A GENERATION OF EXACTLY SOLVABLE ANHARMONIC SYMMETRIC OSCILLATORS

Using the ideas of supersymmetric quantum mechanics, we exactly solve a continuous family of anharmonic potentials, which are the supersymmetric partners of the linear harmonic oscillators. The family includes a series of potentials in which the excited-state energy is the same as that of the harmonic oscillators, but the ground-state energy can be any value lower than the excited states. The shape of the potential is variable, which includes the double-well and triple-well potentials. All the potentials obtained in this paper are free of singularities, and the supersymmetry of the solutions is unbroken.     

两个严格可解双原子分子势函数的散射态解与转动修正

文献［１］中提出了一个新的势函数，其Ｓｃｈｒｄｉｎｇｅｒ方程严格可解。同时提出一个新的变量变换关系，用超几何级数严格求解了双曲型ＰｓｃｈｌＴｅｌｅｒ分子势Ｓｃｈｒｄｉｎｇｅｒ方程的束缚态。文内进一步讨论这两个势函数Ｓｃｈｒｄｉｎｇｅｒ方程散射态的严格解，并求出了Ｓ波的散射相移。文献中关于修正ＰｓｃｈｌＴｅｌｅｒ势及无反射势散射态的结果均作为特例包含在这篇文章更一般的结论之中。此外还用一个简单的方法考虑了转动能修正，对ＨＦ基态转动谱作了具体计算     

Supersymmetry and SWKB Approach to Dirac Equation with Hyperbolic Scarf Potential

By using the supersymmetric quantum mechanics and shape invariance concept, we study the Dirac equation with the hyperbolic Scarf potential and the exact energy spectrum is obtained. Also, we calculate the bound state energy eigenvalues by using the supersymmetric WKB approximation approach so that we get the same results.     

Exact Electronic Bands for a Periodic Pöschl-Teller Potential

We show that supersymmetry is a simple but powerful tool to exactly solve quantum mechanics problems. Here, the supersymmetric approach is used to analyse a quantum system with periodic Pöschl-Teller potential, and to find out the exact energy spectra and the corresponding band structure.     

SUPERSYMMETRIC SU(1|2) GAUDIN MODEL

In this paper, we propose a supersymmetric SU(1|2) Gaudin model and have derived its eigenvalues. We also present the well-defined eigenstates through the quasi-classical limit of the eigenstates in the supersymmetric t-J model.     

Exact Solution of Klein-Gordon Equation for Charged Particle in Magnetic Field with Shape Invariant Method

Based on the shape invariance property we obtain exact solutions of the three-dimensional relativistic Klein Gordon equation for a charged particle moving in the presence of a certain varying magnetic field, and we also show its non-relativistic limit.     

Supersymmetric Quantum Mechanics and SUSY Dependent SU（2） Symmetry for a Spin-1/2 Charged Particle in a Magnetic Field

We find that in a supersymmetric quantum mechanics （SUSY QM） system, in addition to supersymmetric algebra, an associated SU（2） algebra can be obtained by using semiunitary （SUT） operator and projection operator, and the relevant constants of motion can be constructed. Two typical quantum systems are investigated as examples to demonstrate the above finding. The first example is the quantum system of a nonrelativistic charged particle moving in x-y plane and coupled to a magnetic field along z-axis. The second example is provided with the Dirac particle in a magnetic field. Similarly there exists an SUτ（2） SUσ（2） symmetry in the context of the relativistic Pauli Hamilt onian squared. We show that there exists also an SU（2） symmetry associated with the supersvmmetrv of the Dirac particle.     

Dirac Equation and Some Quasi-Exact Solvable Potentials in the Turbiner's Classification

In this paper quasi-exact solvability(QES)of Dirac equation with some scalar potentials based on sl(2)Lie algebra is studied.According to the quasi-exact solvability theory,we construct the configuration of the classes II,IV,V,and X potentials in the Turbiner’s classification such that the Dirac equation with scalar potential is quasi-exactly solved and the Bethe ansatz equations are derived in order to obtain the energy eigenvalues and eigenfunctions.     

Pseudospin symmetric solutions of the Dirac equation with the modified Rosen—Morse potential using Nikiforov—Uvarov method and supersymmetric quantum mechanics approach

Employing the Pekeris-type approximation to deal with the pseudo-centrifugal term, we analytically study the pseudospin symmetry of a Dirac nucleon subjected to equal scalar and vector modified Rosen-Morse potential including the spin-orbit coupling term by using the Nikiforov-Uvarov method and supersymmetric quantum mechanics approach. The complex eigenvalue equation and the total normalized wave functions expressed in terms of Jacobi polynomial with arbitrary spin-orbit coupling quantum number k are presented under the condition of pseudospin symmetry. The eigenvalue equations for both methods reproduce the same result to affirm the mathematical accuracy of analytical calculations. The numerical solutions obtained for different adjustable parameters produce degeneracies for some quantum number.     

