Talk given at the 7-th Lomonosov Conference "Problems of Fundamental Physics", 24-30 August, 1995, see proceedings book (with the minor corrections) Moscow, 1997, p. 54-61
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.
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 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.
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…
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…
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.
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.
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 …
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…
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…
We review the construction of minimally bosonized supersymmetric quantum mechanics and its relation to hidden supersymmetries in pure parabosonic (parafermionic) systems.
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.
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.
A new method for generating exactly solvable Schrödinger equations with a position-dependent mass is proposed. It is based on a relation with some deformed Schrödinger equations, which can be dealt with by using a supersymmetric quantum mechanical approach combined with a deformed shape-invariance condition. The solvability of the latter is shown to impose the form of both the deformed superpotential and the position-dependent mass. The conditions for the existence of bound states are determined. A lot of examples are provided and the corresponding bound-state spectrum and wavefunctions are re…
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.
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.
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.
An anticommuting analogue of Brownian motion, corresponding to fermionic quantum mechanics, is developed, and combined with classical Brownian motion to give a generalised Feynman-Kac-Itô formula for paths in geometric supermanifolds. This formula is applied to give a rigorous version of the proofs of the Atiyah-Singer index theorem based on supersymmetric quantum mechanics. It is also shown how superpaths, parametrised by a commuting and an anticommuting time variable, lead to a manifestly supersymmetric approach to the index of the Dirac operator. After a discussion of the BFV approach to th…
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 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 connection between the strictly isospectral construction in supersymmetric quantum mechanics and the general zero mode solutions of the Schroedinger equation is explained by introducing slightly generalized first-order intertwining operators. We also present a multiple-parameter generalization of the strictly isospectral construction in the same perspective
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.
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…
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.
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…
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 …
We study one dimensional supersymmetric (SUSY) quantum mechanics of a spin 1/2 particle moving in a rotating magnetic field and scalar potential. We also discuss SUSY breaking and it is shown that SUSY breaking essentially depends on the strength and period of the magnetic field. For a purely rotating magnetic field the eigenvalue problem is solved exactly and two band energy spectrum is found.
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…
Supersymmetry applied to quantum mechanics has given new insights in various topics of theoretical physics like analytically solvable potentials, WKB approximation or KdV solitons. Duality plays a central role in many supersymmetric theories such as Yang-Mills theories or strings models. We investigate the possible existence of some duality within supersymmetric quantum mechanics.
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 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…
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.
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 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…
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.
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 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…
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.
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 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…
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…
Electromagnetic waves at grazing incidence onto a planar medium are analogous to zero energy quantum particles incident onto a potential well. In this limit waves are typically completely reflected. Here we explore dielectric profiles supporting optical analogues of `half-bound states', allowing for zero reflection at grazing incidence. To obtain these profiles we use two different theoretical approaches: supersymmetric quantum mechanics, and direct inversion of the Helmholtz equation.
We study a 5d gravity theory with a warped metric and show that two N = 2 supersymmetric quantum-mechanical systems are hidden in the 4d spectrum. The supersymmetry can be regarded as a remnant of higher-dimensional general coordinate invariance and turns out to become a powerful tool to determine the physical 4d spectrum and the allowed boundary conditions. Possible extensions of the N = 2 supersymmetry are briefly discussed.
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.
The ordering problem in quantum systems with position-dependent mass (PDM) is treated by inclusion of the classically fictitious similarity transformation into the kinetic term. This provides a generation of supersymmetry with the first order supercharges from the kinetic term alone, while inclusion of the potential term allows also to generate nonlinear supersymmetry with higher order supercharges. A broad class of finite-gap systems with PDM is obtained by different reduction procedures, and general results on supersymmetry generation are applied to them. We show that elliptic finite-gap sys…
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.
Second order supersymmetric approach is taken to the system describing motion of a quantum particle in a potential endowed with position-dependent effective mass. It is shown that the intertwining relations between second order partner Hamiltonians may be exploited to obtain a simple shape-invariant condition. Indeed a novel relation between potential and mass functions is derived, which leads to a class of exactly solvable model. As an illustration of our procedure, two examples are given for which one obtains whole spectra algebraically. Both shape-invariant potentials exhibit harmonic-oscil…
The Schrödinger equation is shown to be equivalent to a constrained Liouville equation under the assumption that phase space is extended to Grassmann algebra valued variables. For onedimensional systems, the underlying Hamiltonian dynamics has a N=2 supersymmetry. Potential applications to more realistic theories are briefly 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.
The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of $\mathcal N \leq 2$ quantum mechanical systems is $\#P$-complete and therefore intractable. Then, the notions of supersymmetry in the space of qubits and supersymmetric quantum circuits are introduced and some of their properties discussed. In particular, it is shown that these define a nontrivial subclass of quantum algorithms with robustness properties typical of supersymmetric systems. Concrete examples, including the supersymmetric SYK…
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…
We present a framework for the construction of solvable models of optical settings with genuinely two-dimensional landscapes of refractive index. The solutions of the associated non-separable Maxwell equations in paraxial approximation are found with the use of the time-dependent supersymmetry. We discuss peculiar theoretical aspects of the construction. Sufficient conditions for existence of localized states are discussed. Localized solutions vanishing for large $|\vec{x}|$, that we call light dots, as well as the guided modes that vanish exponentially outside the wave guides are constructed.…
