 07131 E.Caliceti, s.Graffi, J.Sjoestrand
 $PT$ symmetric nonselfadjoint operators, diagonalizable and nondiagonalizable, with real discrete spectrum
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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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