Breakup of SUSY Quantum Mechanics in the Limit-Circle Region of the Reflective Kratzer Oscillator (original) (raw)
Related papers
Breakup of SUSY Quantum Mechanics for Kratzer Oscillator
The paper studies violation of conventional rules of SUSY quantum mechanics for the inverse-square-at-origin (IS@O) radial potential V(r) within the limit-circle (LC) range. A special attention is given to transformation properties of the Titchmarsh-Weyl m-function under Darboux deformations of the reflective Kratzer oscillator: centrifugal Kepler-Coulomb (KC) potential plus a Taylor series in r. Since our analysis is based on Fulton's representation of a regular-at-infinity (R@) solution [Math. Nachr. 281, 1418 (2008)] as a superposition of two Frobenius solutions at the origin, we refer to the appropriate expressions as the Titchmarsh-Weyl-Fulton (TWF) functions. Explicit transformation relations are derived for partner TWF functions associated with SUSY pairs of IS@O potentials. It is shown that these relations have a completely different form for Darboux transformations (DTs) keeping the potential within the LC range. As an illustration, we use regular nodeless Frobenius solutions to construct SUSY partners of the radial r-and c-Gauss-reference (GRef) potentials solvable via hypergeometric and confluent hypergeometric functions, respectively. We explicitly demonstrate existence of non-isospectral partners of both radial potentials in the LC region and obtain their discrete energy spectra using the derived closed-form expressions for the TWF functions. The general transformation relations for the TWF function have been verified taking advantage of form-invariance of the radial GRef potentials under double-step DTs with the so-called 'basic' seed solutions (SSs). Similarly we directly ratify that TWF functions for three shape-invariant reflective potentials on the half-line-hyperbolic Pöschl-Teller (h-PT), Eckart/Manning-Rosen (E/MR), and centrifugal KC potentialsdo retain their form under basic DTs.
Non-Central Potentials, Exact Solutions and Laplace Transform Approach
2012
Exact bound state solutions and the corresponding wave functions of the Schr\"odinger equation for some non-central potentials including Makarov potential, modified-Kratzer plus a ring-shaped potential, double ring-shaped Kratzer potential, modified non-central potential and ring-shaped non-spherical oscillator potential are obtained by using the Laplace transform approach. The energy spectrums of the Hartmann potential, modified-Kratzer potential and ring-shaped oscillator potential are also briefly studied as special cases. It is seen that our analytical results for all these potentials are consistent with those obtained by other works. We also give some numerical results obtained for the modified non-central potential for different values of the related quantum numbers.
Progress of Theoretical and Experimental Physics, 2013
An exactly solved ring potential is generated from the central hyperbolic Manning-Rosen potential using the transformation method in the framework of non-relativistic quantum mechanics. The basis of the method is coordinate redesignation and extended transformation, comprising a coordinate transformation supplemented by a functional transformation. Application of the coordinate transformation entailing redesignation of the radial coordinate with polar angle to the Schrödinger radial equation for the exactly solved central hyperbolic Manning-Rosen potential in 3D Euclidean space yields a second-order homogeneous angular differential equation. The functional transformation is indispensable in molding the angular differential equation to the Schrödinger angular equation form and, in the process of retrieving the standard Schrödinger angular equation by invoking a plausible ansatz, a new exactly solved ring potential is generated. The transformed angular wave functions for the generated ring potential are normalizable.
Exact solutions of the radial Schrödinger equation for some physical potentials
Open Physics, 2007
By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial Schrödinger equation for the pseudoharmonic and the Kratzer potentials in two dimensions. The bound-state solutions are easily calculated from this eigenfunction ansatz. The corresponding normalized wavefunctions are also obtained.
The Lambert-W step-potential – an exactly solvable confluent hypergeometric potential
Physics Letters A, 2016
We present an asymmetric step-barrier potential for which the one-dimensional stationary Schrödinger equation is exactly solved in terms of the confluent hypergeometric functions. The potential is given in terms of the Lambert W-function, which is an implicitly elementary function also known as the product logarithm. We present the general solution of the problem and consider the quantum reflection at transmission of a particle above this potential barrier. Compared with the abrupt-step and hyperbolic tangent potentials, which are reproduced by the Lambert potential in certain parameter and/or variable variation regions, the reflection coefficient is smaller because of the lesser steepness of the potential on the particle incidence side. Presenting the derivation of the Lambert potential we show that this is a four-parametric sub-potential of a more general five-parametric one also solvable in terms of the confluent hypergeometric functions. The latter potential, however, is a conditionally integrable one. Finally, we show that there exists one more potential the solution for which is written in terms of the derivative of a bi-confluent Heun function.
Quasi-exactly solvable hyperbolic potential and its anti-isospectral counterpart
Annals of Physics, 2022
We solve the eigenvalue spectra for two quasi exactly solvable (QES) Schrödinger problems defined by the potentials V (x; γ, η) = 4γ 2 cosh 4 (x) + V 1 (γ, η) cosh 2 (x) + η (η − 1) tanh 2 (x) and U (x; γ, η) = −4γ 2 cos 4 (x) − V 1 (γ, η) cos 2 (x) + η (η − 1) tan 2 (x), found by the antiisospectral transformation of the former. We use three methods: a direct polynomial expansion, which shows the relation between the expansion order and the shape of the potential function; direct comparison to the confluent Heun equation (CHE), which has been shown to provide only part of the spectrum in different quantum mechanics problems, and the use of Lie algebras, which has been proven to reveal hidden algebraic structures of this kind of spectral problems.
Modern Physics Letters A
The aim of this work is to introduce a new family of potentials with inverse square singularity which we called the Pöschl–Teller family of potentials. We enforced the matrix representation of the wave operator to be symmetric and (2k[Formula: see text]+[Formula: see text]1) band-diagonal with respect to a square integrable basis set. This, in principle, is only satisfied for specific potential functions within the used basis set. The basis functions we used here are written in terms of Jacobi polynomials, which is the same basis used in the Tridiagonal Representation Approach (TRA). This yield a more general form of Pöschl–Teller potential that can have many terms which could be beneficial for modeling different physical systems where this potential applies. As an illustration, we have studied a specific new five-parameter potential that belongs to this new family and calculated the bound states for both s-wave and l-wave cases using the Asymptotic Iteration Method (AIM). Along the...
It is shown that the regular-at-infinity (R@) solution of the 1D Schrödinger equation with the hyperbolic Pöschl-Teller (h-PT) potential s(s1)sh-2 r(ns+2)(ns+1)ch-2 r, where s and n are positive integers, is expressible in terms of a n-order Heun polynomial Hp n [y;;s] in y thr at an arbitrary negative energy- 2. It was proven that the Heun polynomials in question form a subset of generally complex Lambe-Ward polynomials corresponding to zero value of the accessory parameter. Since the mentioned solution expressed in the new variable y has an almost-everywhere holomorphic (AEH) form it can be used as the factorization function (FF) for 'canonical Liouville-Darboux transformations' (CLDTs) to construct a continuous family of 'shape-invariant' rational potentials 1 V[y;s,n| 1 ] exactly-solvable by the socalled 'Heun-seed' (HpS) Heine polynomials. There are also two (t + = a or a) sequences of infinitely many rational potentials 1 V[y;s,n|t + ,m] generated using CLDTs with nodeless regular-at-origin (R@O) AEH FFs.