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Global Bounds of Schrödinger Heat Kernels with Negative Potentials

✍ Scribed by Qi S. Zhang

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
200 KB
Volume
182
Category
Article
ISSN
0022-1236

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✦ Synopsis

We obtain global in time bounds for the heat kernel G of the Schro dinger operator L=&2+V. The potential V satisfies V(x)t &CÂd(x) b near infinity with b # (0, ). The result can be described as follows. Suppose L is positive and b=2. Then G=G(x, t; y, 0) has a global upper bound which is a standard Gaussian times a polynomial function of x, y, t. However when L is not nonnegative and negative eigenvalues exist G=G(x, t; 0, 0) is bounded below by a standard Gaussian times e ct with c>0. In other words, the growth rate of G(x, t; 0, 0) is comparable with the heat kernel of &2&c, regardless how fast V decays near infinity.

2001 Academic Press

The presence of the functions e &V& t and e &&V& t masks a wealth of information and makes the bounds less useful when t Ä .

An important question arises: Does there exist a global estimate on the heat kernel of &2+V, which reveals an explicit dependence on the potential V ?

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