Recently Scholtz and Geyer proposed a very efficient method to compute metric operators for non-Hermitian Hamiltonians from Moyal products. We develop these ideas further and suggest to use a more symmetrical definition for the Moyal products, because they lead to simpler differential equations. In addition, we demonstrate how to use this approach to determine the Hermitian counterpart for a Pseudo-Hermitian Hamiltonian. We illustrate our suggestions with the explicitly solvable example of the -x^4-potential and the ubiquitous harmonic oscillator in a complex cubic potential.
We show that for the strictly isospectral Hamiltonians, the corresponding coherent states are related by a unitary transformation. As an illustration, we discuss, the example of strictly isospectral one-dimensional harmonic oscillator Hamiltonians and the associated coherent states.
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.
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 discuss a spectrum generating algebra in the supersymmetric quantum mechanical system which is defined as a series of solutions to a specific differential equation. All Hamiltonians have equally spaced eigenvalues, and we realize both positive and negative mode generators of a subalgebra of $W_{1+\infty}$ without use of negative power of raising/lowering operators of the system. All features in the supersymmetric case are generalized to the parasupersymmetric systems of order 2.
The Moyal product is used to cast the equation for the metric of a non-hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form $p^2+V(ix)$ with $V$ polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.
We examine in detail the possibilty of applying Darboux transformation to non Hermitian hamiltonians. In particular we propose a simple method of constructing exactly solvable PT symmetric potentials by applying Darboux transformation to higher states of an exactly solvable PT symmetric potential. It is shown that the resulting hamiltonian and the original one are pseudo supersymmetric partners. We also discuss application of Darboux transformation to hamiltonians with spontaneously broken PT symmetry.
We present a systematic construction of probes into the dynamics of isospectral ensembles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref.[1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the Isospectral twirling of several classes of important quantities in the analysis …
We present, for the isospectral family of oscillator Hamiltonians, a systematic procedure for constructing raising and lowering operators satisfying any prescribed `distorted' Heisenberg algebra (including the $q$-generalization). This is done by means of an operator transformation implemented by a shift operator. The latter is obtained by solving an appropriate partial isometry condition in the Hilbert space. Formal representations of the non-local operators concerned are given in terms of pseudo-differential operators. Using the new annihilation operators, new classes of coherent states are …
We study the Moyal evolution of the canonical position and momentum variables. We compare it with the classical evolution and show that, contrary to what is commonly found in the literature, the two dynamics do not coincide. We prove that this divergence is quite general by studying Hamiltonians of the form $p^2 /2m + V(q)$. Several alternative formulations of Moyal dynamics are then suggested. We introduce the concept of starfunction and use it to reformulate the Moyal equations in terms of a system of ordinary differential equations on the noncommutative Moyal plane. We then use this formula…
We show that the most important measures of quantum chaos like frame potentials, scrambling, Loschmidt echo, and out-of-time-order correlators (OTOCs) can be described by the unified framework of the isospectral twirling, namely the Haar average of a $k$-fold unitary channel. We show that such measures can then be always cast in the form of an expectation value of the isospectral twirling. In literature, quantum chaos is investigated sometimes through the spectrum and some other times through the eigenvectors of the Hamiltonian generating the dynamics. We show that, by exploiting random matrix…
We investigate properties of the most general PT-symmetric non-Hermitian Hamiltonian of cubic order in the annihilation and creation operators as a ten parameter family. For various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral Hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and Moyal products. We elaborate on the subtleties of this approach. For special choices of the ten parameters the Hamiltonian reduces to various models previously studied, such as to the complex cubic pot…
The dynamical algebra associated to a family of isospectral oscillator Hamiltonians is studied through the analysis of its representation in the basis of energy eigenstates. It is shown that this representation becomes similar to that of the standard Heisenberg algebra, and it is dependent of a parameter $w\geq 0$. We name it {\it distorted Heisenberg algebra}, where $w$ is the distortion parameter. The corresponding coherent states for an arbitrary $w$ are derived, and some particular examples are discussed in full detail. A prescription to produce the squeezing, by adequately selecting the i…
By applying algebraic techniques, we construct a two-parametric family of strictly isospectral Hydrogen-like potentials as well as some of its one-parametric limits. An additional one-parametric almost isospectral family of Hydrogen-like potentials is also investigated. It is argued that the construction of a SUSY partner Hamiltonian using a factorization energy $δ$ less than the ground state energy of the departure Hamiltonian is unnecessarily restrictive.
We have recently proposed a strategy to produce, starting from a given hamiltonian $h_1$ and a certain operator $x$ for which $[h_1,xx^\dagger]=0$ and $x^\dagger x$ is invertible, a second hamiltonian $h_2$ with the same eigenvalues as $h_1$ and whose eigenvectors are related to those of $h_1$ by $x^\dagger$. Here we extend this procedure to build up a second hamiltonian, whose eigenvalues are different from those of $h_1$, and whose eigenvectors are still related as before. This new procedure is also extended to crypto-hermitian hamiltonians.
The rationale for introducing non-hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to non-perturbative properties and the metric.
