Denis Borisov, David Krejcirik
The Laplacian in an unbounded tubular neighbourhood of a hyperplane with non-Hermitian complex-symmetric Robin-type boundary conditions is investigated in the limit when the width of the neighbourhood diminishes. We show that the Laplacian converges in a norm resolvent sense to a self-adjoint Schroedinger operator in the hyperplane whose potential is expressed solely in terms of the boundary coupling function. As a consequence, we are able to explain some peculiar spectral properties of the non-Hermitian Laplacian by known results for Schroedinger operators.
http://arxiv.org/abs/1102.5051
Spectral Theory (math.SP); Mathematical Physics (math-ph)
Miloslav Znojil
An array of N subsequent Laguerre polynomials is interpreted as an eigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrum or, better said, of an exactly solvable N-site-lattice cryptohermitian Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th Laguerre polynomial. The two key problems (viz., the one of the ambiguity and the one of the closed-form construction of all of the eligible inner products which make $H$ Hermitian in the respective {\em ad hoc} Hilbert spaces) are discussed. Then, for illustration, the first four simplest, $k-$parametric definitions of inner products with $k=0,k=1,k=2$ and $k=3$ are explicitly displayed. In mathematical terms these alternative inner products may be perceived as alternative Hermitian conjugations of the initial N-plet of Laguerre polynomials. In physical terms the parameter $k$ may be interpreted as a measure of the “smearing of the lattice coordinates” in the model.
http://arxiv.org/abs/1101.1183
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Miloslav Znojil
Non-hermitian quantum graphs possessing real (i.e., in principle, observable) spectra are studied via their discretization. The discretized Hamiltonians are assigned, constructively, an elementary pseudometric and/or a more complicated metric. Both these constructions make the Hamiltonian Hermitian, respectively, in an auxiliary (Krein or Pontryagin) vector space or in a less friendly (but more useful) Hilbert space of quantum mechanics.
http://arxiv.org/abs/1101.1015
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Miloslav Znojil
In an innovative inverse-problem construction the measured, experimental energies $E_1$, $E_2$, …$E_N$ of a quantum bound-state system are assumed fitted by an N-plet of zeros of a classical orthogonal polynomial $f_N(E)$. We reconstruct the underlying Hamiltonian $H$ (in the most elementary nearest-neighbor-interaction form) and the underlying Hilbert space ${\cal H}$ of states (the rich menu of non-equivalent inner products is offered). The Gegenbauer’s ultraspherical polynomials $f_n(x)=C_n^\alpha(x)$ are chosen for the detailed illustration of technicalities.
http://arxiv.org/abs/1011.4803
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Phys. Rev. A 82 (2010) 052113
DOI:10.1103/PhysRevA.82.052113
Miloslav Znojil, Miloš Tater
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.
http://arxiv.org/abs/1011.4806
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)
H. Hernandez-Coronado, D. Krejcirik, P. Siegl
![Transmissions |T|^2 as a function of energy k2 for the step-like potential <i>v</i> with a = Pi/4, epsilon_1 = 0.2, epsilon_3 = 0.5, beta_3 = -100, beta_2 = 0, beta_1 = -120 (continuous red line), and beta_1 = -200 (dashed blue line). See [5] for animated plots of |T|^2 as a function of potential.](http://ptsymmetry.net/wp-content/uploads/2010/11/figure4.png "figure4")
We establish that a perfect-transmission scattering problem can be described by a class of parity and time reversal symmetric operators and hereby we provide a scenario for understanding and implementing the corresponding quasi-Hermitian quantum mechanical framework from the physical viewpoint. One of the most interesting features of the analysis is that the complex eigenvalues of the underlying non-Hermitian problem, associated with a reflectionless scattering system, lead to the loss of perfect-transmission energies as the parameters characterizing the scattering potential are varied. On the other hand, the scattering data can serve to describe the spectrum of a large class of Schroedinger operators with complex Robin boundary conditions.
http://arxiv.org/abs/1011.4281
Mathematical Physics (math-ph); Quantum Physics (quant-ph)