The relevance in Physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics and quantum field theory. %stochastic processesand so on. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of $H$ is obtained using pseudo-bosonic operators.
One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 {\it Commun. Pure Appl. Math.} \tb{13} 609) which may possess a spectral singularity is analyzed from the point of view of non-Hermitian generalization of quantum mechanics. It is shown that $η$ operator, being a second order differential operator, has supersymmetric structure. Asymptotic behavior of eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible…
We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly time-dependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT-symmetry. In particular, for the harmonic oscillator perturbed by a cubic non-Hermitian term, we evaluate explicitly various transition amplitudes, for the situation when these systems are exposed to a monochromatic linearly polarized electric field.
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of unco…
The powerful group theoretical formalism of potential algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by complexifying so(2,1), thereby getting the complex algebra sl(2,\C) or $A_1$. This leads to new types of both PT-symmetric and non-PT-symmetric Hamiltonians.
A harmonic oscillator Hamiltonian augmented by a non-Hermitian \pt-symmetric part and its su(1,1) generalizations, for which a family of positive-definite metric operators was recently constructed, are re-examined in a supersymmetric context. Quasi-Hermitian supersymmetric extensions of such Hamiltonians are proposed by enlarging su(1,1) to a ${\rm su}(1,1/1) \sim {\rm osp}(2/2, \R)$ superalgebra. This allows the construction of new non-Hermitian Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.
We characterize the subclass of quasianti-Hermitian quaternionic Hamiltonian dynamics such that their complex projections are one-parameter semigroup dynamics in the space of complex quasi-Hermitian density matrices. As an example, the complex projection of a spin-1/2 system in a constant quasianti-Hermitian quaternionic potential is considered.
Floquet exceptional points correspond to the coalescence of two (or more) quasi-energies and corresponding Floquet eigenstates of a time-periodic non-Hermitian Hamiltonian. They generally arise when the oscillation frequency satisfies a multiphoton resonance condition. Here we discuss the interplay between Floquet exceptional points and the chiral dynamics observed, over several oscillation cycles, in a wide class of non-Hermitian systems when they are slowly cycled in opposite directions of parameter space.
We show how, given a non-Hermitian Hamiltonian $H$, we can generate new non-Hermitian operators sequentially, producing a virtually infinite chain of non-Hermitian Hamiltonians which are isospectral to $H$ and $H^\dagger$ and whose eigenvectors we can easily deduce in an almost automatic way; no ingredients are necessary other than $H$ and its eigensystem. To set off the chain and keep it running, we use, for the first time in our knowledge, a series of maps all connected to different metric operators. We show how the procedure works in several physically relevant systems. In particular, we ap…
The Double Green Function Formalism has been extensively used in dealing with the thermodynamics of quantum systems which evolved in time under the action of a given self-adjoint Hamiltonian. In this work, we extend the formalism to include pseudo-hermitian Hamiltonians. We apply the formalism to study the PT-symmetry Swanson Hamiltonian at finite temperature, both in the PT-unbroken and in the PT-broken symmetry phase. We analyse the behaviour of the system, which is initially at equilibrium at a given temperature, when it is perturbed by a periodic interaction.
We propose construction of a unique and definite metric ($η_+$), time-reversal operator (T) and an inner product such that the pseudo-Hermitian matrix Hamiltonians are C, PT, and CPT invariant and PT(CPT)-norm is indefinite (definite). Here, P and C denote the generalized symmetries : parity and charge-conjugation respectively. The limitations of the other current approaches have been brought out.
A new class of non-Hermitian Hamiltonians with real spectrum, which are written as a real linear combination of su(2) generators in the form $ H=ωJ_{3}+αJ_{-}+βJ_{+}$, $α\neq β$, is analyzed. The metrics which allows the transition to the equivalent Hermitian Hamiltonian is established. A pseudo-Hermitian supersymmetic extension of such Hamiltonians is performed. They correspond to the pseudo-Hermitian supersymmetric systems of the boson-phermion oscillators. We extend the supercoherent states formalism to such supersymmetic systems via the pseudo-unitary supersymmetric displacement operator m…
This paper deals with the mathematical spectral analysis and physical interpretation of a fermionic system described by a non-Hermitian Hamiltonian possessing real eigenvalues. A statistical thermodynamical description of such a system is considered. Approximate expressions for the energy expectation value and the number operator expectation value, in terms of the absolute temperature $T$ and of the chemical potential $μ$, are obtained, based on the Euler-Maclaurin formula.
