Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

Details Canonical Expansion of PT-Symmetric Operators and Perturbation Theory Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

2004-01-21

arxiv.org/abs/math-ph/0401039v1

Physics

Mathematical Physics

20 pages

Scientific paper

10.1088/0305-4470/37/6/019

Let $H$ be any $\PT$ symmetric Schr\"odinger operator of the type $ -\hbar^2\Delta+(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2(\R^d)$, where $W$ is any odd homogeneous polynomial and $g\in\R$. It is proved that $\P H$ is self-adjoint and that its eigenvalues coincide (up to a sign) with the singular values of $H$, i.e. the eigenvalues of $\sqrt{H^\ast H}$. Moreover we explicitly construct the canonical expansion of $H$ and determine the singular values $\mu_j$ of $H$ through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues $\l_j$ of $H$ by Weyl's inequalities.

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