Let x 0 ∈ R d and (x n ) n∈N be a sequence such that x n → x 0 as n → ∞. Let 0 < 2 x 0 − x n < r 0 for all n ≥ M, M ∈ N. We calculate that E(x 0 , •) − E(x n , •) L 1 (R d ) ≤ E(x 0 , •) − E(x n , •) L 1 (B r 0 (x)) + E(x 0 , •) − E(x n , •) L 1 (R d \B r 0 (x)) .It was shown in[10], Lemma 3.12, page 14 thatm(x, V ) ≥ C m(0, V ) (1 + x m(0, V )) κ for all x ∈ R d (28)for a constant 0 < κ < 1, hence there exists an ε > 0 such that it follows from (27) that|E(x n , y)| ≤ C 1 e −C 2 (1+ε x n −y ) θ x n − y 2−dfor every n ∈ N 0 . Therefore, we obtain thatE(x 0 , •) − E(x n , •) L 1 (B r 0 (x)) ≤ 2r 0 (0) C 1 e −C 2 (1+ε y ) θ y 2−d λ d (dy).…”

Unboundedness of potential dependent Riesz transforms for totally irregular measures

“…In the paper [21], Shen obtained sharp estimates for the fundamental solution of the Schrödinger operator expressed in terms of the Agmon distance. More recently, Mayboroda and Poggi in [16] have generalised these sharp estimates to the operator L V…”