Canonical expansion of -symmetric operators and perturbation theory

doi:10.1088/0305-4470/37/6/019

IOPscience

Journal of Physics A: Mathematical and General

Quick search Find article

Quick search

All journals This journal only

Find article
Volume number: Issue number (if known): Article or page number:

Journal of Physics A: Mathematical and General Volume 37 Number 6 Create an alert RSS this journal

E Caliceti and S Graffi 2004 J. Phys. A: Math. Gen. 37 2239 doi:10.1088/0305-4470/37/6/019

Canonical expansion of ${{\cal P}{\cal T}}$ -symmetric operators and perturbation theory

E Caliceti¹ and S Graffi²

Show affiliations

Tag this article Full text PDF (161 KB)

Abstract References Cited By

Let H be any ${{\cal P}{\cal T}}$ -symmetric Schrödinger operator of the type − planck ²Δ + (x²₁ + sdot sdot sdot + x²_d) + igW(x₁, ..., x_d) on $L^2({\bb R}^d)$ , where W is any odd homogeneous polynomial and $g\in{\bb R}$ . It is proved that ${\cal P} H$ 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 μ_j of H through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues λ_j of H by Weyl's inequalities.