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Miloslav Znojil

Publications and source records attributed to Miloslav Znojil.

At least 163 records · Page 9Linked to original sources
arXiv2008

The complete menu of eligible metrics for a family of toy Hamiltonians $H \neq H^\dagger$ with real spectra

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\dagger$ with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states ${\cal H}^{(physical)}$. The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators $Θ$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $λ\in (-1,1)$ in matrix $H^{(N)}(λ)$, the complete $N-$parametric set of metrics $Θ^{(N)}_{α_1,α_2,...,α_N}(λ)$ is recurrently defined by closed formula.

math-ph↗
arXiv2008

Discrete PT-symmetric models of scattering

One-dimensional scattering mediated by non-Hermitian Hamiltonians is studied. A schematic set of models is used which simulate two point interactions at a variable strength and distance. The feasibility of the exact construction of the amplitudes is achieved via the discretization of the coordinate. By direct construction it is shown that in all our models the probability is conserved. This feature is tentatively attributed to the space- and time-reflection symmetry (a.k.a. PT-symmetry) of our specific Hamiltonians.

quant-ph↗
arXiv2008

Scattering theory with localized non-Hermiticities

In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that and how the well known variability of our ad hoc choice of the metric $Θ$ which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable $Θ$ a fully plausible physical picture of the scattering is recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.

hep-th↗
arXiv2008

PT-symmetric deformations of Calogero models

We demonstrate that Coxeter groups allow for complex PT-symmetric deformations across the boundaries of all Weyl chambers. We compute the explicit deformations for the A_2 and G_2-Coxeter group and apply these constructions to Calogero-Moser-Sutherland models invariant under the extended Coxeter groups. The eigenspecta for the deformed models are real and contain the spectra of the undeformed case as subsystem.

quant-ph↗
arXiv2008

PT-symmetric Sturmians

Sturmian bound states emerging at a fixed energy and numbered by a complete set of real eigencouplings are considered. For Sturm-Schroedinger equations which are manifestly non-Hermitian we outline the way along which the correct probabilistic interpretation of the system can constructively be re-established via a new formula for the metric. PT-symmetrized Coulomb potential is chosen for illustration purposes.

quant-ph↗
arXiv2008

Identification of observables in quantum toboggans

Quantum systems with real energies generated by an apparently non-Hermitian Hamiltonian may re-acquire the consistent probabilistic interpretation via an ad hoc metric which specifies the set of observables in the updated Hilbert space of states. The recipe is extended here to quantum toboggans. In the first step the tobogganic integration path is rectified and the Schroedinger equation is given the generalized eigenvalue-problem form. In the second step the general double-series representation of the eligible metric operators is derived.

quant-ph↗
arXiv2008

Conditional observability versus self-duality in a schematic model

In a simple PT-symmetric model we demonstrate that and how the violation of a reflection symmetry $E_j=-E_{N+1-j}$ of the spectrum (called "self-duality" by Dunne and Shifman) is connected with the loss of the simplicity of the shape of the boundary of the domain D of parameters where the Hamiltonian H is quasi-Hermitian, i.e., where all the bound-state energies are real.

quant-ph↗
arXiv2008

Quantum knots

It is known that besides the usual unitary mappings $Ω= 1/Ω^\dagger$ between the equivalent representations of the physical Hilbert space of Quantum Mechanics (often, Fourier transformations), the generalized non-unitary maps $Ω\neq 1/Ω^\dagger$ can also help to simplify the analysis. We adapt the standard Dirac's notation and recollect the Buslaev's and Grecchi's repulsive quartic oscillator Hamiltonian as an example. Then we propose the whole new class of the models of the similar type, characterized by a complexification of the path ${\cal C}$ of the (obviously, not observable!) "coordinates". An exactly solvable potentialless Schrödinger equation is finally chosen for illustration. In it, the dynamical (i.e., in our example, confining) role of the traditional potentials $V(x)$ is shown to be taken over by the mere topologically nontrivial shape of ${\cal C}$. Our construction evokes several new open questions in physics (${\cal PT}-$symmetric wave packets at a single energy?) as well as in mathematics (a three-Hilbert-space generalized formulation of Quantum Mechanics?).

