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Kernel estimates and the regularized trace of the semigroup generated by a potential perturbation of the bi-Laplacian. (English. Russian original) Zbl 1227.35128

Russ. Math. Surv. 66, No. 3, 635-636 (2011); translation from Usp. Mat. Nauk. 66, No. 3, 205-206 (2011).
From the text: The purpose of this note is to present asymptotic formulae and bounds for the integral kernel \(G_V(x,y,t)\) of the semigroup \(U(t)= \exp(tH_V)\) with generator \(H_V=-P(i\nabla)+V(x)\), where \(P(\xi)= |\xi|^4/4\) and \(V(x)\in L_1(\mathbb R^3)\). The bounds established and the techniques involved can be extended to the case of a positive-definite form, \(P(\xi)\) and a bounded integrable potential in any dimension.

MSC:

35C15 Integral representations of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
32W30 Heat kernels in several complex variables
35K08 Heat kernel
47D08 Schrödinger and Feynman-Kac semigroups
35A08 Fundamental solutions to PDEs

Keywords:

integral kernel
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