We report general properties of N-fold supersymmetry in one-dimensional quantum mechanics. N-fold supersymmetry is characterized by supercharges which are N-th polynomials of momentum. Relations between the anti-commutator of the supercharges and the Hamiltonian, the spectra, the Witten index, the non-renormalization theorems and the quasi-solvability are examined. We also present further investigation about a particular class of N-fold supersymmetric models which we dubbed type A. Algebraic equations which determine a part of spectra of type A models are presented, and the non-renormalization theorem are generalized. Finally, we present a possible generalization of N-fold supersymmetry in multi-dimensional quantum mechanics.
Through a nonperturbative analysis on a sextic triple-well potential, we reveal novel aspects of the dynamical property of the system in connection with N-fold supersymmetry and quasi-solvability.
We report a new type of supersymmetry, "N-fold supersymmetry", in one-dimensional quantum mechanics. Its supercharges are N-th order polynomials of momentum: It reduces to ordinary supersymmetry for N=1, but for other values of N the anticommutator of the supercharges is not the ordinary Hamiltonian, but is a polynomial of the Hamiltonian. (For this reason, the original Hamiltonian is referred to as the "Mother Hamiltonian".) This supersymmetry shares some features with the ordinary variety, the most notable of which is the non-renormalization theorem. An N-fold supersymmetry was earlier found…
We recall the importance of recognizing the different mathematical nature of various concepts relating to PT-symmetric quantum theories. After clarifying the relation between supersymmetry and pseudo-supersymmetry, we prove generically that nonlinear pseudo-supersymmetry, recently proposed by Sinha and Roy, is just a special case of N-fold supersymmetry. In particular, we show that all the models constructed by these authors have type A 2-fold supersymmetry. Furthermore, we prove that an arbitrary one-body quantum Hamiltonian which admits two (local) solutions in closed form belongs to type A …
We investigate particular models which can be N-fold supersymmetric at specific values of a parameter in the Hamiltonians. The models to be investigated are a periodic potential and a parity-symmetric sextic triple-well potential. Through the quantitative analyses on the non-perturbative contributions to the spectra by the use of the valley method, we show how the characteristic features of N-fold supersymmetry which have been previously reported by the authors can be observed. We also clarify the difference between quasi-exactly solvable and quasi-perturbatively solvable case in view of the d…
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 construct a new family of N-fold supersymmetric systems which is referred to as ``type B''. A higher derivative representation of the N-fold supercharge for this new family is given by a deformation of the type A N-fold supercharge. By utilizing the same method as in the sl(2) construction of type A N-fold supersymmetry, we show that this family includes two of the quasi-solvable models of Post-Turbiner type.
We formulate N-fold supersymmetry in quantum mechanical matrix models. As an example, we construct general two-by-two Hermitian matrix 2-fold supersymmetric quantum mechanical systems. We find that there are two inequivalent such systems, both of which are characterized by two arbitrary scalar functions, and one of which does not reduce to the scalar system. The obtained systems are all weakly quasi-solvable.
We investigate supersymmetry in one-dimensional quantum mechanics with point interactions. We clarify a class of point interactions compatible with supersymmetry and present N=2 supersymmetric models on a circle with two point interactions as well as a superpotential. A hidden su(2) structure inherent in the system plays a crucial role to construct the N=2 supercharges. Spontaneous breaking of supersymmetry due to point interactions and an extension to higher N-extended supersymmetry are also discussed.
We investigate one-dimensional quantum mechanical systems which have type A N-fold supersymmetry with two different values of N simultaneously. We find that there are essentially four inequivalent models possessing the property, one is conformal, two of them are hyperbolic (trigonometric) including Rosen-Morse type, and the other is elliptic.
Quantization of the nonlinear supersymmetry faces a problem of a quantum anomaly. For some classes of superpotentials, the integrals of motion admit the corrections guaranteeing the preservation of the nonlinear supersymmetry at the quantum level. With an example of the system realizing the nonlinear superconformal symmetry, we discuss the nature of such corrections and speculate on their possible general origin.
We formulate the framework of N-fold supersymmetry in one-body quantum mechanical systems with position-dependent mass (PDM). We show that some of the significant properties in the constant-mass case such as the equivalence to weak quasi-solvability also hold in the PDM case. We develop a systematic algorithm for constructing an N-fold supersymmetric PDM system. We apply it to obtain type A N-fold supersymmetry in the case of PDM, which is characterized by the so-called type A monomial space. The complete classification and general form of effective potentials for type A N-fold supersymmetry i…
The concept of supersymmetry in a quantum mechanical system is extended, permitting the recognition of many more supersymmetric systems, including very familiar ones such as the free particle. Its spectrum is shown to be supersymmetric, with space-time symmetries used for the explicit construction. No fermionic or Grassmann variables need to be invoked. Our construction extends supersymmetry to continuous spectra. Most notably, while the free particle in one dimension has generally been regarded as having a doubly degenerate continuum throughout, the construction clarifies taht there is a sing…
Type A N-fold supercharge admits a one-parameter family of factorizations into product of N first-order linear differential operators due to an underlying GL(2,C) symmetry. As a consequence, a type A N-fold supersymmetric system can have different intermediate Hamiltonians corresponding to different factorizations. We derive the necessary and sufficient conditions for the latter system to possess intermediate Hamiltonians for the N=2 case. We then show that whenever it has (at least) one intermediate Hamiltonian, it can admit second-order parasupersymmetry and a generalized 2-fold superalgebra…
We continue the previous study on the existence of different intermediate Hamiltonians in type A N-fold supersymmetric systems and carry out an exhaustive investigation on the N=3 case. In contrast with the N=2 case, we find various patterns in the existence of intermediate Hamiltonians due to the presence of two different intermediate positions in a factorized type A 3-fold supercharge. In addition, all the N=3 models are strictly restricted to at most elliptic type, which enables us to make the complete classification of the systems which admit intermediate Hamiltonians. Finally, we show rea…
We construct a new family of quasi-solvable and N-fold supersymmetric quantum systems where each Hamiltonian preserves an exceptional polynomial subspace of codimension 2. We show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the X_2-Laguerre polynomials of the second kind. In addition, we find that the two kinds of the X_2-Laguerre polynomials are ingeniously connected with each other by the N-fold supercharge.
We derive the necessary and sufficient condition for Type A N-fold supersymmetry by direct calculation of the intertwining relation and show the complete equivalence between this analytic construction and the sl(2) construction based on quasi-solvability. An intimate relation between the pair of algebraic Hamiltonians is found. The classification problem on Type A N-fold supersymmetric models is investigated by considering the invariance of both the Hamiltonians and N-fold supercharge under the GL(2,K) transformation. We generalize the Bender-Dunne polynomials to all the Type A N-fold supersym…
Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser models. Because of the additional q^6 (rational models) or sin^2(2q) (trigonometric models) potentials, their quantum versions are not exactly solvable in contrast with Calogero-Moser models. We show that quantum Inozemtsev models can be deformed to be a widest class of partly solvable (or quasi-exactly solvable) multi-particle dynamical systems. They posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A new method for identifying and solving quasi-exactly solvable sy…
The supersymmetric quantum mechanical model based on higher-derivative supercharge operators possessing unbroken supersymmetry and discrete energies below the vacuum state energy is described. As an example harmonic oscillator potential is considered.
We report general forms of one family of the N-fold supersymmetry in one-dimensional quantum mechanics. The N-fold supersymmetry is characterized by the supercharges which are N-th order in differential operators. The family reported here is defined as a particular form of the supercharges and is referred to as ``type A''. We show that a quartic and a periodic potentials, which were previously found to be N-fold supersymmetric by the authors, are realized as special cases of this type A family.
We review the construction of minimally bosonized supersymmetric quantum mechanics and its relation to hidden supersymmetries in pure parabosonic (parafermionic) systems.
We discuss the application of quantum-mechanical supersymmetry to particle traps. The supersymmetric-partner wave functions may be used to describe a valence fermion in a trap system with an isotropic harmonic-oscillator potential. Interactions with the core are incorporated analytically. The close similarity of this approach to the application of supersymmetry in atomic systems is made explicit by means of a radial mapping between the two systems.
