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:2602.17589

States that grow linearly in time, exceptional points, and zero norm states in the simple harmonic oscillator

Philip D. Mannheim

quant-phSubmitted Feb 19, 2026• Announced Feb 20, 2026• 0 citations
arXivHTMLPDF

Abstract

The simple harmonic oscillator has a well-known normalizable, positive energy, bound state spectrum. We show that degenerate with each such positive energy eigenvalue there is a non-normalizable positive energy eigenstate whose eigenfunction is orthogonal to that of the standard energy eigenfunction. This class of states is not built on the vacuum that $a$ annihilates, but is instead built on the vacuum that $a^{\dagger} a$ annihilates. These non-normalizable but nonetheless stationary energy eigenstates are accompanied by yet another set of non-normalizable states, states whose wave functions however are not stationary but instead grow linearly in time. With these states not being energy eigenstates, the eigenbasis of the Hamiltonian is incomplete; with the full Hilbert space containing states that are not energy eigenstates. Thus each energy eigenvalue of the harmonic oscillator is an exceptional point at which the Hamiltonian becomes of non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form. Such non-Hermitian structures occur for Hamiltonians that have an antilinear $PT$ symmetry. As is characteristic of such systems, one can construct a probability conserving inner product that despite the linear in time growth is nonetheless time independent, and not only that, it leads to states with zero norm. In addition, as is again characteristic of $PT$ symmetry, these non-normalizable states can be made normalizable by a continuation into a so-called Stokes wedge domain in the complex plane. In this domain one has a completely consistent quantum theory, one that lives alongside the standard normalizable energy eigenspectrum sector. This thus not quite so simple harmonic oscillator provides an explicit realization of our general contention that antilinearity is more basic to quantum theory than Hermiticity.

Comments
7 pages, revtex 4

Related papers

Fundamental Properties of Eigenstate Wave Functions of PT-Symmetric Anharmonic Oscillators

Jack H. Noble, Ulrich D. Jentschura

The PT-symmetric cubic oscillator with Hamiltonian H_3 = - (1/2) d_x^2 + (1/2) x^2 + i G x^3 is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with L^2(R) boundary conditions. Eigenfunctions of the stationary Schr\{o}dinger equation H_3 psi_n(x) = E_n psi_n(x) are manifestly complex, while the energy eigenvalues E_n are real and positive. Although H_3 does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re psi_n(x) and Im psi_n(x)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when G>0. Furthermore, we find that the eigenfunctions are concentrated in an allowed region where the energy E_n is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex gain and loss terms mutually compensate and lead to the real energy eigenvalues.

•Jan 24, 2013•0 citations

Transition elements for a non-Hermitian quadratic Hamiltonian

Mark S. Swanson

The non-Hermitian quadratic Hamiltonian H=ωa†a+αa2+βa†2 is analyzed, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that ω2−4αβ⩾0, it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for non-Hermitian Hamiltonians. Transition probabilities governed by H are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and non-Hermitian cases to identify the differences in time evolution.

Journal of Mathematical Physics•Jan 22, 2004•186 citations

Exactly solvable PT -symmetric Hamiltonian having no Hermitian counterpart

Carl M. Bender, Philip D. Mannheim

In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian ${H}_{\mathrm{PU}}$ of the model is $\mathcal{P}\mathcal{T}$ symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that ${H}_{\mathrm{PU}}$ becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency $\mathcal{P}\mathcal{T}$ theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schr\"odinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.

arXiv:0804.4190•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 16, 2008•177 citations

Hermitian and pseudo-Hermitian Hamiltonians of SU(1,1) system—Spectrum, exceptional point, quantum–classical correspondence

