We propose a new solvable one-dimensional complex PT-symmetric potential as $V(x)= ig~ \mbox{sgn}(x)~ |1-\exp(2|x|/a)|$ and study the spectrum of $H=-d^2/dx^2+V(x)$. For smaller values of $a,g <1$, there is a finite number of real discrete eigenvalues. As $a$ and $g$ increase, there exist exceptional points (EPs), $g_n$ (for fixed values of $a$) causing a scarcity of real discrete eigenvalues, but there exists at least one. We also show these real discrete eigenvalues as poles of reflection coefficient. We find that the energy-eigenstates $ψ_n(x)$ satisfy (1): PT$ψ_n(x)=1 ψ_n(x)$ and (2): PT$ψ_{E_n}(x)=ψ_{E^*_n}(x)$, for real and complex energy eigenvalues, respectively.
The dilation method is a practical way to experimentally simulate non-Hermitian, especially $\cal PT$-symmetric quantum systems. However, the time-dependent dilation problem cannot be explicitly solved in general. In this paper, we present a simple yet non-trivial exactly solvable dilation problem with two dimensional time-dependent $\cal PT$-symmetric Hamiltonian. Our system is initially set in the unbroken $\cal PT$-symmetric phase and later goes across the so-called exceptional point and enters the broken $\cal PT$-symmetric phase. For this system, the dilated Hamiltonian and the evolution …
A non-unitary version of quantum scattering is studied via an exactly solvable toy model. The model is merely asymptotically local since the smooth path of the coordinate is admitted complex in the non-asymptotic domain. At any real angular-momentum-like parameter the reflection R and transmission T are shown to change with the winding number (i.e., topology) of the path. The points of unitarity appear related to the points of existence of quantum-knot bound states.
The O(N) invariant quartic anharmonic oscillator is shown to be exactly solvable if the interaction parameter satisfies special conditions. The problem is directly related to that of a quantum double well anharmonic oscillator in an external field. A finite dimensional matrix equation for the problem is constructed explicitly, along with analytical expressions for some excited states in the system. The corresponding Niven equations for determining the polynomial solutions for the problem are given.
At the lower edge of the energy continuum the birth of an isolated quantum bound state is studied as caused by an infinitesimally small change of the interaction. In our model a single, asymptotically free massive quantum particle is assumed moving along a discretized real line of coordinates, $x \in \mathbb{Z}$. The motion is assumed controlled by a weakly nonlocal 2J-parametric external potential which is non-Hermitian but PT-symmetric. Mathematically, the bound states are then reinterpreted as Sturmians, i.e., the bound-state energy is treated as a variable real parameter while the value of…
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.
Matrix quasi exactly solvable operators are considered and new conditions are determined to test whether a matrix differential operator possesses one or several finite dimensional invariant vector spaces. New examples of $2\times 2$-matrix quasi exactly solvable operators are constructed with the emphasis set on PT-symmetric Hamiltonians.
For applications to quasi-exactly solvable Schrödinger equations in quantum mechanics, we consider the general conditions that have to be satisfied by the coefficients of a second-order differential equation with at most $k+1$ singular points in order that this equation has particular solutions that are $n$th-degree polynomials. In a first approach, we show that such conditions involve $k-2$ integration constants, which satisfy a system of linear equations whose coefficients can be written in terms of elementary symmetric polynomials in the polynomial solution roots whenver such roots are all …
The solutions of trigonometric Scarf potential, PT/non-PT-symmetric and non-Hermitian q-deformed hyperbolic Scarf and Manning-Rosen potentials are obtained by solving the Schrodinger equation. The Nikiforov-Uvarov method is used to obtain the real energy spectra and corresponding eigenfunctions.
The generalized Sinh-Gordon potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. The quasi exact solutions of energy eigenvalues and eigenfunctions of the generalized Sinh-Gordon potential are found for n=0,1 states.
The spectrum of a one-dimensional Hamiltonian with potential $V(x)=ix^2$ for negative $x$ and $V(x)=-ix^2$ for positive $x$ is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.