Potential algebras are extended from Hermitian to non-Hermitian Hamiltonians and shown to provide an elegant method for studying the transition from real to complex eigenvalues for a class of non-Hermitian Hamiltonians associated with the complex Lie algebra A$_1$.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-compon…
Being chosen as a differential operator of a special form, metric $η$ operator becomes unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits to determine both the metric operator and corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it becomes a superpartner of another Hermitian Hamiltonian.
We propose an extended version of supersymmetric quantum mechanics which can be useful if the Hamiltonian of the physical system under investigation is not Hermitian. The method is based on the use of two, in general different, superpotentials. Bi-coherent states of the Gazeau-Klauder type are constructed and their properties are analyzed. Some examples are also discussed, including an application to the Black-Scholes equation, one of the most important equations in Finance.
The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads to a challenging open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives the exact asymptotic behavior of the eigenfunction.
We consider a two-level system such as a two-level atom, interacting with a cavity field mode in the rotating wave approximation, when the atomic transition frequency or the field mode frequency is periodically driven in time. We show that in both cases, for an appropriate choice of the modulation parameters, the state amplitudes in a generic $n${-}excitation subspace obey the same equations of motion that can be obtained from a \emph{static} non-Hermitian Jaynes-Cummings Hamiltonian with ${\mathcal PT}$ symmetry, that is with an imaginary coupling constant. This gives further support to recen…
Diagonalizable pseudo-Hermitian Hamiltonians with real and discrete spectra, which are superpartners of Hermitian Hamiltonians, must be $η$-pseudo-Hermitian with Hermitian, positive-definite and non-singular $η$ operators. We show that despite the fact that an $η$ operator produced by a supersymmetric transformation, corresponding to the exact supersymmetry, is singular, it can be used to find the eigenfunctions of a Hermitian operator equivalent to the given pseudo-Hermitian Hamiltonian. Once the eigenfunctions of the Hermitian operator are found the operator may be reconstructed with the hel…
We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors.
We present a new approach to study a class of non-Hermitian (1+1)-dimensional Dirac Hamiltonian in the presence of local Fermi velocity. We apply the well known Nikiforov-Uvarov method to solve such a system. We discuss applications and explore the solvability of both $\mathcal{PT}$-symmetric and non-$\mathcal{PT}$ symmetric classes of potentials. In the former case we obtain the solution of a harmonic oscillator in the presence of a linear vector potential while in the latter case we solve the shifted harmonic oscillator problem.
A class of non-Hermitian quadratic su(2) Hamiltonians having an anti-linear symmetry is constructed. This is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra. In fact, different realisations for this type of symmetry are obtained, including the natural occurrence of charge conjugation together with parity and time reversal. Once specified the underlying anti-linear symmetry of the Hamiltonian, the former, if unbroken, leads to a purely real spectrum and the latter can be mapped to a Hermitian counterpart by, amongst other possibilities, a s…
Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: $ e^{-αp} x e^{αp} = x+ i α$, to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.
In this letter we will use higher-order supersymmetric quantum mechanics to obtain several families of complex solutions of the Painleve IV equation with real parameters. We shall also study the algebraic structure, the eigenfunctions and the energy spectra of the corresponding non-hermitian Hamiltonians.
We give a simple proof of the fact that every diagonalizable operator that has a real spectrum is quasi-Hermitian and show how the metric operators associated with a quasi-Hermitian Hamiltonian are related to the symmetry generators of an equivalent Hermitian Hamiltonian.
A generalized non-Hermitian oscillator Hamiltonian is proposed that consists of additional linear terms which break PT-symmetry explicitly. The model is put into an equivalent Hermitian form by means of a similarity transformation and the criterion of N-fold supersymmetry with a position-dependent mass is shown to reside in it.
