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arXiv:0705.4218 · Journal of Physics A Mathematical and Theoretical

symmetric non-self-adjoint operators, diagonalizable and non-diagonalizable, with a real discrete spectrum

Emanuela Caliceti, S. Graffi, Johannes Sjöstrand

Published Aug 1, 2007• 29 citations
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Abstract

Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. $\ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2$ 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.

DOI
10.1088/1751-8113/40/33/014

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We notice contradictory statements of Z.ahmed on PT\mathcal{PT}PT symmetry potential V=x24+igx∣x∣V=\frac{x^{2}}{4}+igx|x|V=4x2​+igx∣x∣ (g=2) The curve (dash-dot) shows only one zero whose corresponding energy eigenvalue is 1.72.This graphical representation contradicts the discussion (presented below the graph )which claims number of real discrete states between $0\leg\le2.3$ are 3.as g=2g=2g=2 <g=2.3g=2.3g=2.3 ,one should get number of real discrete states 3 instead of 1.However present calculation matrix diagonalisation pethod reflects 3.secondly the statement on complex Harmonic Oscillator VH=x@+ixV_{H}=x^{@}+ixVH​=x@+ix (which entails complete real discrete spectrum) as a singular example is not acceptable as we notice many complex ∣mathcalPT|mathcal{PT}∣mathcalPT symmetric potentials can be constructed ,which would reflect infinitr real discrete energy eigenvalues.
A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the PTPTPT-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.
Abstract Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.