Using purely physical arguments it is claimed that for ID Schrodinger operators with complex PT- Symmatric potentials having a purely real attractive potential well and a purely imaginary repulsive part,bound state eigenvalues will be discrete and real.This has been illustrated with several potentials possessing similar properties.
We notice contradictory statements of Z.ahmed on PT symmetry potential 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=2 <g=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@+ix (which entails complete real discrete spectrum) as a singular example is not acceptable as we notice many complex ∣mathcalPT symmetric potentials can be constructed ,which would reflect infinitr real discrete energy eigenvalues.
We investigate the approximate bound state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number ℓ.
We study a class of PT-symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in \cite{BoMe14} that, when the potential is analytic, the eigenvalues stay real for a perturbation of size O(1). We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.