The simplest purely imaginary and piecewise constant $\cal PT$-symmetric potential located inside a larger box is studied. Unless its strength exceeds a certain critical value, all the spectrum of its bound states remains real and discrete. We interpret such a model as an initial element of the generalized non-Hermitian Witten's hierarchy of solvable Hamiltonians and construct its first supersymmetric (SUSY) partner in closed form.
Within quantum mechanics which works with parity-pseudo-Hermitian Hamiltonians we study the tunneling in a symmetric double well formed by two delta functions with complex conjugate strengths. The model is exactly solvable and exhibits several interesting features. Besides an amazingly robust absence of any PT symmetry breaking, we observe a quasi-degeneracy of the levels which occurs all over the energy range including the high-energy domain. This pattern is interpreted as a manifestation of certain "quantum beats".
The ${\cal PT}$ symmetric version of the generalised Ginocchio potential, a member of the general exactly solvable Natanzon potential class is analysed and its properties are compared with those of ${\cal PT}$ symmetric potentials from the more restricted shape-invariant class. It is found that the ${\cal PT}$ symmetric generalised Ginocchio potential has a number of properties in common with the latter potentials: it can be generated by an imaginary coordinate shift $x\to x+{\rm i}ε$; its states are characterised by the quasi-parity quantum number; the spontaneous breakdown of ${\cal PT}$ sym…
Non-hermitian, $\mathcal{PT}$-symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, $\mathcal{PT}$ potentials for a non-relativistic particle confined in a circular geometry. We show that the $\mathcal{PT}$ symmetry threshold can be tuned by introducing a second gain-loss potential or its hermitian counterpart. Our results explicitly demonstrate that $\mathcal{PT}$ breaking in two dimensions has a rich phase diagram, …
The ${\cal PT}$ symmetric version of the generalised Ginocchio potential, a member of the general exactly solvable Natanzon potential class is analysed and its properties are compared with those of ${\cal PT}$ symmetric potentials from the more restricted shape-invariant class. It is found that the ${\cal PT}$ symmetric generalised Ginocchio potential has a number of properties in common with the latter potentials: it can be generated by an imaginary coordinate shift $x\to x+{\rm i}ε$; its states are characterised by the quasi-parity quantum number; the spontaneous breakdown of ${\cal PT}$ sym…
We investigate complex PT-symmetric potentials, associated with quasi-exactly solvable non-hermitian models involving polynomials and a class of rational functions. We also look for special solutions of intertwining relations of SUSY Quantum Mechanics providing a partnership between a real and a complex PT-symmetric potential of the kind mentioned above. We investigate conditions sufficient to ensure the reality of the full spectrum or, for the quasi-exactly solvable systems, the reality of the energy of the finite number of levels.
By applying the higher order Darboux algorithm to an exactly solvable non Hermitian ${\cal{PT}}$ symmetric potential, we obtain a hierarchy of new exactly solvable non Hermitian ${\cal{PT}}$ symmetric potentials with real spectra. It is shown that the symmetry underlying the potentials so generated and the original one is {\it nonlinear pseudo supersymmetry}. We also show that this formalism can be used to generate a larger class of new solvable potentials when applied to non Hermitian systems.
Morse potential $V_M(x)= g^2\exp (2x)-g(2h+1)\exp(x)$ is defined on the full line, $-\infty<x<\infty$ and it defines an exactly solvable 1-d quantum mechanical system with finitely many discrete eigenstates. By taking its right half $0\le x<\infty$ and glueing it with the left half of its mirror image $V_M(-x)$, $-\infty<x\le0$, the symmetric Morse potential $V(x)= g^2\exp (2|x|)-g(2h+1)\exp(|x|)$ is obtained. The quantum mechanical system of this piecewise analytic potential has infinitely many discrete eigenstates with the corresponding eigenfunctions given by the Whittaker W function. The e…
It is well known that using the conventional non-Hermitian but ${\cal PT}-$symmetric Bose-Hubbard Hamiltonian with real spectrum one can realize the Bose-Einstein condensation (BEC) process in an exceptional-point limit of order $N$. Such an exactly solvable simulation of the BEC-type phase transition is, unfortunately, incomplete because the standard version of the model only offers an extreme form of the limit characterized by a minimal geometric multiplicity $K=1$. In our paper we describe a rescaled and partitioned direct-sum modification of the linear version of the Bose-Hubbard model whi…
We present a class of confining potentials which allow one to reduce the one-dimensional Schroodinger equation to a named equation of mathematical physics, namely either Bessel's or Whittaker's differential equation. In all cases, we provide closed form expressions for both the symmetric and antisymmetric wavefunction solutions, each along with an associated transcendental equation for allowed eigenvalues. The class of potentials considered contains an example of both cusp-like single wells and a double-well.
