Related papers: PT invariant Non-Hermitian Potentials with Real QE…
We show that the quasi-exactly solvable eigenvalues of the Schr\"odinger equation for the PT-invariant potential $V(x) = -(\zeta \cosh 2x -iM)^2$ are complex conjugate pairs in case the parameter M is an even integer while they are real in…
We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex PT-invariant, double sinh-Gordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES…
We show that the complex $\cal PT$-symmetric periodic potential $V(x) = - ({\rm i} \xi \sin 2x + N)^2$, where $\xi$ is real and $N$ is a positive integer, is quasi-exactly solvable. For odd values of $N \ge 3$, it may lead to exceptional…
We consider a PT Symmetric Partner to Khare Mandal's recently proposed non-Hermitian potential with complex eigen values. Our potential is Quasi-Exactly solvable and is shown to possess only real eigen values.
Comparison between the exact value of the spectral zeta function, $Z_{H}(1)=5^{-6/5}[3-2\cos(\pi/5)]\Gamma^2(1/5)/\Gamma(3/5)$, and the results of numeric and WKB calculations supports the conjecture by Bessis that all the eigenvalues of…
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…
A new two-parameter family of quasi-exactly solvable quartic polynomial potentials $V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x$ is introduced. Until now, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial…
We solve the eigenvalue spectra for two quasi exactly solvable (QES) Schr\"odinger problems defined by the potentials $V(x;\gamma,\eta) = 4\gamma^{2}\cosh^{4}(x) + V_{1}(\gamma,\eta) \cosh^{2}(x) + \eta \left( \eta-1 \right)\tanh^{2}(x)$…
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 propose the notion of $E_{2}$-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the…
A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in closed form. An entirely new class of QES Hamiltonians having sextic polynomial…
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…
Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schr\"odinger operator is Fermi isospectral to the zero potential. This provides a negative…
We show that there are two different families of (weakly) orthogonal polynomials associated to the quasi-exactly solvable Razavy potential $V(x)=(\z \cosh 2x-M)^2$ ($\z>0$, $M\in\mathbf N$). One of these families encompasses the four sets…
We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters…
Using the NU method [A.F.Nikiforov, V.B.Uvarov, Special Functions of Mathematical Physics, Birkhauser,Basel,1988], we investigated the real eigenvalues of the complex and/or $PT$- symmetric, non-Hermitian and the exponential type systems,…
The one-dimensional Schroedinger's equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential's parameters, we show that the decatic polynomial potential…
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form $H=p^2+V(x)$, where $V(x)$ is odd in $x$, pure imaginary, and vanishes as $|x|\to\infty$. Five…
The supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian deformed Morse and P\"{o}schl-Teller potentials are obtained by solving the Schr\"{o}dinger equation. The Hamiltonian hierarchy method is used to get the real energy…
Searching for non-Hermitian (parity-time)$\mathcal{PT}$-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian…