David J Fernandez C, Asish Ganguly
We obtain exact solutions of the one-dimensional Schrodinger equation for some families of associated Lame potentials with arbitrary energy through a suitable ansatz, which may be appropriately extended for other such a families. The formalism of supersymmetric quantum mechanics is used to generate new exactly solvable potentials.
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The general solution of the stationary Schrodinger equation for the associated Lame potentials with an arbitrary real energy is found. The supersymmetric partners are generated by employing seeds solutions for factorization energies inside the gaps.
A systematic procedure to derive exact solutions of the associated Lame equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are used to generate new exactly solvable potentials; some of them exhibit an interesting property of periodicity defects.
Supersymmetry transformations of first and second order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. It is studied also the way in which the eigenfunctions of the initial Hamiltonian are transformed. The first and certain second order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators. Since systems with this kind of operators are linked with the Painlevé IV equation, several solutions of this non-linear second-order differential equation will be simply found.
In a box of size $L$, a spatially antisymmetric square-well potential of a purely imaginary strength ${\rm i}g$ and size $l < L$ is interpreted as an initial element of the SUSY hierarchy of solvable Hamiltonians, the energies of which are all real for $g < g_c(l)$. The first partner potential is constructed in closed form and discussed.
Using the formalism of supersymmetric quantum mechanics, we obtain a large number of new analytically solvable one-dimensional periodic potentials and study their properties. More specifically, the supersymmetric partners of the Lame potentials ma(a+1)sn^2(x,m) are computed for integer values a=1,2,3,.... For all cases (except a=1), we show that the partner potential is distinctly different from the original Lame potential, even though they both have the same energy band structure. We also derive and discuss the energy band edges of the associated Lame potentials pm sn^2(x,m)+qm cn^2(x,m)/ dn^…
Supersymmetric partners of the trigonometric Poschl-Teller potentials
本文通过一阶和二阶超对称变换从三角波薛定谔势构造具有可控能谱的新哈密顿量，系统研究了修改能级与边界行为。
The $\cal PT$-symmetric complexified Scarf II potential $V(x)= - V_1 \sech^{2}x + {\rm i} V_2 \sech x \tanh x$, $V_1>0$ , $V_{2}\neq 0$ is revisited to study the interplay among its coupling parameters. The existence of an isolated real and positive energy level that has been recently identified as a spectral singularity or zero-width resonance is here demonstrated through the behaviour of the corresponding wavefunctions and some property of the associated pseudo-norms is pointed out. We also construct four different rationally-extended supersymmetric partners to $V(x)$, which are $\cal PT$-sy…
Quantum Hamilton-Jacobi formalism is used to give a proof for Gozzi's criterion that for eigenstates of the supersymmetric partners, corresponding to same energy, the difference in the number of nodes is equal to one when supersymmetry (SUSY) is unbroken and is zero when SUSY is broken. We show that this proof is also applicable to the case, where isospectral deformation is involved.
The Parity-Time ($\mathcal{PT}$) symmetric potentials are derived by non-Hermitian supersymmetric quantum mechanics for square well and barrier. These $\mathcal{PT}$-supersymmetric square well and barrier. The partners have complex partners. The partners are isospectral with real energies. $\mathcal{PT}$-symmetry is only unbroken for the bound states.
We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator $-d^2/dx^2$ on $L^2[-a,a]$, $a>0$, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging t…
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 $H_…
We connect the classical free damping cases by means of Rosner's construction in supersymmetric quantum mechanics. Starting with the critical damping, one can obtain in the underdamping case a chirping of instantaneous physical frequency ω^{2}(t) \propto ω_{u}^{2}sech^2(ω_{u}t), whereas in the overdamped case the "chirping" is of the (unphysical) type ω^{2}(t)\proptoω_{o}^{2}sec^{2}(ω_{o}t), where ω_{u}$ and $ω_{o} are the underdamped and overdamped frequency parameters, respectively
We address the problem of identifying the (nonstationary) quantum systems that admit supersymmetric dynamical invariants. In particular, we give a general expression for the bosonic and fermionic partner Hamiltonians. Due to the supersymmetric nature of the dynamical invariant the solutions of the time-dependent Schrödinger equation for the partner Hamiltonians can be easily mapped to one another. We use this observation to obtain a class of exactly solvable time-dependent Schrödinger equations. As applications of our method, we construct classes of exactly solvable time-dependent generalized …
The supersymmetric Quantum Mechanics approach is applied to embed bound states in the energy gaps of periodic potentials. The mechanism to generate periodicity defects in the first Lame potential is analyzed. The related bound states are explicitly derived.
Irreducible second-order Darboux transformations are applied to the periodic Schrodinger's operators. It is shown that for the pairs of factorization energies inside of the same forbidden band they can create new non-singular potentials with periodicity defects and bound states embedded into the spectral gaps. The method is applied to the Lame and periodic piece-wise transparent potentials. An interesting phenomenon of translational Darboux invariance reveals nonlocal aspects of the supersymmetric deformations.
The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to hypergeometric type operators. It is based on some solvable Riccati equations and leads to a unitary description of the quantum systems exactly solvable in terms of orthogonal polynomials or associated special functions.
Green's functions for reflectionless potentials are constructed and analyzed. Green's functions for power law potentials, their Super Symmetric partners and sum rules for eigenvalues are examined. The SUSY partner potentials to power law potentials which have an additional bound state at $E=0$ are constructed.
Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. If applied to the harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg algebras is obtained. In this paper it will be shown that the SUSY partner Hamiltonians of the harmonic oscillator can produce evolution loops. The corresponding geometric phases will be as well studied.
Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators $H^{(\pm)}$ is chosen antilinear. Secondly, both these components of a super-Hamiltonian ${\cal H}$ are defined along certain topologically nontrivial complex curves $r^{(\pm)}(x)$ which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map ${\cal T}$ between "tobogganic" partner curves $r^{(+)}(x)$ and $r^{(-)}(x)$ is emphasized.
The supersymmetrical approach is used to analyse a class of two-dimensional quantum systems with periodic potentials. In particular, the method of SUSY-separation of variables allowed us to find a part of the energy spectra and the corresponding wave functions (partial solvability) for several models. These models are not amenable to conventional separation of variables, and they can be considered as two-dimensional generalizations of Lame, associated Lame, and trigonometric Razavy potentials. All these models have the symmetry operators of fourth order in momenta, and one of them (the Lame po…
A Half Bound State (HBS) $ψ_*(x)$ can be defined as a single, conditional, zero-energy, continuous solution of the one dimensional Schr{ö}dinger equation for a scattering potential well $V(x)$ ($s.t ~ V(\pm \infty)=0$). The non-normalizable and solitary HBS of a potential satisfies Neumann boundary condition that $ψ'_*(\pm \infty)=0$ and it can have $n$ (= 0,1,2,...) number of nodes indicating $n$ number of bound states in $V(x)$ below $E=0$. Here we show that starting with a nodeless HBS, we can construct a (supersymmetric) pair of finite potentials (well, double wells, well-barrier): $V_{\pm…
A Wronskian differential formula, useful for applying the confluent second-order SUSY transformations to arbitrary potentials, will be obtained. This expression involves a parametric derivative with respect to the factorization energy which, in many cases, is simpler for calculations than the previously found integral equation. This alternative mechanism shall be applied to the free particle and the single-gap Lame potential.
We introduce a method for constructing Darboux (or supersymmetric) pairs of pseudoscalar and scalar Dirac potentials that are associated with exceptional orthogonal polynomials. Properties of the transformed potentials and regularity conditions are discussed. As an application, we consider a pseudoscalar Dirac potential related to the Schroeodinger model for the rationally extended radial oscillator. The pseudoscalar partner potentials are constructed under first- and second-order Darboux transformations.
As it has been proven, the determination of general one-dimensional Schrödinger Hamiltonians having third-order differential ladder operators requires to solve the Painlevé IV equation. In this work, it will be shown that some specific subsets of the higher-order supersymmetric partners of the harmonic oscillator possess third-order differential ladder operators. This allows us to introduce a simple technique for generating solutions of the Painlevé IV equation. Finally, we classify these solutions into three relevant hierarchies.
In this work the supersymmetric technique is applied to the truncated oscillator to generate Hamiltonians ruled by second and third-order polynomial Heisenberg algebras, which are connected to the Painlevé IV and Painlevé V equations respectively. The aforementioned connection is exploited to produce particular solutions to both non-linear differential equations and the Bäcklund transformations relating them.