We show that complex Lie algebras (in particular sl(2,C)) provide us with an elegant method for studying the transition from real to complex eigenvalues of a class of non-Hermitian Hamiltonians: complexified Scarf II, generalized Pöschl-Teller, and Morse. The characterizations of these Hamiltonians under the so-called pseudo-Hermiticity are also discussed.
In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\cP\cT$-symmetric wrong-sign quartic Hamiltonian $H=\half p^2-gx^4$ has the same spectrum as the conventional Hermitian Hamiltonian $\tilde H=\half p^2+4g x^4-\sqrt{2g} x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken pa…
Recently Bender, Brody, Jones and Meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small, when the time-evolution operator is taken to be non-Hermitian but PT-symmetric. Here we demonstrate that such phenomena can also be obtained for non-Hermitian Hamiltonians for which PT-symmetry is completely broken, i.e. dissipative systems. We observe that the effect of a tunable passage time can be achieved by projecting between orthogonal eigenstates by means of a time-evolut…
The eigenvalue problem of the spherically symmetric oscillator Hamiltonian is revisited in the context of canonical raising and lowering operators. The Hamiltonian is then factorized in terms of two not mutually adjoint factorizing operators which, in turn, give rise to a non-Hermitian radial Hamiltonian. The set of eigenvalues of this new Hamiltonian is exactly the same as the energy spectrum of the radial oscillator and the new square-integrable eigenfunctions are complex Darboux-deformations of the associated Laguerre polynomials.
Using techniques of supersymmetric quantum mechanics, scattering properties of Hermitian Hamiltonians, which are related to non-Hermitian ones by similarity transformations, are studied. We have found that the scattering matrix of the Hermitian Hamiltonian coincides with the phase factor of the non-unitary scattering matrix of the non-Hermitian Hamiltonian. The possible presence of a spectral singularity in a non-Hermitian Hamiltonian translates into a pronounced resonance in the scattering cross section of its Hermitian counterpart. This opens a way for detecting spectral singularities in sca…
We discuss the dynamical quantum systems which turn out to be bi-unitary with respect to the same alternative Hermitian structures in a infinite-dimensional complex Hilbert space. We give a necessary and sufficient condition so that the Hermitian structures are in generic position. Finally the transformations of the bi-unitary group are explicitly obtained.
We present a large class of non-Hermitian non-PT-symmetric two-component Dirac Hamiltoninas with real energy spectra. These Hamiltonians are invariant under the combined action of "charge" conjugation (two-component transpose) and space-parity. Examples are given from the two subclasses of these systems having localized and/or continuum states with real energies.
We analyse a class of non-Hermitian Hamiltonians, which can be expressed bilinearly in terms of generators of a sl(2,R)-Lie algebra or their isomorphic su(1,1)-counterparts. The Hamlitonians are prototypes for solvable models of Lie algebraic type. Demanding a real spectrum and the existence of a well defined metric, we systematically investigate the constraints these requirements impose on the coupling constants of the model and the parameters in the metric operator. We compute isospectral Hermitian counterparts for some of the original non-Hermitian Hamiltonian. Alternatively we employ a gen…
The family of metric operators, constructed by Musumbu {\sl et al} (2007 {\sl J. Phys. A: Math. Theor.} {\bf 40} F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian $\cal PT$-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily $\cal PT$-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some exa…
For an arbitrary possibly non-Hermitian matrix Hamiltonian H, that might involve exceptional points, we construct an appropriate parameter space M and the lines bundle L^n over M such that the adiabatic geometric phases associated with the eigenstates of the initial Hamiltonian coincide with the holonomies of L^n. We examine the case of 2 x 2 matrix Hamiltonians in detail and show that, contrary to claims made in some recent publications, geometric phases arising from encircling exceptional points are generally geometrical and not topological in nature.
A transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator $η$ are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.
