Category Nuclear Physics Institute in Rez

Quantum star-graph analogues of PT-symmetric square wells. II: Spectra

Miloslav Znojil

For non-Hermitian equilateral q-pointed star-shaped quantum graphs of paper I [Can. J. Phys. 90, 1287 (2012), arXiv 1205.5211] we show that due to certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.

http://arxiv.org/abs/1411.3828
Quantum Physics (quant-ph); Spectral Theory (math.SP)

Action-at-a-distance in a solvable quantum model

Miloslav Znojil

Among quantum systems with finite Hilbert space a specific role is played by systems controlled by non-Hermitian Hamiltonian matrices \(H\neq H^\dagger\) for which one has to upgrade the Hilbert-space metric by replacing the conventional form \(\Theta^{(Dirac)}=I\) of this metric by a suitable upgrade \(\Theta^{(non−Dirac)}\neq I\) such that the same Hamiltonian becomes self-adjoint in the new, upgraded Hilbert space, \(H=H\ddagger=\Theta^{−1}H^\dagger\Theta\). The problems only emerge in the context of scattering where the requirement of the unitarity was found to imply the necessity of a non-locality in the interaction, compensated by important technical benefits in the short-range-nonlocality cases. In the present paper we show that an why these technical benefits (i.e., basically, the recurrent-construction availability of closed-form Hermitizing metrics \(\Theta^{(non−Dirac)}\) can survive also in certain specific long-range-interaction models.

http://arxiv.org/abs/1410.3583
Quantum Physics (quant-ph)

Solvable non-Hermitian discrete square well with closed-form physical inner product

Miloslav Znojil

A non-Hermitian N−level quantum model with two free real parameters is proposed in which the bound-state energies are given as roots of an elementary trigonometric expression and in which they are, in a physical domain of parameters, all real. The wave function components are expressed as closed-form superpositions of two Chebyshev polynomials. In any eligible physical Hilbert space of finite dimension \(N<\infty\) our model is constructed as unitary with respect to an underlying Hilbert-space metric \(\Theta\neq I\). The simplest version of the latter metric is finally constructed, at any dimension N=2,3,…, in closed form. This version of the model may be perceived as an exactly solvable N−site lattice analogue of the \(N=\infty\) square well with complex Robin-type boundary conditions. At any \(N<\infty\) our closed-form metric becomes trivial (i.e., equal to the most common Dirac’s metric \(\Theta(Dirac)=I\)) at the special, Hermitian-Hamiltonian-limit parameters.

http://arxiv.org/abs/1409.3788
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Comment on letter “Local PT-symmetry violates the no-signaling principle” by Yi-Chan Lee et al, Phys. Rev. Lett. 112, 130404 (2014)

Miloslav Znojil

It is shown that the toy-model-based considerations of loc. cit. (see also arXiv:1312.3395) are based on an incorrect, manifestly unphysical choice of the Hilbert space of admissible quantum states. A two-parametric family of all of the eligible correct and potentially physical Hilbert spaces of the model is then constructed. The implications of this construction are discussed. In particular, it is emphasized that contrary to the conclusions of loc. cit. there is no reason to believe that the current form of the PT-symmetric quantum theory should be false as a fundamental theory.

http://arxiv.org/abs/1404.1555
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

Three solvable matrix models of a quantum catastrophe

Geza Levai, Frantisek Ruzicka, Miloslav Znojil

Three classes of finite-dimensional models of quantum systems exhibiting spectral degeneracies called quantum catastrophes are described in detail. Computer-assisted symbolic manipulation techniques are shown unexpectedly efficient for the purpose.

http://arxiv.org/abs/1403.0723
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Pseudospectra in non-Hermitian quantum mechanics

D. Krejcirik, P. Siegl, M. Tater, J. Viola

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.

http://arxiv.org/abs/1402.1082

Spectral Theory (math.SP); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

The large-g observability of the low-lying energies in the strongly singular potentials \(V(x)=x^2+g^2/x^6\) after their PT-symmetric regularization

Miloslav Znojil

The elementary quadratic plus inverse sextic interaction containing a strongly singular repulsive core in the origin is made regular by a complex shift of coordinate \(x=s−i\epsilon\). The shift \(\epsilon>0\) is fixed while the value of s is kept real and potentially observable, \(s∈(−\infty,\infty)\). The low-lying energies of bound states are found in closed form for the large couplings g. Within the asymptotically vanishing \(\mathcal{O}(g^{−1/4})\) error bars these energies are real so that the time-evolution of the system may be expected unitary in an {\em ad hoc} physical Hilbert space.

http://arxiv.org/abs/1401.1435
Quantum Physics (quant-ph)

Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding

Miloslav Znojil

The practical use of non-Hermitian (i.e., typically, PT-symmetric) phenomenological quantum Hamiltonians is discussed as requiring an explicit reconstruction of the ad hoc Hilbert-space metrics which would render the time-evolution unitary. Just the N-dimensional matrix toy models Hamiltonians are considered, therefore. For them, the matrix elements of alternative metrics are constructed via solution of a coupled set of polynomial equations, using the computer-assisted symbolic manipulations for the purpose. The feasibility and some consequences of such a model-construction strategy are illustrated via a discrete square well model endowed with multi-parametric close-to-the-boundary real bidiagonal-matrix interaction. The degenerate exceptional points marking the phase transitions are then studied numerically. A way towards classification of their unfoldings in topologically non-equivalent dynamical scenarios is outlined.

http://arxiv.org/abs/1305.4822
Quantum Physics (quant-ph)

Can unavoided level crossing disguise phase transition?

Miloslav Znojil

The answer is yes. Via an elementary, exactly solvable crypto-Hermitian example it is shown that inside the interval of admissible couplings the innocent-looking point of a smooth unavoided crossing of the eigenvalues of Hamiltonian $H$ may carry a dynamically nontrivial meaning of a phase-transition boundary or “quantum horizon”. Passing this point requires a change of the physical Hilbert-space metric $\Theta$, i.e., a thorough modification of the form and of the interpretation of the operators of all observables.

http://arxiv.org/abs/1303.4876

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Non-Hermitian star-shaped quantum graphs

Miloslav Znojil

A compact review is given, and a few new numerical results are added to the recent studies of the q-pointed one-dimensional star-shaped quantum graphs. These graphs are assumed endowed with certain specific, manifestly non-Hermitian point interactions, localized either in the inner vertex or in all of the outer vertices and carrying, in the latter case, an interesting zero-inflow interpretation.

http://arxiv.org/abs/1303.4331
Quantum Physics (quant-ph)