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arXiv:1101.1183 · Journal of Physics A Mathematical and Theoretical

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

Miloslav Znojil

Published Jan 24, 2011• 22 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/7/075302

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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.
A new Hermitizable quantum model is proposed in which the bound-state energies are real and given as roots of an elementary trigonometric expression while the wave function components are expressed as superpositions of two Chebyshev polynomials. As an -site lattice version of square well with complex Robin-type two-parametric boundary conditions the model is unitary with respect to the Hilbert space metric Θ which becomes equal to the most common Diracʼs metric in the conventional textbook Hermitian–Hamiltonian limit. This metric is constructed in closed form at all .
Two sample constructions of all the Hilbert spaces for a given operator HHH of an observable
An elementary set of non-Hermitian NNN by NNN matrices H(N)(λ)≠[H(N)(λ)]†H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\daggerH(N)(λ)=[H(N)(λ)]† 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 H(physical){\cal H}^{(physical)}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\ThetaΘ) is addressed. The problem is shown exactly solvable and, for every size N=2,4,...N=2,4,...N=2,4,... and parameter λ∈(−1,1)\lambda \in (-1,1)λ∈(−1,1) in matrix H(N)(λ)H^{(N)}(\lambda)H(N)(λ), the complete N−N-N−parametric set of metrics Θα1,α2,...,αN(N)(λ)\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)Θα1​,α2​,...,αN​(N)​(λ) is recurrently defined by closed formula.
The complete menu of eligible metrics for a family of toy Hamiltonians H≠H†H \neq H^\daggerH=H† with real spectra
An elementary set of non-Hermitian NNN by NNN matrices H(N)(λ)≠[H(N)(λ)]†H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\daggerH(N)(λ)=[H(N)(λ)]† 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 H(physical){\cal H}^{(physical)}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 ΘΘΘ) is addressed. The problem is shown exactly solvable and, for every size N=2,4,...N=2,4,...N=2,4,... and parameter λ∈(−1,1)λ\in (-1,1)λ∈(−1,1) in matrix H(N)(λ)H^{(N)}(λ)H(N)(λ), the complete N−N-N−parametric set of metrics Θα1,α2,...,αN(N)(λ)Θ^{(N)}_{α_1,α_2,...,α_N}(λ)Θα1​,α2​,...,αN​(N)​(λ) is recurrently defined by closed formula.
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 $H\ensuremath{\ne}{H}^{\ifmmode\dagger\else\textdagger\fi{}}$. 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 $\ensuremath{\Theta}\ensuremath{\ne}I$, (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.
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.