We construct isospectral partner potentials of a complex PT-invariant potential, viz., V(x) = V_1 sech ^2 x - i V_2 sech x tanh x using Darboux's method. Oneset of isospectral potentials are obatined which can be termed 'Satellite potentials', in the sense that they are pf the same form as the original potential. In a particular case, the supersymmetric partner potential has the same spectrum, including the zero energy ground state, a fact which cannot occur in conventional supersymmetric quantum mechanics with real potential. An explicit example of a non-trivial set of isospectral potential i…
New supersymmetric partners of the modified Poschl-Teller and the Dirac's delta well potentials are constructed in closed form. The resulting one-parametric potentials are shown to be interrelated by a limiting process. The range of values of the parameters for which these potentials are free of singularities is exactly determined. The construction of higher order supersymmetric partner potentials is also investigated.
We present here a supersymmetric (SUSY) approach for determining excitation energies within the context of a quantum Monte Carlo scheme. By using the fact that SUSY quantum mechanics gives rises to a series of isospectral Hamiltonians, we show that Monte Carlo ground-state calculations in the SUSY partners can be used to reconstruct accurately both the spectrum and states of an arbitrary Schrödinger equation. Since the ground-state of each partner potential is node-less, we avoid any ``node''-problem typically associated with the Monte Carlo technique. While we provide an example of using this…
In supersymmetric quantum mechanics the emergence of a singularity may lead to the breakdown of isospectrality between partner potentials. One of the regularization recipes is based on a topologically nontrivial, multisheeted complex deformations of the line of coordinate $x$ giving the so called quantum toboggan models (QTM). The consistent theoretical background of this recipe is briefly reviewed. Then, certain supersymmetric QTM pairs are shown exceptional and reducible to doublets of non-singular ordinary differential equations a.k.a. Sturm-Schrödinger equations containing a weighted energ…
We study the scattering produced by a one dimensional hyperbolic step potential, which is exactly solvable and shows an unusual interest because of its asymmetric character. The analytic continuation of the scattering matrix in the momentum representation has a branch cut and an infinite number of simple poles on the negative imaginary axis which are related with the so called anti-bound states. This model does not show resonances. Using the wave functions of the anti-bound states, we obtain supersymmetric (SUSY) partners which are the series of Rosen Morse II potentials. We have computed the …
Factorization of quantum mechanical Hamiltonians has been a useful technique for some time. This procedure has been given an elegant description by supersymmetric quantum mechanics, and the subject has become well-developed. We demonstrate that the existence of raising and lowering operators for the harmonic oscillator (and many other potentials) can be extended to their supersymmetric partners. The use of double supersymmetry (or a factorization chain) is used to obtain non-singular isospectral potentials, and the explicit expressions for the ladder operators, wave functions and probability d…
Using algebraic tools of supersymmetric quantum mechanics we construct classes of conditionally exactly solvable potentials being the supersymmetric partners of the linear or radial harmonic oscillator. With the help of the raising and lowering operators of these harmonic oscillators and the SUSY operators we construct ladder operators for these new conditionally solvable systems. It is found that these ladder operators together with the Hamilton operator form a non-linear algebra which is of quadratic and cubic type for the SUSY partners of the linear and radial harmonic oscillator, respectiv…
We analyze the one dimensional scattering produced by all variations of the Pöschl-Teller potential, i.e., potential well, low and high barriers. We show that the Pöschl-Teller well and low barrier potentials have no resonance poles, but an infinite number of simple poles along the imaginary axis corresponding to bound and antibound states. A quite different situation arises on the Pöschl-Teller high barrier potential, which shows an infinite number of resonance poles and no other singularities. We have obtained the explicit form of their associated Gamow states. We have also constructed ladde…
In the case of a one-dimensional nonsingular Hamiltonian $H$ and a singular supersymmetric partner $H_a$, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied …
The higher order supersymmetric partners of a stationary periodic potential are studied. The transformation functions associated to the band edges do not change the spectral structure. However, when the transformation is implemented for factorization energies inside of the forbidden bands, the final potential will have again the initial band structure but it can have bound states encrusted into the gaps, giving place to localized periodicity defects.
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.
We review the current status of one dimensional periodic potentials and also present several new results. It is shown that using the formalism of supersymmetric quantum mechanics, one can considerably enlarge the limited class of analytically solvable one-dimensional periodic potentials. Further, using the Landen transformations as well as cyclic identities for Jacobi elliptic functions discovered by us recently, it is shown that a linear superposition of Lamé (as well as associated Lamé) potentials are also analytically solvable. Finally, using anti-isospectral transformations, we also obtain…
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.
