Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1011.4803 · Physical Review A

Gegenbauer-solvable quantum chain model

Miloslav Znojil

Published Nov 16, 2010• 51 citations
arXivHTMLPDF

Abstract

An $N$-level quantum model is proposed in which the energies are represented by an $N$-plet of zeros of a suitable classical orthogonal polynomial. The family of Gegenbauer polynomials $G(n,a,x)$ is selected for illustrative purposes. The main obstacle lies in the non-Hermiticity (aka crypto-Hermiticity) of Hamiltonians $H\ensuremath{\ne}{H}^{\ifmmode\dagger\else\textdagger\fi{}}$. We managed to (i) start from elementary secular equation $G(N,a,{E}_{n})=0$, (ii) keep our $H$, in the nearest-neighbor-interaction spirit, tridiagonal, (iii) render it Hermitian in an ad hoc, nonunique Hilbert space endowed with metric $\ensuremath{\Theta}\ensuremath{\ne}I$, (iv) construct eligible metrics in closed forms ordered by increasing nondiagonality, and (v) interpret the model as a smeared $N$-site lattice.

DOI
10.1103/physreva.82.052113

Related papers

NEW CONCEPT OF SOLVABILITY IN QUANTUM MECHANICS

Miloslav Znojil

In quite a few recent quantum models one is allowed to make a given Hamiltonian H self-adjoint only after an ad hoc generalization of Hermitian conjugation, H†→H‡:= Θ −1H†Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠ I the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of H) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.

arXiv:1302.2035•Acta Polytechnica•Oct 24, 2013•1 citation

An exactly solvable quantum-lattice model with a tunable degree of nonlocality

Miloslav Znojil

A non-Hermitian N -site-lattice Hamiltonian H with Laguerre-polynomial right eigenvectors and real energies is made self-adjoint in an ad hoc Hilbert space . The necessary physical inner products are defined via alternative k -parametric (2 k + 1)-diagonal metrics Θ = Θ k constructed, in the closed form, at k = 0, k = 1, k = 2 and k = 3. The value of k is interpreted as a degree of nonlocality of the model.

arXiv:1101.1183•Journal of Physics A Mathematical and Theoretical•Jan 24, 2011•22 citations

Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models

Miloslav Znojil

Abstract In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that <?CDATA $H^\dagger \Theta = \Theta\,H$?> H † Θ = Θ H , the exactly solvable M-level bound-state models with arbitrary <?CDATA $M\leqslant \infty$?> M ⩽ ∞ are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just <?CDATA $(2M-1)$?> ( 2 M − 1 ) -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.

arXiv:2307.03439•Journal of Physics A Mathematical and Theoretical•Jul 19, 2023•2 citations

3 New Concept of Solvability in Quantum Mechanics

Miloslav Znojil

In quite a few recent quantum models one is allowed to make a given Hamiltonian H self-adjoint only after an ad hoc generalization of Hermitian conjugation, H†→H‡:= Θ −1H†Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠ I the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of H) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.

•Aug 14, 2016•1 citation

Quasi-Hermitian quantum mechanics and a new class of user-friendly matrix Hamiltonians

Olaf Lechtenfeld, Miloslav Znojil

In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state $ψ$ is controlled, in Hilbert space ${\cal L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces $\mathfrak{h}$ by its isospectral but non-Hermitian avatar $H = Ω^{-1}\mathfrak{h}Ω$ with $Ω^\daggerΩ= Θ\neq I$. Although acting in another, manifestly unphysical Hilbert space ${\cal H}$, the amended Hamiltonian $H \neq H^\dagger$ can be perceived as self-adjoint with respect to the amended inner-product metric $Θ$. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of $H$ we introduce and describe a specific new family of Hamiltonians $H$ for which the metrics $Θ$ become available in closed form.

