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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.

##### MSC:
 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
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