Discrete multiparametric 1D quantum well with PT-symmetric long-range boundary conditions is proposed and studied. As a nonlocal descendant of the square well families endowed with Dirac (i.e., Hermitian) and with complex Robin (i.e., non-Hermitian but still local) boundary conditions, the model is shown characterized by the survival of solvability in combination with an enhanced spectral-design flexibility. The solvability incorporates also the feasibility of closed-form constructions of the physical Hilbert-space inner products rendering the time-evolution unitary.
We introduce a very simple, exactly solvable PT-symmetric non-Hermitian model with real spectrum, and derive a closed formula for the metric operator which relates the problem to a Hermitian one.
For any central potential V in D dimensions, the angular Schroedinger equation remains the same and defines the so called hyperspherical harmonics. For non-central models, the situation is more complicated. We contemplate two examples in the plane: (1) the partial differential Calogero's three-body model (without centre of mass and with an impenetrable core in the two-body interaction), and (2) the Smorodinsky-Winternitz' superintegrable harmonic oscillator (with one or two impenetrable barriers). These examples are solvable due to the presence of the barriers. We contemplate a small complex s…
The family of complex PT-symmetric sextic potentials is studied to show that for various cases the system is essentially quasi-solvable and possesses real, discrete energy eigenvalues. For a particular choice of parameters, we find that under supersymmetric transformations the underlying potential picks up a reflectionless part.
We study the spectrum in such a PT-symmetric square well of a diameter L where the "strength of the non-Hermiticity" is controlled by the two parameters, viz., by an imaginary coupling ig and by the distance d of its onset from the origin. We solve this problem and confirm that the spectrum is discrete and real in a non-empty interval of g. Surprisingly, a specific distinction between the bound states is found in their asymptotic stability/instability with respect to an unlimited growth of g. In our model, all of the low-lying levels remain asymptotically unstable at the small d and finite L w…
In this paper, we introduce a family of sextic potentials that are exactly solvable, and for the first time, a family of triple-well potentials with their whole energy spectrum and wavefunctions using supersymmetry method. It was suggested since three decades ago that all "additive" or "translational" shape invariant superpotentials formed by two combination of functions have been found and their list was already exhausted by the well-known exactly solvable potentials that are available in most textbooks and furthermore, there are no others. We have devised a new family of superpotentials form…
Bound states generated by K coupled PT-symmetric square wells are studied in a series of models where the Hamiltonians are assumed $R-$pseudo-Hermitian and $R^2-$symmetric. Specific rotation-like generalized parities $R$ are considered such that $R^N=I$ at some integers N. We show that and how our assumptions make the models exactly solvable and quasi-Hermitian. This means that they possess the real spectra as well as the standard probabilistic interpretation.
We consider optical systems where propagation of light can be described by a Dirac-like equation with $PT$-symmetric Hamiltonian. In order to construct exactly solvable configurations, we extend the confluent Crum-Darboux transformation for the one-dimensional Dirac equation. The properties of the associated intertwining operators are discussed and the explicit form for higher-order transformations is presented. We utilize the results to derive a multi-parametric class of exactly solvable systems where the balanced gain and loss represented by the $PT$-symmetric refractive index can imply loca…
Searching for non-Hermitian (parity-time)$\mathcal{PT}$-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian Hamiltonian model which is given as $\hat{\mathcal{H}}=ω(\hat{b}^\dagger\hat{b}+\frac{1}{2})+ α(\hat{b}^{2}-(\hat{b}^\dagger)^{2})$ where $ω$ and $α$ are real constants, $\hat{b}$ and $\hat{b^\dagger}$ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of $\mathcal{PT}$ symmetry has been attracting a growing intere…
Associated Lamé potentials $V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\cn^2 (x,m)}/{\dn^2(x,m)}$ are used to construct complex, PT-invariant, periodic potentials using the anti-isospectral transformation $x \to ix+β$, where $β$ is any nonzero real number. These PT-invariant potentials are defined by $V^{PT}(x) \equiv -V(ix+β)$, and have a different real period from $V(x)$. They are analytically solvable potentials with a finite number of band gaps, when $a$ and $b$ are integers. Explicit expressions for the band edges of some of these potentials are given. For the special case of the complex potential $V…
The Dirac equation is considered in the background of potentials of several types, namely scalar and vector-potentials as well as "Dirac-oscillator" potential or some of its generalisations. We investigate the radial Dirac equation within a quite general spherically symmetric form for these potentials and we analyse some exactly and quasi exactly solvable properties of the underlying matricial linear operators.
