The PT Symmeter

Miloslav Znojil

In a pre-selected Hilbert space of quantum states the unitarity of the evolution is usually guaranteed via a pre-selection of the generator (i.e., of the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use of both of these pre-selections is overrestrictive. One should be allowed to make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity scenario with nontrivial metric, the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of Hamiltonian) requires a generalization allowing for a non-numerical form of metric. A few illustrative solvable quantum models of this type are presented.

http://arxiv.org/abs/1302.2035
Quantum Physics (quant-ph)

Miloslav Znojil

The quantum-catastrophe (QC) benchmark Hamiltonians of paper I (M. Znojil, J. Phys. A: Math. Theor. 45 (2012) 444036) are reconsidered, with the infinitesimal QC distance \(\lambda\) replaced by the total time $\tau$ of the fall into the singularity. Our amended model becomes unique, describing the complete QC history as initiated by a Hermitian and diagonalized N-level oscillator Hamiltonian at \(\tau=0\). In the limit \(\tau \to 1\) the system finally collapses into the completely (i.e., N-times) degenerate QC state. The closed and compact Hilbert-space metrics are then calculated and displayed up to N=7. The phenomenon of the QC collapse is finally attributed to the manifest time-dependence of the Hilbert space and, in particular, to the emergence and to the growth of its anisotropy. A quantitative measure of such a time-dependent anisotropy is found in the spread of the N-plet of the eigenvalues of the metric. Unexpectedly, the model appears exactly solvable — at any multiplicity N, the N-plet of these eigenvalues is obtained in closed form.

http://arxiv.org/abs/1212.0734
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Miloslav Znojil

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 ad hoc choice of the inner product in the physical Hilbert space of quantum bound states (i.e., via an 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.

http://arxiv.org/abs/1206.6000
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Miloslav Znojil

A new exactly solvable model of a quantum system is proposed, living on an equilateral q-pointed star graph (q is arbitrary). The model exhibits a weak and spontaneously broken form of \({\cal PT}-\)symmetry, offering a straightforward generalization of one of the standard solvable square wells with \(q=2\) and unbroken \({\cal PT}-\)symmetry. The kinematics is trivial, Kirchhoff in the central vertex. The dynamics is one-parametric (viz., \(\alpha-\)dependent), prescribed via complex Robin boundary conditions (i.e., the interactions are non-Hermitian and localized at the outer vertices of the star). The (complicated, trigonometric) secular equation is shown reducible to an elementary and compact form. This renders the model (partially) exactly solvable at any \(q \geq 2\) — an infinite subset of the real roots of the secular equation proves q-independent and known (i.e., inherited from the square-well \(q=2\) special case). The systems with \(q=4m-2\) are found anomalous, supporting infinitely many (or, at m=1, one) additional real m-dependent and \(\alpha-\)dependent roots.

http://arxiv.org/abs/1205.5211
Quantum Physics (quant-ph)

Miloslav Znojil

For the description of quantum evolution, the use of a manifestly time-dependent quantum Hamiltonian \(\mathfrak{h}(t) =\mathfrak{h}^\dagger(t)\) is shown equivalent to the work with its simplified, time-independent alternative \(G\neq G^\dagger\). A tradeoff analysis is performed recommending the latter option. The physical unitarity requirement is shown fulfilled in a suitable ad hocrepresentation of Hilbert space.

http://arxiv.org/abs/1204.5989
Quantum Physics (quant-ph); General Relativity and Quantum Cosmology (gr-qc)

Miloslav Znojil

Besides the standard quantum version of the Coulomb/Kepler problem, an alternative quantum model with not too dissimilar phenomenological (i.e., spectral and scattering) as well as mathematical (i.e., exact-solvability) properties may be formulated and solved. Several aspects of this model are described. The paper is made self-contained by explaining the underlying innovative quantization strategy which assigns an entirely new role to symmetries.

http://arxiv.org/abs/1204.1257
Quantum Physics (quant-ph)