The general solution of SUSY intertwining relations for three-dimensional Schrödinger operators is built using the class of second order supercharges with nondegenerate constant metric. This solution includes several models with arbitrary parameters. We are interested only in quantum systems which are not amenable to separation of variables, i.e. can not be reduced to lower dimensional problems. All constructed Hamiltonians are partially integrable - each of them commutes with a symmetry operator of fourth order in momenta. The same models can be considered also for complex values of parameter…
We show that a polynomial H(N) of degree N of a harmonic oscillator hamiltonian allows us to devise a fully solvable continuous quantum system for which the first N discrete energy eigenvalues can be chosen at will. In general such a choice leads to a re-ordering of the associated energy eigenfunctions of H such that the number of their nodes does not increase monotonically with increasing level number. Systems H have certain universal features, we study their basic behaviours.
The generalized Swanson Hamiltonian $H_{GS} = w (\tilde{a}\tilde{a}^†+ 1/2) + α\tilde{a}^2 + β\tilde{a}^{†^2}$ with $\tilde{a} = A(x)d/dx + B(x)$, can be transformed into an equivalent Hermitian Hamiltonian with the help of a similarity transformation. It is shown that the equivalent Hermitian Hamiltonian can be further transformed into the harmonic oscillator Hamiltonian so long as $[\tilde{a},\tilde{a}^†]=$ constant. However, the main objective of this paper is to show that though the commutator of $\tilde{a}$ and $\tilde{a}^†$ is constant, the generalized Swanson Hamiltonian is not necessar…
We systematically formulate a hierarchy of isospectral Hamiltonians in one-dimensional supersymmetric quantum mechanics on an interval and on a circle, in which two successive Hamiltonians form N=2 supersymmetry. We find that boundary conditions compatible with supersymmetry are severely restricted. In the case of an interval, a hierarchy of, at most, three isospectral Hamiltonians is possible with unique boundary conditions, while in the case of a circle an infinite tower of isospectral Hamiltonians can be constructed with two-parameter family of boundary conditions.
We examine the properties and consequences of pseudo-supersymmetry for quantum systems admitting a pseudo-Hermitian Hamiltonian. We explore the Witten index of pseudo-supersymmetry and show that every pair of diagonalizable (not necessarily Hermitian) Hamiltonians with discrete spectra and real or complex-conjugate pairs of eigenvalues are isospectral and have identical degeneracy structure except perhaps for the zero eigenvalue if and only if they are pseudo-supersymmetric partners. This implies that pseudo-supersymmetry is the basic framework for generating non-Hermitian PT-symmetric and non…
A technique for constructing an infinite tower of pairs of PT-symmetric Hamiltonians, $\hat{H}_n$ and $\hat{K}_n$ (n=2,3,4,...), that have exactly the same eigenvalues is described. The eigenvalue problem for the first Hamiltonian $\hat{H}_n$ of the pair must be posed in the complex domain, so its eigenfunctions satisfy a complex differential equation and fulfill homogeneous boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions of the second Hamiltonian $\hat{K}_n$ of the pair obey a real differential equation and satisfy boundary conditions on the real axis. This equi…
The Hilbert-Schmidt operator formulation of non-commutative quantum mechanics in 2D Moyal plane is shown to allow one to construct Schwinger's SU(2) generators. Using this the SU(2) symmetry aspect of both commutative and non-commutative harmonic oscillator are studied and compared. Particularly, in the non-commutative case we demonstrate the existence of a critical point in the parameter space of mass and angular frequency where there is a manifest SU(2) symmetry for a unphysical harmonic oscillator Hamiltonian built out of commuting (unphysical yet covariantly transforming under SU(2)) posit…
We show that the supersymmetric algebra of Witten's quantum mechanics is invariant under a given point canonical transformation. It is shown that Witten's supersymmetric quantum mechanics can be isospectral or not to the seed Hamiltonian depending on the space coordinate you work on. We illustrate our results by generating a new class of exactly solvable supersymmetric partner Hamiltonians which are not isospectral to the seed Hamiltonian.
In this paper, we construct isospectral Hamiltonians without shape invariant potentials for the relativistic quantum mechanical potentials such as the Dirac Oscillator and Hydrogen-like atom.
Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.
The recently observed quantum states of neutrons bound in a gravitational field are analyzed in the framework of one-parameter isospectral hamiltonians. Potentials isospectral to the usual Newton potential are explicitly constructed for the first time, then constrained using measured properties of the neutron gravitational states. The corresponding wave functions and the neutron fluxes are also calculated and analyzed in a simple model, including the ground state and the excited state contributions. The constructed isospectral potentials are discussed as candidates for a possible modification …
A relativistic phase-space representation for a class of observables with matrix-valued Weyl symbols proportional to the identity matrix (charge-invariant observables)is proposed. We take into account the nontrivial charge structure of the position and momentum operators. The evolution equation coincides with its analog in relativistic quantum mechanics with nonlocal Hamiltonian under conditions where particle-pair creation does not take place (free particle and constant magnetic field). The differences in the equations are connected with peculiarities of the constraints on the initial conditi…
A set of factorization energies is introduced, giving rise to a generalization of the Schrödinger (or Infeld and Hull) factorization for the radial hydrogen-like Hamiltonian. An algebraic intertwining technique involving such factorization energies leads to derive $n$-parametric families of potentials in general almost-isospectral to the hydrogen-like radial Hamiltonians. The construction of SUSY partner Hamiltonians with ground state energies greater than the corresponding ground state energy of the initial Hamiltonian is also explicitly performed.
