Quasi-exactly solvable quartic potentials with centrifugal and Coulombic terms

PT symmetric complex potential V(r) = - r^4 + i a r^3 + b r^2 + i c r + i d/r + e/r^2 is studied. Arbitrarily large multiplets of its closed bound-state solutions with real energies are shown obtainable quasi-exactly (i.e., with a certain relationship between their charges and energies) from a single underlying finite-dimensional secular equation.Comment: 13 pages, 1 figure, submitted to J. Phys. A: Math. Ge

Quantum catastrophes: a case study

The bound-state spectrum of a Hamiltonian H is assumed real in a non-empty domain D of physical values of parameters. This means that for these parameters, H may be called crypto-Hermitian, i.e., made Hermitian via an {\it ad hoc} choice of the inner product in the physical Hilbert space of quantum bound states (i.e., via an {\it ad hoc} construction of the so called metric). The name of quantum catastrophe is then assigned to the N-tuple-exceptional-point crossing, i.e., to the scenario in which we leave domain D along such a path that at the boundary of D, an N-plet of bound state energies degenerates and, subsequently, complexifies. At any fixed $N \geq 2$, this process is simulated via an N by N benchmark effective matrix Hamiltonian H. Finally, it is being assigned such a closed-form metric which is made unique via an N-extrapolation-friendliness requirement.Comment: 23 p

CPT-symmetric discrete square well

A new version of an elementary PT-symmetric square well quantum model is proposed in which a certain Hermiticity-violating end-point interaction leaves the spectrum real in a large domain of couplings $\lambda\in (-1,1)$. Within this interval we employ the usual coupling-independent operator P of parity and construct, in a systematic Runge-Kutta discrete approximation, a coupling-dependent operator of charge C which enables us to classify our P-asymmetric model as CPT-symmetric or, equivalently, hiddenly Hermitian alias cryptohermitian.Comment: 12 pp., presented to conference PHHQP IX (http://www.math.zju.edu.cn/wjd/

Supersymmetric quantum mechanics living on topologically nontrivial Riemann surfaces

Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators $H^{(\pm)}$ is chosen antilinear. Secondly, both these components of a super-Hamiltonian ${\cal H}$ are defined along certain topologically nontrivial complex curves $r^{(\pm)}(x)$ which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map ${\cal T}$ between "tobogganic" partner curves $r^{(+)}(x)$ and $r^{(-)}(x)$ is emphasized.Comment: 14p

Non linear pseudo-bosons versus hidden Hermiticity. II: The case of unbounded operators

Parallels between the notions of nonlinear pseudobosons and of an apparent non-Hermiticity of observables as shown in paper I (arXiv: 1109.0605) are demonstrated to survive the transition to the quantum models based on the use of unbounded metric in the Hilbert space of states.Comment: 21 p

Matching method and exact solvability of discrete PT-symmetric square wells

Discrete PT-symmetric square wells are studied. Their wave functions are found proportional to classical Tshebyshev polynomials of complex argument. The compact secular equations for energies are derived giving the real spectra in certain intervals of non-Hermiticity strengths Z. It is amusing to notice that although the known square well re-emerges in the usual continuum limit, a twice as rich, upside-down symmetric spectrum is exhibited by all its present discretized predecessors.Comment: 25 pp, 3 figure

Identification of observables in quantum toboggans

Quantum systems with real energies generated by an apparently non-Hermitian Hamiltonian may re-acquire the consistent probabilistic interpretation via an ad hoc metric which specifies the set of observables in the updated Hilbert space of states. The recipe is extended here to quantum toboggans. In the first step the tobogganic integration path is rectified and the Schroedinger equation is given the generalized eigenvalue-problem form. In the second step the general double-series representation of the eligible metric operators is derived.Comment: 25 p

Comment on `Solution of the Dirac equation for the Woods-Saxon potential with spin and pseudospin symmetry' [J. Y. Guo and Z-Q. Sheng, Phys. Lett. A 338 (2005) 90]

Out of the four bound-state solutions presented in loc. cit., only one (viz., the spin-symmetric one, in the low-mass regime) is shown compatible with the physical boundary conditions. We clarify the problem, correct the method and offer another, "missing" (viz., pseudospin-symmetric) new solution with certain counterintuitive "repulsion-generated" property.Comment: 6 p

Maximal couplings in PT-symmetric chain-models with the real spectrum of energies

The domain ${\cal D}$ of all the coupling strengths compatible with the reality of the energies is studied for a family of non-Hermitian $N$ by $N$ matrix Hamiltonians $H^{(N)}$ with tridiagonal and ${\cal PT}-$symmetric structure. At all dimensions $N$, the coordinates are found of the extremal points at which the boundary hypersurface $\partial {\cal D}$ touches the circumscribed sphere (for odd $N=2M+1$) or ellipsoid (for even $N=2K$).Comment: 18 pp., 2 fig