an:01143857
Zbl 0898.34077
Sordoni, Vania
Gaussian decay for the eigenfunctions of a Schrödinger operator with magnetic field constant at infinity
EN
Commun. Partial Differ. Equations 23, No. 1-2, 223-242 (1998).
0360-5302 1532-4133
1998
j
34L20 78A30 34L40 47E05
eigenfunction's decay; Schrödinger operator with magnetic field
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.
G.E.Karadzhov (Sofia)