Ni Liu, Meng Luo, Z. D. Wang +1 more

We in this paper study the relation of hermiticity and energy spectrum for Hamiltonians consisting of $SU(1,1)$ generators. In contrast with the common belief, the transition from real to imaginary spectra can appear at an exceptional point for the Hermitian Hamiltonian of a conservative system. The imaginary domain of spectrum resulted from an inverted potential well unbounded from below. An outward force applies on the particle, which moves acceleratingly away from the central point similar to the minisuperspace model of expanding universe. The Hamiltonian remains Hermitian beyond the exceptional point in a price that the boson-operator realization of $SU(1,1)$ generator $\widehat{S}_{z}$ becomes non-Hermitian. Oppositely a non-Hermitian Hamiltonian (called pseudo-Hermitian) possesses real eigenvalues in the entire region of interaction constant, which increases the gradient of potential well. While the Hermitian interaction decreases the gradient continuously to zero, namely the exceptional point, where eigenstates are degenerate. We extend the non-Hermitian Hamiltonian to the periodically time-dependent and two-dimensional system. The probability density of wave function coincides with classical orbits, which are derived from the corresponding classical-counterpart of the non-Hermitian quantum Hamiltonian.

Results in Physics•Dec 20, 2023•2 citations

Imaginary eigenvalues of Hermitian Hamiltonian with an inverted potential well and transition to the real spectrum at exceptional point by a non-Hermitian interaction

Ni Liu, Zuohong Wang, Meng Luo +1 more

Abstract We in this paper study the hermiticity of Hamiltonian and energy spectrum for the SU (1, 1) systems. The Hermitian Hamiltonian can possess imaginary eigenvalues if the potential is unbounded below like the Gamow state in nuclear physics. We provide a rigorous quantization for the Hermitian Hamiltonian of inverted harmonic oscillator with the imaginary eigenvalues, which are derived in algebraic method with imaginary frequency boson operators. Dual sets of mutually orthogonal eigenstates are required corresponding respectively to the complex conjugate eigenvalues. Arbitrary order eigenfunctions are generated from the normalized ground state wave functions, which are spatially non-localized. The Hamiltonian including a non-Hermitian interaction term can be converted by similarity transformation to the Hermitian one with an effective potential of reduced slope, which can be manipulated by the interaction constant. The transformation operator is Hermitian different from the unitary transformation in ordinary quantum mechanics. The effective potential vanishes at a critical value of coupling strength called the exceptional point (EP), where all eigenstates are degenerate with zero eigenvalue. The spectrum becomes real beyond the EP just opposite to the usual transition direction. The classical counterpart of the quantum Hamiltonian with non-Hermitian interaction is a complex function of canonical variables. It becomes by the canonical transformation of variables a real function. The quantum–classical correspondence exists precisely for the system with imaginary eigenvalues.

Physica Scripta•Feb 26, 2026•0 citations

Antilinearity rather than Hermiticity as a guiding principle for quantum theory

Philip D. Mannheim

Abstract Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT . We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT -symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.

