07-131 E.Caliceti, s.Graffi, J.Sjoestrand
$PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum (53K, Latex 2e) May 29, 07
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Abstract. Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. $\ds H_0=\,a^\ast_1a_1+\ldots +a^\ast_da_d+d/2$ 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 ope\-rator $\ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1$ has real discrete spectrum but is not diagonalizable.

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