I examine quantum mechanical Hamiltonians with partial supersymmetry, and explore two main applications. First, I analyze a theory with a logarithmic spectrum, and show how to use partial supersymmetry to reveal the underlying structure of this theory. This method reveals an intriguing equivalence between two formulations of this theory, one of which is one-dimensional, and the other of which is infinite-dimensional. Second, I demonstrate the use of partial supersymmetry as a tool to obtain the asymptotic energy levels in non-relativistic quantum mechanics in an exceptionally easy way. In the …
In this work, a spin $\frac 12$ relativistic particle described by a generalized potential containing both the Coulomb potential and a Lorentz scalar potential in Dirac equation is investigated in terms of the generalized ladder operators from supersymmetry in quantum mechanics. This formalism is applied for the generalized Dirac-Coulomb problem, which is an exactly solvable potential in relativistic quantum mechanics. We obtain the energy eigenvalues and calculate explicitly the energy eigenfunctions for the ground state and the first excited state.
We study possible factorizations of supersymmetric (SUSY) transformations in the one-dimensional quantum mechanics into chains of elementary Darboux transformations with nonsingular coefficients. A classification of irreducible (almost) isospectral transformations and of related SUSY algebras is presented. The detailed analysis of SUSY algebras and isospectral operators is performed for the third-order case.
The well-known supersymmetric constructions such as Witten's supersymmetric quantum mechanics, Spiridonov-Rubakov parasupersymmetric quantum mechanics, and higher-derivative SUSY of Andrianov et al. are extended to the nonstationary Schrödinger equation. All these constructions are based on the time-dependent Darboux transformation. The superalgebra over the conventional Lie algebra is constructed. Examples of time-dependent exactly solvable potentials are given.
We show that the Ginzburg-Landau expansion of the grand potential for the Bogoliubov-de Gennes Hamiltonian is determined by the integrable nonlinear equations of the AKNS hierarchy, and that this provides the natural mathematical framework for a hidden nonlinear quantum mechanical supersymmetry underlying the dynamics.
We investigate $N$-extended supersymmetry in one-dimensional quantum mechanics on a circle with point singularities. For any integer $n$, $N=2n+1$ supercharges are explicitly constructed in terms of discrete transformations, and a class of singularities compatible with supersymmetry is clarified. In our formulation, the supersymmetry can be reduced to $M$-extended supersymmetry for any integer $M<N$. The degeneracy of the spectrum and spontaneous supersymmetry breaking are also studied.
In this paper, we continue to study factorization of supersymmetric (SUSY) transformations in one-dimensional Quantum Mechanics into chains of elementary Darboux transformations with nonsingular coefficients. We define the class of potentials that are invariant under the Darboux - Crum transformations and prove a number of lemmas and theorems substantiating the formulated formerly conjectures on reducibility of differential operators for spectral equivalence transformations. Analysis of the general case is performed with all the necessary proofs.
New types of irreducible second order Darboux transformations for the one dimensional Schroedinger equation are described. The main feature of such transformations is that the transformation functions have the eigenvalues grater then the ground state energy of the initial (or reference) Hamiltonian. When such a transformation is presented as a chain of two first order transformations, an intermediate potential is singular and therefore intermediate Hamiltonian can not be Hermitian while the final potential is regular and the final Hamiltonian is Hermitian. Second derivative supersymmetric quan…
Emergent supersymmetry in a time-space inverted quantum mechanics
Universidade Federal do Paraná（巴拉那联邦大学）；Centro Interdisciplinar de Ciência, Tecnologia e Inovação (CICTI)（跨学科科学与技术创新中心）；Max-Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
Semiclassical methods provide important tools for approximating solutions in quantum mechanics. In several cases these methods are intriguingly exact rather than approximate, as has been shown by direct calculations on particular systems. In this paper we prove that the long-conjectured exactness of the supersymmetry-based semiclassical quantization condition for broken supersymmetry is a consequence of the additive shape invariance for the corresponding potentials.
