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Communications in Mathematical Physics

, Volume 210, Issue 2, pp 371–398 | Cite as

Large Time Behavior of¶Schrödinger Heat Kernels and Applications

  • Qi S. Zhang

Abstract:

We obtain global in time and qualitatively sharp bounds for the heat kernel G of the Schrödinger operator −Δ +V. The potential V satisfies V(x)∼±C/d(x) b near infinity with b∈ (0, 2). When V≥ 0 our result can be described as follows: G is bounded from above and below by the multiples of standard Gaussians with a weight function. If b>2 then the weight is bounded between two positive {\it constants}; if b=2, the weight is bounded between two positive functions of t, d(x) and d(y), which have polynomial decay; if b<2, the weight is bounded between two positive functions of t, d(x) and d(y), which have exponential decay. Up to now satisfactory bounds for heat kernels can only be found when b>2 or b<0. An application to a semilinear elliptic problem is also given.

Keywords

Weight Function Potential Versus Exponential Decay Large Time Positive Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Qi S. Zhang
    • 1
  1. 1.Department of Mathematics, University of Memphis, Memphis, TN 38152, USA.¶E-mail: zhang@msci.memphis.eduUS

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