A conditionally exactly solvable potential, the supersymmetric partner of the harmonic oscillator is investigated in the PT-symmetric setting. It is shown that a number of properties characterizing shape-invariant and Natanzon-class potentials generated by an imaginary coordinate shift $x-{\rm i}ε$ also hold for this potential outside the Natanzon class.
We start from a seven parameters (six continuous and one discrete) family of non-central exactly solvable potential in three dimensions and construct a wide class of ten parameters (six continuous and four discrete) family of rationally extended exactly solvable non-central real as well as PT symmetric complex potentials. The energy eigenvalues and the eigenfunctions of these extended non-central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these po- tentials are either associated with the exceptional orthogonal polynomials (EOPs) or some type of new polyn…
We use a Lie algebraic technique to construct complex quasi exactly solvable potentials with real spectrum. In particular we obtain exact solutions of a complex sextic oscillator potential and also a complex potential belonging to the Morse family.
The spectrum of complex PT-symmetric potential, $V(x)=igx$, is known to be null. We enclose this potential in a hard-box: $V(|x| \ge 1) =\infty $ and in a soft-box: $V(|x|\ge 1)=0$. In the former case, we find real discrete spectrum and the exceptional points of the potential. The asymptotic eigenvalues behave as $E_n \sim n^2.$ The solvable purely imaginary PT-symmetric potentials vanishing asymptotically known so far do not have real discrete spectrum. Our solvable soft-box potential possesses two real negative discrete eigenvalues if $|g|<(1.22330447)$. The soft-box potential turns out to b…
Supersymmetrical intertwining relations of second order in derivatives allow to construct a two-dimensional quantum model with complex potential, for which {\it all} energy levels and bound state wave functions are obtained analytically. This model {\it is not amenable} to separation of variables, and it can be considered as a specific complexified version of generalized two-dimensional Morse model with additional $\sinh^{-2}$ term. The energy spectrum of the model is proved to be purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensi…
Recently, a class of PT-invariant quantum mechanical models described by the non-Hermitian Hamiltonian $H=p^2+x^2(ix)^ε$ was studied. It was found that the energy levels for this theory are real for all $ε\geq0$. Here, the limit as $ε\to\infty$ is examined. It is shown that in this limit, the theory becomes exactly solvable. A generalization of this Hamiltonian, $H=p^2+x^{2M}(ix)^ε$ (M=1,2,3,...) is also studied, and this PT-symmetric Hamiltonian becomes exactly solvable in the large-εlimit as well. In effect, what is obtained in each case is a complex analog of the Hamiltonian for the square …
In this paper we present a novel quasi-exactly solvable model with symmetric inverted potentials which are unbounded from below. The quasi-exactly solvable states are shown to be total transmission (or reflectionless) modes. From these modes even and odd wavefunctions can be constructed which are normalizable and flux-zero. Under the procedure of self-adjoint extension, a discrete spectrum of bound states can be obtained for these inverted potentials and the solvable part of the spectrum is the quasi-exactly solvable states we have discovered.
Below we study theoretically and numerically the asymptotics of the algebraic part of the spectrum for the quasi-exactly solvable sextic potential, its level crossing points, and its monodromy in the complex plane of its parameter. We also discuss connection between the quasi-exactly solvable sextic and the classical quartic potential.
We construct an exactly solvable PT-symmetric example of Sturmian bound states which exist in the absence of any confining potential. Their origin is purely topological -- these states live on certain nontrivial contours of complex coordinates.
