Related papers: PT-symmetric potentials having continuous spectra

The spectrum of a one-dimensional Hamiltonian with potential $V(x)=ix^2$ for negative $x$ and $V(x)=-ix^2$ for positive $x$ is analyzed. The Schr\"odinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of…

The $\cal PT$-symmetric complexified Scarf II potential $V(x)= - V_1 \sech^{2}x + {\rm i} V_2 \sech x \tanh x$, $V_1>0$ , $V_{2}\neq 0$ is revisited to study the interplay among its coupling parameters. The existence of an isolated real and…

The spectrum of complex PT-symmetric potential, $V(x)=igx$, is known to be null. We enclose this potential in a hard-box: $V(|x| \ge 1) =\infty $ and in a soft-box: $V(|x|\ge 1)=0$. In the former case, we find real discrete spectrum and the…

The spectrum of the Hermitian Hamiltonian $H=p^2+V(x)$ is real and discrete if the potential $V(x)\to\infty$ as $x\to\pm\infty$. However, if $V(x)$ is complex and PT-symmetric, it is conjectured that, except in rare special cases, $V(x)$…

The PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\epsilon$ ($\epsilon$ real) exhibits a phase transition at $\epsilon=0$. When $\epsilon\geq0$, the eigenvalues are all real, positive, discrete, and grow as $\epsilon$ increases. However, when…

Version 1: The well known Eckart's singular s-wave potential is PT-symmetrically regularized and continued to the whole real line. The new model remains exactly solvable and its bound states remain proportional to Jacobi polynomials. Its…

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian $H=p^2+x^2(ix)^\epsilon$ has real, positive, and discrete eigenvalues for all $\epsilon\geq 0$. These eigenvalues are analytic continuations of the harmonic-oscillator…

The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of ${\cal PT}$ symmetry, one obtains new infinite classes of complex Hamiltonians…

Non-Hermitian PT-symmetric quantum-mechanical Hamiltonians generally exhibit a phase transition that separates two parametric regions, (i) a region of unbroken PT symmetry in which the eigenvalues are all real, and (ii) a region of broken…

We investigate the parametric evolution of the real discrete spectrum of several complex PT symmetric scattering potentials of the type $V(x)=-V_1 F_e(x) + i V_2 F_o(x), V_1>0, F_e(x)>0$ by varying $V_2$ slowly. Here $e,o$ stand for even…

We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum.…

The Schroedinger eigenvalue problems for the Whittaker-Hill potential $Q_{2}(x)=\tfrac{1}{2} h^2\cos4x+4h\mu\cos2x$ and the periodic complex potential $Q_{1}(x)=\tfrac{1}{4}h^2{\rm e}^{-4ix}+2h^2\cos2x$ are studied using their realizations…

We propose a new solvable one-dimensional complex PT-symmetric potential as $V(x)= ig~ \mbox{sgn}(x)~ |1-\exp(2|x|/a)|$ and study the spectrum of $H=-d^2/dx^2+V(x)$. For smaller values of $a,g <1$, there is a finite number of real discrete…

We investigate complex PT-symmetric potentials, associated with quasi-exactly solvable non-hermitian models involving polynomials and a class of rational functions. We also look for special solutions of intertwining relations of SUSY…

Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential,…

We discuss three Hamiltonians, each with a central-field part $H_{0}$ and a PT-symmetric perturbation $igz$. When $H_{0}$ is the isotropic Harmonic oscillator the spectrum is real for all $g$ because $H$ is isospectral to $H_{0}+g^{2}/2$.…

This paper investigates finite-dimensional representations of PT-symmetric Hamiltonians. In doing so, it clarifies some of the claims made in earlier papers on PT-symmetric quantum mechanics. In particular, it is shown here that there are…

The fourth, missing example of an exactly solvable complex potential with PT symmetry V(x) = [V(-x)]^* defined on a bent contour and leading, at the real energies, to the Jacobi polynomial wave functions is found in a generalized Hulthen…

So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf II potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: $E^n_{\pm}(\lambda)$ with $n=0,1,2...$.…

It is shown that the PT symmetric Hamiltonian with the periodic potential V(x) = 4 cos^2 x + 4 i V_0 sin 2x can be mapped into a Hermitian Hamiltonian for $V_0<0.5$, by a similarity transformation. It is also shown that there exist a second…