The PT Symmeter

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)

Miloslav Znojil

Non-Hermitian ring-shaped discrete lattices share the appeal with their more popular linear predecessors. Their dynamics controlled by the nearest-neighbor interaction is equally phenomenologically interesting. In comparison, the innovative nontriviality of their topology may be expected to lead to new spectral effects. Some of them are studied here via solvable examples. Main attention is paid to the perturbation-caused removals of spectral degeneracy at exceptional points.

http://arxiv.org/abs/1108.0620
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Miloslav Znojil

The one-dimensional real line of coordinates is replaced, for simplification or approximation purposes, by an N-plet of the so called Gauss-Hermite grid points. These grid points are interpreted as the eigenvalues of a tridiagonal matrix \(\mathfrak{q}_0\) which proves rather complicated. Via the “zeroth” Dyson-map \(\Omega_0\) the “operator of position” \(\mathfrak{q}_0\) is then further simplified into an isospectral matrix \(Q_0\) which is found optimal for the purpose. As long as the latter matrix appears non-Hermitian it is not an observable in the manifestly “false” Hilbert space \({\cal H}^{(F)}:=\mathbb{R}^N\). For this reason the optimal operator \(Q_0\) is assigned the family of its isospectral avatars \(\mathfrak{h}_\alpha\), \(\alpha=(0,)\,1,2,…\). They are, by construction, selfadjoint in the respective \(\alpha-\)dependent image Hilbert spaces \({\cal H}^{(P)}_\alpha\) obtained from \({\cal H}^{(F)}\) by the respective “new” Dyson maps \(\Omega_\alpha\). In the ultimate step of simplification, the inner product in the F-superscripted space is redefined in an {\it ad hoc}, $\alpha-$dependent manner. The resulting “simplest”, S-superscripted representations \({\cal H}^{(S)}_\alpha\) of the eligible physical Hilbert spaces of states (offering different dynamics) then emerge as, by construction, unitary equivalent to the (i.e., indistinguishable from the) respective awkward, P-superscripted and \(\alpha-\)subscripted physical Hilbert spaces.

http://arxiv.org/abs/1107.1770
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Miloslav Znojil

A toy-model quantum system is proposed. At a given integer \(N\) it is defined by the pair of \(N\) by \(N\) real matrices \((H,\Theta)\) of which the first item \(H\) specifies an elementary, diagonalizable non-Hermitian Hamiltonian \(H \neq H^\dagger\) with the real and explicit spectrum given by the zeros of the \(N-\)th Chebyshev polynomial of the first kind. The second item \(\Theta\neq I\) must be (and is being) constructed as the related Hilbert-space metric which specifies the (in general, non-unique) physical inner product and which renders our toy-model Hamiltonian selfadjoint, i.e., compatible with the Dieudonne equation \(H^\dagger \Theta= \Theta\,H\). The elements of the (in principle, complete) set of the eligible metrics are then constructed in closed band-matrix form. They vary with our choice of the \(N-\)plet of optional parameters, \(\Theta=\Theta(\vec{\kappa})>0\) which must be (and are being) selected as lying in the positivity domain of the metric, \(\vec{\kappa} \in {\cal D}^{(physical)}\).

http://arxiv.org/abs/1105.1863
Quantum Physics (quant-ph); High Energy Physics – Lattice (hep-lat); Mathematical Physics (math-ph)

Miloslav Znojil

The “Hermitizability” problem of quantum theory is explained, discussed and illustrated via the discrete-lattice cryptohermitian quantum graphs. In detail, the description of the domain ${\cal D}$ of admissible parameters is provided for the “circular-model” three-parametric quantum Hamiltonian $H$ using periodic boundary conditions. It is emphasized that even in such an elementary system the weak- and strong-coupling subdomains of ${\cal D}$ become, unexpectedly, non-empty and disconnected.

http://arxiv.org/abs/1103.4001
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)