Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

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.Comment: 20 page

PTPT symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum

Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. \ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2 is the quantum harmonic oscillator with rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real coupling constant. We show that if $|g|<\rho$, $\rho$ being an explicitly determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we show that the operator \ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1 has real discrete spectrum but is not diagonalizable.Comment: 20 page