quant-ph↗
arXiv2008

Matrix Hamiltonians with an algebraic guarantee of unbroken PT-symmetry

Quantum bound-state energies are assumed generated by PT-symmetric Hamiltonians H where P is, typically, parity. It is known that their spectrum only remains real and observable (i.e., in the language of physics, the PT-symmetry remains unbroken) inside a certain domain D of couplings. We show that the boundary of this domain (i.e., certain stability and observability horizon formed by the Kato's exceptional points) remains algebraic (i.e., we determine it by closed formulae) for a certain toy-model family of N-dimensional anharmonic-oscillator-related matrix Hamiltonians with dimensions between N=2 and N=11.

math-ph↗
arXiv2008

On the Role of the Normalization Factors $κ_n$ and of the Pseudo-Metric ${\cal P} \neq {\cal P}^\dagger$ in Crypto-Hermitian Quantum Models

Among ${\cal P}$-pseudo-Hermitian Hamiltonians $H ={\cal P}^{-1} H^\dagger \cal P}$ with real spectra, the ''weakly pseudo-Hermitian" ones (i.e., those employing non-self-adjoint ${\cal P} \neq {\cal P}^\dagger$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever ${\cal P} \neq {\cal P}^\dagger$, the current involutive operator of charge ${\cal C}$ gets complemented by a nonequivalent alternative involutive quasiparity operator ${\cal Q}$. We show how, in this language, the standard quantum mechanics is restored via the two alternative inner products in the physical Hilbert space of states, with $<ψ_1|{\cal PQ}|ψ_2>=< ψ_1|{\cal CP}|ψ_2>$.

math-ph↗
arXiv2007

Reply to Comment on ``Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution''

In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.

quant-ph↗
arXiv2007

Which operator generates time evolution in Quantum Mechanics?

We attract attention to an interesting family of quantum systems where the generator $H_{(gen)}$ of time-evolution of wave functions is not equal to the Hamiltonian $H$. We describe the origin of the difference $H_{(gen)}-H$ and interpret it as a carrier of a compressed information about the other relevant observables.

quant-ph↗
arXiv2007

Time-dependent quasi-Hermitian Hamiltonians and the unitarity of quantum evolution

We show that the consequences of an introduction of a manifest time-dependence in a pseudo-Hermitian Hamiltonian H=H(t) are by far less drastic than suggested by A. Mostafazadeh in Phys. Lett. B 650 (2007) 208 (arXiv:0706.1872v2 [quant-ph]). In particular, the unitarity of the evolution does not necessitate the time-independence of the metric $η_+=η_+(t)$.

quant-ph↗
arXiv2007

Horizons of stability in matrix Hamiltonians

Non-Hermitian Hamiltonians H possess the real (i.e., observable) spectra inside certain specific domains of parameters D. In general, the determination of their observability-horizon boundaries is difficult. We list the pseudo-Hermitian real N by N matrix Hamiltonians for which the prototype horizons are defined by closed analytic formulae.

quant-ph↗
arXiv2007

Quantum toboggans: models exhibiting a multisheeted PT symmetry

A generalization of the concept of PT-symmetric Hamiltonians H=p^2+V(x) is described. It uses analytic potentials V(x) (with singularities) and a generalized concept of PT-symmetric asymptotic boundary conditions. Nontrivial toboggans are defined as integrated along topologically nontrivial paths of coordinates running over several Riemann sheets of wave functions.

quant-ph↗
arXiv2007

Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime

A generic PT-symmetric Hamiltonian is assumed tridiagonalized and truncated to N dimensions, and its up-down symmetrized special cases with J=[N/2] real couplings are considered. In the strongly non-Hermitian regime the secular equation gets partially factorized at all N. This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity domain in which all the spectrum of energies remains real and observable.

math-ph↗