A number of new L$\acute{e}$vi-Leblond type equations admitting four component spinor solutions have been proposed. The pair of linearized equations thus obtained in each case lead to Hamiltonians with characteristic features like L-S coupling and supersymmetry. The relevant momentum operators have often been understood in terms of Clifford algebraic bases producing Schr$\ddot{o}$dinger Hamiltonians with L-S coupling. As for example Hamiltonians representing Rashba effect or three dimensional harmonic oscillator have been constructed. The supersymmetric nature of one dimensional oscillator has…
An elementary introduction is given to the subject of Supersymmetry in Quantum Mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct new exactly solvable n Hamiltonians having n-1,n-2,...,0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one-dimensional harmonic oscillator problem to a class of potentia…
We investigate N-extended supersymmetry in one-dimensional quantum mechanics on a circle with point singularities. For any integer n, N=2n supercharges are explicitly constructed and a class of point singularities compatible with supersymmetry is clarified. Key ingredients in our construction are n sets of discrete transformations, each of which forms an su(2) algebra of spin 1/2. The degeneracy of the spectrum and spontaneous supersymmetry breaking are briefly discussed.
When several inequivalent supercharges form a closed superalgebra in Quantum Mechanics it entails the appearance of hidden symmetries of a Super-Hamiltonian. We examine this problem in one-dimensional QM for the case of periodic potentials and potentials with finite number of bound states. After the survey of the results existing in the subject the algebraic and analytic properties of hidden-symmetry differential operators are rigorously elaborated in the Theorems and illuminated by several examples.
We investigate a new realization of extended quantum-mechanical supersymmetry. We first show that an $\mathcal{N}=2$ quantum-mechanical supersymmetry is hidden in the four-dimensional (4D) spectrum of the Kaluza-Klein decomposition for the higher dimensional Dirac field, that is, Kaluza-Klein mode functions of 4D right-handed spinors and 4D left-handed ones form $\mathcal{N}=2$ supermultiplets. In addition to $\mathcal{N}=2$ supersymmetry, we discover that an $\mathcal{N}$-extended supersymmetry ($\mathcal{N} = d+2\ (d+1)$ for $d=$ even (odd) extra dimensions) is further hidden in the 4D spect…
Nonlinear (Polynomial, N-fold) SUSY approach to preparation of quantum systems with pre-planned spectral properties is reviewed. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. Possible extensions of SUSY in one dimension are described. They include (no more than) ${\cal N} =2$ extended SUSY with two nilpotent SUSY charges which generate the hidden symmetry acting as a central charge. Embedding stationary quantum systems into a non-stationary SUSY QM is shown to yield new insight on quantum orbits and on spectrum generating al…
We study quantum mechanics in the stochastic formulation, using the functional integral approach. The noise term enters the classical action as a local contribution of anticommuting fields. The partition function is not invariant under ${\mathcal N}=1$ SUSY, but can be obtained from a, manifestly, supersymmetric expression, upon fixing a local fermionic symmetry, called $κ-$symmetry. The kinetic term for the fermions is a total derivative and can contribute only on the boundaries. We define combinations that scale appropriately, as the lattice spacing is taken to zero and the lattice size to i…
It is shown how to derive fractional supersymmetric quantum mechanics of order k as a superposition of k-1 copies of ordinary supersymmetric quantum mechanics.
Motivated by the duality of normalizable states and the presence of the quasi-parity quantum number q=+/-1 in PT symmetric (non-Hermitian) quantum mechanical potential models, the relation of PT symmetry and supersymmetry (SUSY) is studied. As an illustrative example the PT invariant version of the Scarf II potential is presented, and it is shown that the "bosonic" Hamiltonian has two different "fermionic" SUSY partner Hamiltonians (potentials) generated from the ground-state solutions with q=1 and q=-1. It is shown that the "fermionic" potentials cease to be PT invariant when the PT symmetry …
Two known 2-dim SUSY quantum mechanical constructions - the direct generalization of SUSY with first-order supercharges and Higher order SUSY with second order supercharges - are combined for a class of 2-dim quantum models, which {\it are not amenable} to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained - the explicit expressions of quantum symmetry operators and of classical integrals of motion are given…
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 study the Nonlinear (Polynomial, N-fold,...) Supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Super-Hamiltonian projection on the zero-mode subspace of a supercharge. We show that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that…
We construct a double degenerate supersymmetry in one dimensional quantum mechanics. Here the energy levels satisfy the conditions $E_{0,1}^{((-)}=0$ and $E_{n,n+1}^{(+)}=E_{n+2,n+3}^{(-)}$.The corresponding SUSY Hamiltonians$(H^{(\pm)}$) are double degenerate in nature. the method has been tested on power law potentials.
