The PT Symmeter

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)

Miloslav Znojil

A given Hamiltonian matrix H with real spectrum is assumed tridiagonal and non-Hermitian. Its possible Hermitizations via an amended, ad hoc inner-product metric are studied. Under certain reasonable assumptions, all of these metrics are shown obtainable as recurrent solutions of the hidden Hermiticity constraint called Dieudonne equation. In this framework even the two-parametric Jacobi-polynomial real- and asymmetric-matrix N-site lattice Hamiltonian is found tractable non-numerically at all N.

http://arxiv.org/abs/1201.2263
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Miloslav Znojil

A non-unitary version of quantum scattering is studied via an exactly solvable toy model. The model is merely asymptotically local since the smooth path of the coordinate is admitted complex in the non-asymptotic domain. At any real angular-momentum-like parameter the reflection R and transmission T are shown to change with the winding number (i.e., topology) of the path. The points of unitarity appear related to the points of existence of quantum-knot bound states.

http://arxiv.org/abs/1112.2644
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Miloslav Znojil

Two discrete N-level alternatives to the popular imaginary cubic oscillator are proposed and studied. In a certain domain \({\cal D}\) of parameters \(a\) and \(z\) of the model, the spectrum of energies is shown real (i.e., potentially, observable) and the unitarity of the evolution is shown mediated by the construction of a (non-unique) physical, ad hoc Hilbert space endowed with a nontrivial, Hamiltonian-dependent inner-product metric \(\Theta\). Beyond \({\cal D}\) the complex-energy curves are shown to form a “Fibonacci-numbered” geometric pattern and/or a “topologically complete” set of spectral loci. The dynamics-determining construction of the set of the eligible metrics is shown tractable by a combination of the computer-assisted algebra with the perturbation and extrapolation techniques. Confirming the expectation that for the local potentials the effect of the metric cannot be short-ranged.

http://arxiv.org/abs/1111.0484
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Miloslav Znojil

An extension of the scope of quantum theory is proposed in a way inspired by the recent heuristic as well as phenomenological success of the use of non-Hermitian Hamiltonians which are merely required self-adjoint in a Krein space with an indefinite metric (chosen, usually, as the operator of parity). In nuce, the parity-like operators are admitted to represent the mere indefinite metric in a Pontryagin space. A constructive version of such a generalized quantization strategy is outlined and, via a non-numerical illustrative example, found feasible.

http://arxiv.org/abs/1110.1218
Mathematical Physics (math-ph)