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 with 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 the sharpness of our results. The only previously known estimates of this type pertain to the classical Schrödinger operator −Δ + V .
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Received by the editors February 7, 2018, and, in revised form, August 22, 2018. 2010 Mathematics Subject Classification. Primary 35J10; Secondary 35J15, 35J08, 35B40, 35E05, 35Q60, 35R03, 46N20, 47N20, 81Q10, 81Q12. The first author was supported in part by NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089, the Simons Fellowship, and Simons Foundation grant 563916, SM. Both authors would like to thank the Mathematical Sciences Research Institute (NSF grant DMS 1440140) for its support and hospitality.
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