The appearances of complex eigenvalues in the spectra of PT-symmetric quantum-mechanical systems are usually associated with a spontaneous breaking of PT. In this letter we discuss a family of models for which this phenomenon is also linked with an explicit breaking of supersymmetry. Exact level-crossings are located, and connections with N-fold supersymmetry and quasi-exact solvability in certain special cases are pointed out.
It is shown that the supersymmetric quantum mechanics has an octonionic generalization. The generalization is based on the inclusion of quaternions into octonions. The elements from the coset octonions/quaternions are unobservables bacause they cannot be considered as quantum operators as a consequence of their non-associative properties. The idea that the octonionic generalization of the supersymmetric quantum mechanics describes an observable particle formed with unobservable ``particles'' is presented.
We study an analogue of the equations describing TE and TM modes in a planar waveguide with an arbitrary continuous dependence of the electric permittivity and magnetic permeability on coordinates with the stationary Schrödinger equation. The effective potential energies involved in the Schrödinger equation for TE and TM modes are found. In general, the effective potential energies for TE and TM modes are different but in the limit of a weak dependence of the permittivity and permeability on coordinates they coincide. In the case when the product of a position-dependent permittivity and permea…
In Foldy-Wouthuysen representation, we deduce the effective quantum mechanics for a particle confined to a curved surface by using the thin-layer quantization scheme. We find that the spin effect caused by confined potential as the results of relativistic correction in the non-relativistic limit. Furthermore, the spin connection appeared in curved surface which depends on curvature contributes a Zeeman-like gap in the relativistic correction term. In addition, the confined potential also induces a curvature-independent energy shift, which is from the zitterbewegung effect. As an example, we ap…
We introduce the confluent version of the quantum-mechanical supersymmetry (SUSY) formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results.
This work analyzes monolayer graphene in external electromagnetic fields, which is described by the Dirac equation with minimal coupling. Supersymmetric quantum mechanics allows building new Dirac equations with modified magnetic fields. Here, we will use complex factorization energies and iterate the method in order to arrive at Hermitian graphene Hamiltonians. Finally, we compare these results with the matrix supersymmetric quantum mechanics approach.
We show that some simple well studied quantum mechanical systems without fermion (spin) degrees of freedom display, surprisingly, a hidden supersymmetry. The list includes the bound state Aharonov-Bohm, the Dirac delta and the Poschl-Teller potential problems, in which the unbroken and broken N=2 supersymmetry of linear and nonlinear (polynomial) forms is revealed.
Over the years, supersymmetric quantum mechanics has evolved from a toy model of high energy physics to a field of its own. Although various examples of supersymmetric quantum mechanics have been found, systems that have a natural realization are scarce. Here, we show that the extension of the conventional Cooper-pair box by a 4pi-periodic Majorana-Josephson coupling realizes supersymmetry for certain values of the ratio between the conventional Josephson and the Majorana- Josephson coupling strength. The supersymmetry we find is a "hidden" minimally bosonized supersymmetry that provides a non…
A supersymmetric generalization of the Lieb-Liniger-Yang dynamics governing $N$ massive bosons moving on a line with delta interactions among them at coinciding points is developed. The analysis of the delicate balance between integrability and supersymmetry, starting from the exactly solvable non supersymmetric LLY system, is one of the paper main concerns. Two extreme regimes of the $N$ parameter are explored: 1) For few bosons we fall in the realm of supersymmetric quantum mechanics with a short number of degrees of freedom, e.g., the SUSY P$\ddot{\rm o}$sch-Teller potentials if $N=1$. 2) F…
Relativistic quantum mechanics of a Proca (spin-1) particle in Riemannian spacetimes is constructed. Covariant equations defining electromagnetic interactions of a Proca particle with the anomalous magnetic moment and the electric dipole moment in Riemannian spacetimes are formulated. The relativistic Foldy-Wouthuysen transformation with allowance for terms proportional to the zero power of the Planck constant is performed. The Hamiltonian obtained agrees with the corresponding Foldy-Wouthuysen Hamiltonians derived for scalar and Dirac particles and with their classical counterpart. The unific…