We consider a PT Symmetric Partner to Khare Mandal's recently proposed non-Hermitian potential with complex eigen values. Our potential is Quasi-Exactly solvable and is shown to possess only real eigen values.
We consider quasinormal modes with complex energies from the point of view of the theory of quasi-exactly solvable (QES) models. We demonstrate that it is possible to find new potentials which admit exactly solvable or QES quasinormal modes by suitable complexification of parameters defining the QES potentials. Particularly, we obtain one QES and four exactly solvable potentials out of the five one-dimensional QES systems based on the $sl(2)$ algebra.
A new two-parameter family of quasi-exactly solvable quartic polynomial potentials $V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x$ is introduced. Until now, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial potential is sextic. This belief is based on the assumption that the Hamiltonian must be Hermitian. However, it has recently been discovered that there are huge classes of non-Hermitian, ${\cal PT}$-symmetric Hamiltonians whose spectra are real, discrete, and bounded below [physics/9712001]. Replacing Hermiticity by the weaker condition of ${\cal PT}$ symmetr…
We deform the real potential of Poeschl and Teller by a shift of its coordinate in imaginary direction. We show that the new model remains exactly solvable. Its bound states are constructed in closed form. Wave functions are complex and proportional to Jacobi polynomials. Some of them diverge in the Hermitian limit. In contrast, all their energies prove real and shift-independent. In this sense the lost Hermiticity of our family of Hamiltonians seems replaced by their accidental PT symmetry.
This paper gives a new perspective on how to solve the second-order linear differential equation written in normal form. Extending the argument of the potential to a complex number leads to solving exactly the Schrödinger equation when the potential is complex using the factorization method. This method leads to solving two Riccati nonlinear equations and by constructing the only possible superpotential, the factorization method gives the eigenvalues and eigenfunctions in closed form for potentials satisfying the shape invariance property. Extending the potential to the complex argument has le…
A general formalism is worked out for the description of one-dimensional scattering by non-local separable potentials and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. The case of a solvable Yamaguchi potential is discussed in detail.
We complexify one of the Natanzon's exactly solvable potentials in PT symmetric manner and discover that it supports the pairs of bound states with the same number of nodal zeros. This could indicate that the Sturm Liouville oscillation theorem does not admit an immediate generalization.
Exact solutions of Schrodinger equation for PT-/non-PT-symmetric and non-Hermitian Morse and Poschl-Teller potentials are obtained with the position-dependent effective mass by applying a point canonical transformation method. Three kinds of mass distributions are used in order to construct exactly solvable target potentials and obtain energy spectrum and corresponding wave functions.
A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in closed form. An entirely new class of QES Hamiltonians having sextic polynomial potentials is constructed. These new Hamiltonians are different from the sextic QES Hamiltonians in the literature because their eigenfunctions obey PT-symmetric rather than Hermitian boundary conditions. These new Hamiltonians present a novel problem that is not encountered when the Hamiltonian is Hermitian: It is necessary to distinguish between the parametric…
We obtained a new class of exactly-solvable potentials by means of the hypergeometric equation for Schrodinger equation, which different from the exactly-solvable potentials introduced by Bose and Natanzon. Using the new class of solvable potentials, we can obtain the corresponding complex PT-invariant potentials. This method can also apply to the other Fuchs equations.
We propose a general method for constructing quasi-exactly solvable potentials with three analytic eigenstates. These potentials can be real or complex functions but the spectrum is real. A comparison with other methods is also performed.
We study a wide class of solvable PT symmetric potentials in order to identify conditions under which these potentials have regular solutions with complex energy. Besides confirming previous findings for two potentials, most of our results are new. We demonstrate that the occurrence of conjugate energy pairs is a natural phenomenon for these potentials. We demonstrate that the present method can readily be extended to further potential classes.
The recently proposed PT-symmetric quantum mechanics works with complex potentials which possess, roughly speaking, a symmetric real part and an anti-symmetric imaginary part. We propose and describe a new exactly solvable model of this type. It is defined as a specific analytic continuation of the shape-invariant potential of Morse. In contrast to the latter well-known example, all the new spectrum proves real, discrete and bounded below. All its three separate subsequences are quadratic in n.
