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:0803.0403 · Journal of Physics A Mathematical and Theoretical

Identification of observables in quantum toboggans

Miloslav Znojil

Published May 8, 2008• 10 citations
arXivHTMLPDF

Abstract

Schrödinger equations for ‘quantum toboggans’ with real energies are given the generalized eigenvalue-problem form H ψ = EW ψ, where H ≠ H † and W ≠ W † ≠ I . The consistent probabilistic interpretation of these models is provided. The new double-series formula for the necessary ad hoc metric Θ = Θ( H , W ) is derived which defines the acceptable inner products in the physical Hilbert space of states. The formula degenerates to the usual single series in the non-tobogganic trivial-weight limit W → I .

DOI
10.1088/1751-8113/41/21/215304

Related papers

Quantum toboggans: models exhibiting a multisheetedPT—symmetry

Miloslav Znojil

A generalization of the concept of PT —symmetric Hamiltonians H = p 2 + V ( x ) is presented. For the usual analytic potentials V ( x ) (with singularities) and for the recently widely accepted ' PT —symmetric' asymptotic boundary conditions for wave functions ψ( x ) (selected inside a pair of complex wedges generalizing the usual x → ± ∞ asymptotics), non-equivalent quantum toboggans are defined as integrated along topologically different paths C of coordinates x ε Cl .

arXiv:0710.1485•Journal of Physics Conference Series•Aug 1, 2008•7 citations

Two sample constructions of all the Hilbert spaces for a given operator $H$ of an observable

Miloslav Znojil

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\dagger$ 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 $\Theta$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $\lambda \in (-1,1)$ in matrix $H^{(N)}(\lambda)$, the complete $N-$parametric set of metrics $\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)$ is recurrently defined by closed formula.

•Jun 26, 2008•0 citations

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

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

Miloslav Znojil

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.

arXiv:0806.4295•Jun 26, 2008•0 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

Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA ${\mathcal{L}}$?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA ${\mathcal{H}}$?> H , the amended Hamiltonian H ≠ H † 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.

Journal of Physics Conference Series•Dec 1, 2023•0 citations
Schrödinger equations for ‘quantum toboggans’ with real energies are given the generalized eigenvalue-problem form H ψ = EW ψ, where H ≠ H † and W ≠ W † ≠ I . The consistent probabilistic interpretation of these models is provided. The new double-series formula for the necessary ad hoc metric Θ = Θ( H , W ) is derived which defines the acceptable inner products in the physical Hilbert space of states. The formula degenerates to the usual single series in the non-tobogganic trivial-weight limit W → I .
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.
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.
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.
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 the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA L{\mathcal{L}}L?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA H{\mathcal{H}}H?> H , the amended Hamiltonian H ≠ H † 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.