arXiv:1512.04915•Journal of Physics A Mathematical and Theoretical•Apr 26, 2018•43 citations
The simple harmonic oscillator has a well-known normalizable, positive energy, bound state spectrum. We show that degenerate with each such positive energy eigenvalue there is a non-normalizable positive energy eigenstate whose eigenfunction is orthogonal to that of the standard energy eigenfunction. This class of states is not built on the vacuum that aaa annihilates, but is instead built on the vacuum that a†aa^{\dagger} aa†a annihilates. These non-normalizable but nonetheless stationary energy eigenstates are accompanied by yet another set of non-normalizable states, states whose wave functions however are not stationary but instead grow linearly in time. With these states not being energy eigenstates, the eigenbasis of the Hamiltonian is incomplete; with the full Hilbert space containing states that are not energy eigenstates. Thus each energy eigenvalue of the harmonic oscillator is an exceptional point at which the Hamiltonian becomes of non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form. Such non-Hermitian structures occur for Hamiltonians that have an antilinear PTPTPT symmetry. As is characteristic of such systems, one can construct a probability conserving inner product that despite the linear in time growth is nonetheless time independent, and not only that, it leads to states with zero norm. In addition, as is again characteristic of PTPTPT symmetry, these non-normalizable states can be made normalizable by a continuation into a so-called Stokes wedge domain in the complex plane. In this domain one has a completely consistent quantum theory, one that lives alongside the standard normalizable energy eigenspectrum sector. This thus not quite so simple harmonic oscillator provides an explicit realization of our general contention that antilinearity is more basic to quantum theory than Hermiticity.
The PT-symmetric cubic oscillator with Hamiltonian H_3 = - (1/2) d_x^2 + (1/2) x^2 + i G x^3 is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with L^2(R) boundary conditions. Eigenfunctions of the stationary Schr\{o}dinger equation H_3 psi_n(x) = E_n psi_n(x) are manifestly complex, while the energy eigenvalues E_n are real and positive. Although H_3 does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re psi_n(x) and Im psi_n(x)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when G>0. Furthermore, we find that the eigenfunctions are concentrated in an allowed region where the energy E_n is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex gain and loss terms mutually compensate and lead to the real energy eigenvalues.
The non-Hermitian quadratic Hamiltonian H=ωa†a+αa2+βa†2 is analyzed, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that ω2−4αβ⩾0, it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for non-Hermitian Hamiltonians. Transition probabilities governed by H are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and non-Hermitian cases to identify the differences in time evolution.
In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian HPU{H}_{\mathrm{PU}}HPU​ of the model is PT\mathcal{P}\mathcal{T}PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that HPU{H}_{\mathrm{PU}}HPU​ becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT\mathcal{P}\mathcal{T}PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schr\"odinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.
We in this paper study the relation of hermiticity and energy spectrum for Hamiltonians consisting of SU(1,1)SU(1,1)SU(1,1) generators. In contrast with the common belief, the transition from real to imaginary spectra can appear at an exceptional point for the Hermitian Hamiltonian of a conservative system. The imaginary domain of spectrum resulted from an inverted potential well unbounded from below. An outward force applies on the particle, which moves acceleratingly away from the central point similar to the minisuperspace model of expanding universe. The Hamiltonian remains Hermitian beyond the exceptional point in a price that the boson-operator realization of SU(1,1)SU(1,1)SU(1,1) generator S^z\widehat{S}_{z}Sz​ becomes non-Hermitian. Oppositely a non-Hermitian Hamiltonian (called pseudo-Hermitian) possesses real eigenvalues in the entire region of interaction constant, which increases the gradient of potential well. While the Hermitian interaction decreases the gradient continuously to zero, namely the exceptional point, where eigenstates are degenerate. We extend the non-Hermitian Hamiltonian to the periodically time-dependent and two-dimensional system. The probability density of wave function coincides with classical orbits, which are derived from the corresponding classical-counterpart of the non-Hermitian quantum Hamiltonian.
Abstract We in this paper study the hermiticity of Hamiltonian and energy spectrum for the SU (1, 1) systems. The Hermitian Hamiltonian can possess imaginary eigenvalues if the potential is unbounded below like the Gamow state in nuclear physics. We provide a rigorous quantization for the Hermitian Hamiltonian of inverted harmonic oscillator with the imaginary eigenvalues, which are derived in algebraic method with imaginary frequency boson operators. Dual sets of mutually orthogonal eigenstates are required corresponding respectively to the complex conjugate eigenvalues. Arbitrary order eigenfunctions are generated from the normalized ground state wave functions, which are spatially non-localized. The Hamiltonian including a non-Hermitian interaction term can be converted by similarity transformation to the Hermitian one with an effective potential of reduced slope, which can be manipulated by the interaction constant. The transformation operator is Hermitian different from the unitary transformation in ordinary quantum mechanics. The effective potential vanishes at a critical value of coupling strength called the exceptional point (EP), where all eigenstates are degenerate with zero eigenvalue. The spectrum becomes real beyond the EP just opposite to the usual transition direction. The classical counterpart of the quantum Hamiltonian with non-Hermitian interaction is a complex function of canonical variables. It becomes by the canonical transformation of variables a real function. The quantum–classical correspondence exists precisely for the system with imaginary eigenvalues.
Abstract Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT . We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT -symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.