Approximate Solution of D-Dimensional Klein-Gordon Equation with Hulthén-Type Potential via SUSYQM

Approximate analytical solutions of the D-dimensional Klein-Gordon equation are obtained for the scalar and vector general Hulthén-type potential and position-dependent mass with any l by using the concept of supersymmetric quantum mechanics (SUSYQM). The problem is numerically
discussed for some cases of parameters.     

Supersymmetric Quantum Mechanics and SUSY Dependent SU(2) Symmetry for a Spin-1/2 Charged Particle in a Magnetic Field

We find that in a supersymmetric quantum mechanics (SUSY QM) system, in addition to supersymmetric algebra, an associated SU(2) algebra can be obtained by using semiunitary (SUT) operator and projection operator, and the relevant constants of motion can be constructed. Two typical quantum systems are investigated as examples to demonstrate the above finding. The first example is the quantum system of a nonrelativistic charged particle moving in x-y plane and coupled to a magnetic field along z axis. The second example is provided with the Dirac particle in a magnetic field. Similarly there exists an SU_τ(2) \otimes SU_σ(2) symmetry in the context of the relativistic Pauli Hamiltonian squared. We show that there exists also an SU(2) symmetry associated with the supersymmetry of the Dirac particle.     

Dirac Equation and Some Quasi-Exact Solvable Potentials in the Turbiner’s Classification

In this paper quasi-exact solvability (QES) of Dirac equation with some scalar potentials based on sl(2) Lie algebra is studied. According to the quasi-exact solvability theory, we construct the configuration of the classes II, IV, V, and X potentials in the Turbiner's classification such that the Dirac equation with scalar potential is quasi-exactly solved and the Bethe ansatz equations are derived in order to obtain the energy eigenvalues and eigenfunctions.     

BETHE ANSATZ FOR SUPERSYMMETRIC MODEL WITH Uq[osp(1|2)] SYMMETRY

Using the algebraic Bethe ansatz method, we obtain the eigenvalues of the transfer matrix of the supersymmetric model with U_q[osp(1|2)] symmetry under periodic boundary and twisted boundary condition.     

Approximate solutions of Klein-Gordon equation with improved Manning-Rosen potential in D-dimensions using SUSYQM

In this paper, we present solutions of the Klein–Gordon equation for the improved Manning–Rosen potential for arbitrary l state in d-dimensions using the supersymmetric shape invariance method. We obtained the energy levels and the corresponding wave functions expressed in terms of Jacobi polynomial in a closed form for arbitrary l state. We also calculate the oscillator strength for the potential.     

精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量     

Supersymmetric quantum mechanics and the index theorem

We consider the classical mechanics of the spinning particle and investigate which Abelian interactions can be added without breaking supersymmetry. A quantum theory is presented. The well known index theorem for the Dirac operator is extended to take into account the effect of anti-symmetric Abelian tensor fields. Furthermore interactions with non-Abelian anti-symmetric tensor fields are investigated. It turns out in both cases that these fields do not give any non-trivial contributions to the index.     

Jack Superpolynomials: Physical and Combinatorial Definitions

Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland Hamiltonian. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this quantum-mechanical problem. But Jack superpolynomials can also be defined more combinatorially, starting from the multiplicative bases of symmetric superpolynomials, enforcing orthogonality with respect to a one-parameter deformation of the combinatorial scalar product. Both constructions turn out to be equivalent.     

一种新型双势和周期势模型的构造及其精确解

提出了一种构造并精确求解新型势函数的方法.采用这种方法.讨论了一维二次型函数的通解,将这种二次型函数分段连接构造出一种新型一维双势阱模型和新型一维周期势模型,在此基础上得出了这两种新型势模型的精确解.     

Low frequency asymptotics for the spin-weighted spheroidal equation in the case of s=1/2

The spin-weighted spheroidal equation in the case of s=1/2 is thoroughly studied by using the perturbation method from the supersymmetric quantum mechanics.The first-five terms of the superpotential in the series of parameter β are given.The general form for the n-th term of the superpotential is also obtained,which could also be derived from the previous terms W k,k < n.From these results,it is easy to obtain the ground eigenfunction of the equation.Furthermore,the shape-invariance property in the series of parameter β is investigated and is proven to be kept.This nice property guarantees that the excited eigenfunctions in the series form can be obtained from the ground eigenfunction by using the method from the supersymmetric quantum mechanics.We show the perturbation method in supersymmetric quantum mechanics could completely solve the spin-weight spheroidal wave equations in the series form of the small parameter β.