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Dancer, E. N.; López-Gómez, J.

Semiclassical analysis of general second order elliptic operators on bounded domains. (English) Zbl 0965.35105

Trans. Am. Math. Soc. 352, No. 8, 3723-3742 (2000).
The authors analyze the limiting behavior (as \(h\to 0\)) of the fundamental energy and the ground state of the following linear boundary value problem: \[ h^2L\phi+V(x)\phi=\sigma\phi \quad \text{in } D, \qquad \phi=0 \quad \text{ on } \partial D, \] where \(D\) is a bounded domain of \({\mathbb R}^N\) with \(C^{0,1}\) boundary. \(L\) is a second-order uniformly elliptic operator of the form \[ L = \sum_{i,j=1}^N a_{ij}(x)\partial_i\partial_j +\sum_{i=1}^N b_i(x)\partial_i +c(x), \] with \[ a_{ij}\in C(D), \qquad b_i,c \in L^\infty (D) \] and \(V\in L^\infty (D)\).
It is shown that \(\lim_{h\to 0} \sigma_1^D[h^2L+V]= \text{ess}\inf_D V\) and that the limit of ground states is concentrated in the region \(\{x\in D:V(x)> \text{ess}\inf_D\;V \}\). Decay rates are also investigated in the given general setting.
Reviewer: Michael Perelmuter (Kyïv)

Cited in 17 Documents

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35B25 Singular perturbations in context of PDEs

Keywords:

fundamental energy; ground state; decay rates
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References:

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