arXiv:2308.15043•Aug 29, 2023•0 citations

Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions

Miloslav Znojil

Abstract A family of exactly solvable quantum square wells with discrete coordinates and with certain non-stationary Hermiticity-violating Robin boundary conditions is proposed and studied. Manifest non-Hermiticity of the model in conventional Hilbert space H friendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space H physical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ(t)) as well as the time-dependent observables (say, Λ(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrödinger-equation generator) and Σ(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in H physical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + Σ(t). The applicability of these ideas is illustrated via an N by N matrix model. At N = 2, closed formulae are presented not only for the measurable instantaneous energy spectrum but also for all of the eligible time-dependent physical inner-product metrics Θ(N=2)(t), for the related Dyson maps Ω(N=2)(t), for the Coriolis force Σ(N=2)(t) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrödinger-equation generator G (N=2)(t).

arXiv:2401.10682•Physica Scripta•Feb 26, 2024•6 citations
An NNN-level quantum model is proposed in which the energies are represented by an NNN-plet of zeros of a suitable classical orthogonal polynomial. The family of Gegenbauer polynomials G(n,a,x)G(n,a,x)G(n,a,x) is selected for illustrative purposes. The main obstacle lies in the non-Hermiticity (aka crypto-Hermiticity) of Hamiltonians H\ensuremath≠H\ifmmode†\else\textdagger\fiH\ensuremath{\ne}{H}^{\ifmmode\dagger\else\textdagger\fi{}}H\ensuremath=H\ifmmode†\else\textdagger\fi. We managed to (i) start from elementary secular equation G(N,a,En)=0G(N,a,{E}_{n})=0G(N,a,En​)=0, (ii) keep our HHH, in the nearest-neighbor-interaction spirit, tridiagonal, (iii) render it Hermitian in an ad hoc, nonunique Hilbert space endowed with metric \ensuremathΘ\ensuremath≠I\ensuremath{\Theta}\ensuremath{\ne}I\ensuremathΘ\ensuremath=I, (iv) construct eligible metrics in closed forms ordered by increasing nondiagonality, and (v) interpret the model as a smeared NNN-site lattice.
In quite a few recent quantum models one is allowed to make a given Hamiltonian H self-adjoint only after an ad hoc generalization of Hermitian conjugation, H†→H‡:= Θ −1H†Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠ I the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of H) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.
Abstract In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that <?CDATA H†Θ=Θ HH^\dagger \Theta = \Theta\,HH†Θ=ΘH?> H † Θ = Θ H , the exactly solvable M-level bound-state models with arbitrary <?CDATA M⩽∞M\leqslant \inftyM⩽∞?> M ⩽ ∞ are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just <?CDATA (2M−1)(2M-1)(2M−1)?> ( 2 M − 1 ) -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.
In quite a few recent quantum models one is allowed to make a given Hamiltonian H self-adjoint only after an ad hoc generalization of Hermitian conjugation, H†→H‡:= Θ −1H†Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠ I the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of H) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.
In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψψψ is controlled, in Hilbert space L{\cal L}L, by a Hamiltonian h\mathfrak{h}h which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces h\mathfrak{h}h by its isospectral but non-Hermitian avatar H=Ω−1hΩH = Ω^{-1}\mathfrak{h}ΩH=Ω−1hΩ with Ω†Ω=Θ≠IΩ^\daggerΩ= Θ\neq IΩ†Ω=Θ=I. Although acting in another, manifestly unphysical Hilbert space H{\cal H}H, the amended Hamiltonian H≠H†H \neq H^\daggerH=H† can be perceived as self-adjoint with respect to the amended inner-product metric ΘΘΘ. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of HHH we introduce and describe a specific new family of Hamiltonians HHH for which the metrics ΘΘΘ become available in closed form.
Abstract A family of exactly solvable quantum square wells with discrete coordinates and with certain non-stationary Hermiticity-violating Robin boundary conditions is proposed and studied. Manifest non-Hermiticity of the model in conventional Hilbert space H friendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space H physical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ(t)) as well as the time-dependent observables (say, Λ(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrödinger-equation generator) and Σ(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in H physical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + Σ(t). The applicability of these ideas is illustrated via an N by N matrix model. At N = 2, closed formulae are presented not only for the measurable instantaneous energy spectrum but also for all of the eligible time-dependent physical inner-product metrics Θ(N=2)(t), for the related Dyson maps Ω(N=2)(t), for the Coriolis force Σ(N=2)(t) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrödinger-equation generator G (N=2)(t).