SUSY partners and $S$-matrix poles of the one dimensional Rosen-Morse II Hamiltonian
本文分析了具有Rosen-Morse II势的一维哈密顿量的散射矩阵极点（包括冗余极点、反束缚极点及束缚态），并利用基态波函数或反束缚态、冗余态的解执行不同的超对称变换，研究了这些变换的性质。
We consider supersymmetric quantum mechanical models with both local and nonlocal potentials. We present a nonlocal deformation of exactly solvable local models. Its energy eigenfunctions and eigenvalues are determined exactly. We observe that both our model Hamiltonian and its supersymmetric partner may have normalizable zero-energy ground states, in contrast to local models with nonperiodic or periodic potentials.
In this project, we will develop the foundations of quantum mechanics using the methods of supersymmetry. We will discuss the use of the superpotential to derive the supersymmetric partner of a potential in one dimension, and explore several key examples with an emphasis on shape invariant potentials. We will then discuss the modeling of supersymmetric quantum systems using matrices and operators, and how it relates to the eigenstate thermalization hypothesis.
Supersymmetric Quantum Mechanics, multiphoton algebras and coherent states
本文研究了具有无穷离散谱的一维哈密顿量及其 k 阶 SUSY 伴的多光子代数，发现其为单光子代数的多项式形变，并探讨了 Barut-Girardello 相干态的不确定性关系。
Flat-band engineering of quasi-one-dimensional systems via supersymmetric transformations
本研究提出一种基于超对称变换的准一维系统谱设计方法，可将赝自旋1/2模型扩展为赝自旋1模型，按需引入平带与离散能级，并以Su-Schriefer-Heeger链转换为短柱晶格的两个示例进行验证。
The concept of supersymmetry developed in particle physics has been applied to various fields of modern physics. In quantum mechanics, the supersymmetric systems refer to the systems involving two supersymmetric partner Hamiltonians, whose energy levels are degeneracy except one of the systems has an extra ground state possibly, and the eigenstates of the partner systems can be mapped onto each other. Recently, an interferometric scheme has been proposed to show this relationship in ultracold atoms [Phys. Rev.A 96, 043624 (2017)]. Here this approach is generalized to linear optics for observin…
Time-dependent Darboux transformation and supersymmetric hierarchy of Fokker-Planck equations
本文提出一种基于非平稳薛定谔方程对应关系及Darboux变换的方法，求解恒定扩散非平稳漂移的Fokker-Planck方程，并构建超对称层级。
Semiclassical approximations are implemented in the calculation of position and width of low energy resonances for radial barriers. The numerical integrations are delimited by t/T<<8, with t the period of a classical particle in the barrier trap and T the resonance lifetime. These energies are used in the construction of `haired' short range potentials as the supersymmetric partners of a given radial barrier. The new potentials could be useful in the study of the transient phenomena which give rise to the Moshinsky's diffraction in time.
We present a classical analog of quantum optical deformed oscillators in arrays of waveguides. The normal modes of these one-dimensional photonic crystals are given in terms of Jacobi polynomials. We show that it is possible to attack the problem via factorization by exploiting the quantum optical analogy. This allows us to provide an unbroken supersymmetric partner of the proposed Jacobi lattices. Thanks to the underlying SU(1,1) group symmetry of the lattices, we present the analytic propagators and impulse functions for these one-dimensional photonic crystals.
We propose time-dependent Darboux (supersymmetric) transformations that provide a scheme for the calculation of explicitly time-dependent solvable non-Hermitian partner Hamiltonians. Together with two Hermitian Hamilitonians the latter form a quadruple of Hamiltonians that are related by two time-dependent Dyson equations and two intertwining relations in form of a commutative diagram. Our construction is extended to the entire hierarchy of Hamiltonians obtained from time dependent Darboux-Crum transformations. As an alternative approach we also discuss the intertwining relations for Lewis-Rie…
Supersymmetry allows one to build a hierarchy of Hamiltonians that share the same spectral properties and which are pairwise connected through common superpotentials. The iso-spectral properties of these Hamiltonians imply that the dynamics and therefore control of different eigenstates are connected through supersymmetric intertwining relations. In this work we explore how this enables one to study general dynamics, shortcuts to adiabaticity (STA) and quantum speed limits for distinct states of different supersymmetric partner potentials by using the infinite box as an example.
