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Gaussian decay for the eigenfunctions of a Schrödinger operator with magnetic field constant at infinity. (English) Zbl 0898.34077
Let \(P\) be a Schrödinger operator with a magnetic field, \(P=(D_x-A(x))^ 2 +V(x)\), and \(A(x)=Bx+r(x)\), where the matrix \(B\) is antisymmetric, \(x=(x',x'')\), \(x' \in \text{Ran }B\), \(x''\in \text{Ker }B\). The functions \(V(x)\) and \(r(x)\) are smooth and have analytic bounded extension in a sector. Let \(u\) be the eigenfunction of \(P\) corresponding to an eigenvalue \(E\) below the bottom \(\lambda_o\) of the essential spectrum. The author proves that for every \(\alpha''<\sqrt{\lambda_o-E}\) there exist constants \(\alpha'>0\), \(c>0\) such that \(| u(x)| \leq c \exp(-\alpha'| x'| ^2-\alpha''| x''|)\). A similar estimate is proven for the semiclassical case.

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
78A30 Electro- and magnetostatics
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
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