Non-Hermitian Hamiltonians are generally sensitive to boundary conditions, and their spectra and wave functions under open boundary conditions are not necessarily predicted by the Bloch band theory for periodic boundary conditions. To elucidate such a non-Bloch feature, recent works have developed a non-Bloch band theory that works even under arbitrary boundary conditions. Here, it is demonstrated that the standard non-Bloch band theory breaks down in the symplectic class, in which non-Hermitian Hamiltonians exhibit Kramers degeneracy because of reciprocity. Instead, a modified non-Bloch band …
We propose a procedure to obtain exact analytical solutions to the time-dependent Schrödinger equations involving explicit time-dependent Hermitian Hamitonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.
By modifying and generalizing known supersymmetric models we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of $\frac{d}{dx}$. Also, by changing the derivative $\frac{d}{dx}$ by the Dunkl derivative in the creation and annihilation-type operators of …
This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identi…
An algorithm for finding the bound-state eigenvalues and eigenfunctions of a Hermitian Hamiltonian operator using Green's method, developed by Waxman\cite{W98},has been extended to include non-Hermitian Hamiltonian operators.
We introduce, for the first time, bicoherent-state path integration as a method for quantizing non-hermitian systems. Bicoherent-state path integrals arise as a natural generalization of ordinary coherent-state path integrals, familiar from hermitian quantum physics. We do all this by working out a concrete example, namely, computation of the propagator of a certain quasi-hermitian variant of Swanson's model, which is not invariant under conventional $PT$-transformation. The resulting propagator coincides with that of the propagator of the standard harmonic oscillator, which is isospectral wit…
We formulate a systematic algorithm for constructing a whole class of Hermitian position-dependent-mass Hamiltonians which, to lowest order of perturbation theory, allow a description in terms of PT-symmetric Hamiltonians. The method is applied to the Hermitian analogue of the PT-symmetric cubic anharmonic oscillator. A new example is provided by a Hamiltonian (approximately) equivalent to a PT-symmetric extension of the one-parameter trigonometric Poschl-Teller potential.
For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys Θ^2=-1, but no such a degeneracy exists when Θ^2=+1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when Θ^2=+1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states w…
In this talk I present a simple and unified approach to both exact and quasi-exact solvabilities of the one-dimensional Schrödinger equation. It is based on the prepotential together with Bethe ansatz equations. This approach gives the potential as well as the eigenfunctions and eigenvalues simultaneously. In this approach the system is completely defined by the choice of the change of variables, and the so-called zero-th order prepotential. We illustrate the approach by several examples of Hermitian and non-Hermitian Hamiltonians with real energies. The method can be easily extended to the co…
We show that a quantum system possessing an exact antilinear symmetry, in particular PT-symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We apply our general results to PT-symmetry in finite-dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT-symmetric field theories are equivalent to certain nonlocal Hermiti…
A complexified von Roos Hamiltonian is considered and a Hermitian first-order intertwining differential operator is used to obtain the related position dependent mass $η$-weak-pseudo-Hermitian Hamiltonians. Using a Liouvillean-type change of variables, the $η$-weak-pseudo-Hermitian von Roos Hamiltonians H(x) are mapped into the traditional Schrodinger Hamiltonian form H(q), where exact isospectral correspondence between H(x) and H(q) is obtained. Under a user-friendly position dependent mass settings, it is observed that for each exactly-solvable $η$-weak-pseudo-Hermitian reference-Hamiltonian…
We investigate some questions on the construction of $η$ operators for pseudo-Hermitian Hamiltonians. We give a sufficient condition which can be exploited to systematically generate a sequence of $η$ operators starting from a known one, thereby proving the non-uniqueness of $η$ for a particular pseudo-Hermitian Hamiltonian. We also study perturbed Hamiltonians for which $η$'s corresponding to the original Hamiltonian still work.
We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe Ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians. It is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wave function with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter the non-Hermitian is broken PT symmetry…
In this work we address systems described by time-dependent non-Hermitian Hamiltonians under time-dependent Dyson maps. We shown that when starting from a given time-dependent non-Hermitian Hamiltonian which is not itself an observable, an infinite chain of gauge linked time-dependent non-observable non-Hermitian Hamiltonians can be derived from it. The matrix elements of the observables associated with all these non observable Hamiltonians are, however, all linked to each other, and in the particular case where global gauges exist, these matrix elements becomes all identical to each other. In…
Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P.