We present an algorithm to design networks that feature pretty good state transfer (PGST), which is of interest for high-fidelity transfer of information in quantum computing. Realizations of PGST networks have so far mostly relied either on very special network geometries or imposed conditions such as transcendental on-site potentials. However, it was recently shown [Eisenberg et al., arXiv:1804.01645] that PGST generally arises when a network's eigenvectors and the factors $P_{\pm}$ of its characteristic polynomial $P$ fulfill certain conditions, where $P_{\pm}$ correspond to eigenvectors wh…
Some exactly solvable potentials in the position dependent mass background are generated whose bound states are given in terms of Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These potentials are shown to be shape invariant and isospectral to the potentials whose bound state solutions involve classical Laguerre or Jacobi polynomials.
The modified factorization technique of a quantum system characterized by position-dependent mass Hamiltonian is presented. It has been shown that the singular superpotential defined in terms of a mass function and a excited state wave function of a given position-dependent mass Hamiltonian can be used to construct non-singular isospectral Hamiltonians. The method has been illustrated with the help of a few examples.
A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of $η$-weak-pseudo-Hermitian position-dependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target $η$-weak-pseudo-Hermitian PDM -- Hamiltonians' map is suggested. Some $η$-weak-pseudo-Hermitian PT -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-levels crossing and flown-away states phenomena are reported for the resulting Scarf II spectrum. Some of the corresponding $η$-weak-pseudo-Hermitian Scarf II- and periodic-type-isospectral models…
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…
It is shown that if the C operator for a PT-symmetric Hamiltonian with simple eigenvalues is not unique, then it is unbounded. Apart from the special cases of finite-matrix Hamiltonians and Hamiltonians generated by differential expressions with PT-symmetric point interactions, the usual situation is that the C operator is unbounded. The fact that the C operator is unbounded is significant because, while there is a formal equivalence between a PT-symmetric Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the two Hamiltonians are isospectral, the Hilbert spaces are inequ…
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…
In 1956, Freeman Dyson discovered that the practical solution of Schrödinger equation may be facilitated when one replaces the standard self-adjoint Hamiltonian $\mathfrak{h}=\mathfrak{h}^\dagger$ by its manifestly non-Hermitian isospectral avatar $H =Ω^{-1} \mathfrak{h} Ω$ with $Ω^\daggerΩ=Θ\neq I$. The standard unitary interpretation of the evolution can be then achieved either by the ``operator transformation'' (OT) reconstruction of $\mathfrak{h}$, or via a metric-amendment (MA) change of the Hilbert space, ${\cal H}_{auxiliary}\to {\cal H}_{physical}$. In our present letter we describe an…
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 extend the standard intertwining relations used in Supersymmetrical (SUSY) Quantum Mechanics which involve real superpotentials to complex superpotentials. This allows to deal with a large class of non-hermitean Hamiltonians and to study in general the isospectrality between complex potentials. In very specific cases we can construct in a natural way "quasi-complex" potentials which we define as complex potentials having a global property such as to lead to a Hamiltonian with real spectrum. We also obtained a class of complex transparent potentials whose Hamiltonian can be intertwined to a …
New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., $ H_- = {\cal{A}}^{\dagger} {\cal{A}} + α{\cal{A}} ^2 + β{\cal{A}}^{\dagger 2} $, where $ αβ$ are real constants, with $ α\neq β$, and ${\cal{A}}^{\dagger}$ and ${\cal{A}}$ are generalized creation and annihilation operators. It is shown that the initial Hamiltonian $H_-$, and its partner $H_+$, are related by pseudo supersymmetry, and they share all the eigen energies except for the ground state. This pseudo supersymmetric extension enlarges the class of non Hermitian Hamiltonians $H_…
The dynamical algebra associated to a family of Isospectral Oscillator Hamiltonians, named {\it Distorted Heisenberg Algebra} because its dependence on a distortion parameter $W \geq 0$, has been recently studied. The connection of this algebra with the Hilbert space of entire functions of growth (1/2, 2) is analized.
A method to construct non-Dirac-hermitian supersymmetric quantum system that is isospectral with a Dirac-hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving both bosonic and fermionic degrees of freedom in terms of non-Dirac-hermitian operators which are hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. A pseudo-hermitian realization of the Clifford algebra for a pre-determined positive-definite metric is used to construct supersymmetric systems with one or many…
In conventional Schrödinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to H=Ω\,\mathfrak{h}\,Ω\neq H^\dagger$ induces the change ${\cal L} \to {\cal K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\neq H^\dagger$. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into the third, correct physical Hilbert space ${\cal H}$, unitarily equivalent to ${\c…