Two dimensional effective Hamiltonian for a massless Dirac electron interacting with a hyperbolic magnetic field is discussed within PT symmetry. Factorization method and polynomial procedures are used to solve Dirac equation for the constant Fermi velocity and the effective potential which is complex Scarf II potential. The more general effective Scarf II potential models are also obtained within pseudo-supersymmetry. Finally, an extension of Panella and Roy's work [12] to the both PT symmetric and real Scarf II partner potentials are given using the position dependent Fermi velocity.
Exact solutions of the inhomogeneous nonlinear Schrödinger equation through supersymmetric potentials
本文通过超对称方法构造非均匀非线性薛定谔方程的精确解，利用波斯-泰勒势作为示例展示其应用。
We present the general ideas on SuperSymmetric Quantum Mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found,there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved.
Supersymmetry is a technique that allows us to extract information about the states and spectra of quantum mechanical systems which may otherwise be unsolvable. In this paper we reconstruct Ioffe's set of states for the singular non-separable two-dimensional Morse potential using supersymmetry from a non-degenerate set of states constructed for the initial separable Morse Hamiltonian. We define generalised coherent states, compute their uncertainty relations, and we find that the singularity in the partner Hamiltonian significantly affects the localisation of the coherent state wavefunction.
Using the Darboux method and its relation with supersymmetric quantum mechanics we construct all SUSY partners of the harmonic oscillator. With the help of the SUSY transformation we introduce ladder operators for these partner Hamiltonians and shown that they close a quadratic algebra. The associated coherent states are constructed and discussed in some detail.
We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator, associated with a position-dependent effective mass, by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass, uniform field interactions and to ob…
In this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of $N=2$ supersymmetric quantum mechanics. To motivate its application within supersymmetric quantum cosmology, we present a case study to illustrate the value of this promising tool. Concretely, we take a spatially flat FRW model in the presence of a single scalar field, minimally coupled to gravity. Then, we extract the associated Schrödinger-Wheeler-DeWitt equation, allowing for a particular scope of factor ordering. Subsequently, we compute the corresponding supersy…
The origin of pseudospin symmetry (PSS) and its breaking mechanism are explored by combining supersymmetry (SUSY) quantum mechanics, perturbation theory, and the similarity renormalization group (SRG) method. The Schrödinger equation is taken as an example, corresponding to the lowest-order approximation in transforming a Dirac equation into a diagonal form by using the SRG. It is shown that while the spin-symmetry-conserving term appears in the single-particle Hamiltonian $H$, the PSS-conserving term appears naturally in its SUSY partner Hamiltonian $\tilde{H}$. The eigenstates of Hamiltonian…
Starting with a time-independent Hamiltonian $h$ and an appropriately chosen solution of the von Neumann equation $i\dotρ(t)=[ h,ρ(t)]$ we construct its binary-Darboux partner $h_1(t)$ and an exact scattering solution of $i\dotρ_1(t)=[h_1(t),ρ_1(t)]$ where $h_1(t)$ is time-dependent and not isospectral to $h$. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where $h$ corresponds to a 1-D harmonic oscillator. The resulting $h_1(t)$ represents a scattering of a soliton-like pul…
We derive the analytical eigenvalues and eigenstates of a family of potentials wells with exponential form (FPWEF). We provide a brief summary of the supersymmetry formalism applied to quantum mechanics and illustrate it by producing from the FPWEF another class of exact solutions made of their isospectral partners. Interestingly, a subset of the supersymmetric partners provides a class of exactly solvable double well potentials. We use the exact solutions of the FPWEF to test the robustness and accuracy of different approximation schemes. We determine (i) the ground state through variational …
We find theoretical results on energy eigenvalues and corresponding supersymmetric Hamiltonians reflect contradictory behavior for negative values of A. furthermore the resulting supersymmetric partners potentials can be model scattering states instead of bound states, However following the literature ( Flu 1979);J and A (J. Physics A 2020), we suggest a correct form of super potential , which remains valid for both positive and negative values of constants. A from this in complex space also the eigenvalues remain invariant without discussion of T-symmetry as the previous discussion appears in…
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