scholarly journals Lie Symmetry and the Bethe Ansatz Solution of a New Quasi-Exactly Solvable Double-Well Potential

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
M. Baradaran ◽  
H. Panahi
Keyword(s):  
Bethe Ansatz ◽  
Algebraic Approach ◽  
Lie Symmetry ◽  
Wave Functions ◽  
Ansatz Method ◽  
Exactly Solvable ◽  
Bethe Ansatz Solution ◽  
Double Well Potential ◽  
Potential Parameters ◽  
Good Agreement

We study the Schrödinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions, and the allowed values of the potential parameters are obtained using two different methods, the Bethe ansatz method and the Lie algebraic approach. Some numerical results are reported and it is shown that the results are in good agreement with each other and with those obtained previously via a different method.

scholarly journals Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
M. Baradaran ◽  
H. Panahi
Keyword(s):  
Bethe Ansatz ◽  
Numerical Evaluation ◽  
Energy Levels ◽  
Algebraic Approach ◽  
Wave Functions ◽  
Algebraic Equations ◽  
Ansatz Method ◽  
Exact Solvability ◽  
Exactly Solvable ◽  
Bistable Potential

Applying the Bethe ansatz method, we investigate the Schrödinger equation for the three quasi-exactly solvable double-well potentials, namely, the generalized Manning potential, the Razavy bistable potential, and the hyperbolic Shifman potential. General exact expressions for the energies and the associated wave functions are obtained in terms of the roots of a set of algebraic equations. Also, we solve the same problems using the Lie algebraic approach of quasi-exact solvability through the sl(2) algebraization and show that the results are the same. The numerical evaluation of the energy spectrum is reported to display explicitly the energy levels splitting.

scholarly journals Algebraic Approach to Exact Solution of the (2 + 1)-Dimensional Dirac Oscillator in the Noncommutative Phase Space

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
H. Panahi ◽  
A. Savadi
Keyword(s):  
Exact Solution ◽  
Phase Space ◽  
Algebraic Approach ◽  
Wave Functions ◽  
Dirac Oscillator ◽  
Noncommutative Phase Space ◽  
Energy Eigenvalues ◽  
Good Agreement

We study the (2 + 1)-dimensional Dirac oscillator in the noncommutative phase space and the energy eigenvalues and the corresponding wave functions of the system are obtained through the sl(2) algebraization. It is shown that the results are in good agreement with those obtained previously via a different method.

Relativistic particles with a nonpolynomial oscillator potential in a spacelike dislocation

2020 ◽  
Vol 35 (21) ◽  
pp. 2050108
Author(s):  
P. Sedaghatnia ◽  
H. Hassanabadi ◽  
G. J. Rampho
Keyword(s):  
Lie Algebra ◽  
Wave Functions ◽  
Oscillator Potential ◽  
Exactly Solvable ◽  
Potential Parameters ◽  
Relativistic Particles

We study the relativistic particles with a nonpolynomial oscillator potential in a spacelike dislocation and the solution of the system is obtained by using of quasi-exactly solvable method. The energy and wave functions of the system are described and the permissible values of the potential parameters are examined through the sl(2) Lie algebra.

Thermodynamics of the Repulsive Lieb–Liniger Model

2019 ◽  
pp. 633-640
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Finite Temperature ◽  
Bose Gas ◽  
Main Ingredient ◽  
Ansatz Method ◽  
One Dimensional ◽  
Bethe Ansatz Solution ◽  
The One

This chapter discusses how the Bethe ansatz solution of the one-dimensional Bose gas with repulsive δ‎-function interaction is extended to finite temperatures, the thermody- namic Bethe ansatz. The excitations of this system consist of particle and hole excitations, which can be described by the corresponding densities of Bethe ansatz roots. It shows how these Bethe ansatz root densities are used to define an appropriate expression for the entropy of the system of Bose particles, which is the main ingredient for the extension of the Bethe ansatz method to finite temperature.

scholarly journals QUASI-EXACTLY SOLVABLE DEFORMATIONS OF GAUDIN MODELS AND "QUASI-GAUDIN ALGEBRAS"

1998 ◽  
Vol 13 (04) ◽  
pp. 281-292 ◽  
Author(s):  
A. G. USHVERIDZE
Keyword(s):  
Bethe Ansatz ◽  
Complete Integrability ◽  
Integrable Models ◽  
Exact Solvability ◽  
New Class ◽  
Algebraic Bethe Ansatz ◽  
Exactly Solvable ◽  
Algebraic Deformation ◽  
Bethe Ansatz Solution ◽  
Gaudin Models

A new class of completely integrable models is constructed. These models are deformations of the famous integrable and exactly solvable Gaudin models. In contrast with the latter, they are quasi-exactly solvable, i.e. admit the algebraic Bethe ansatz solution only for some limited parts of the spectrum. An underlying algebra responsible for both the phenomena of complete integrability and quasi-exact solvability is constructed. We call it "quasi-Gaudin algebra" and demonstrate that it is a special non-Lie-algebraic deformation of the ordinary Gaudin algebra.

The Bethe Ansatz Solution of a One-Dimensional Fermion Model with Boundary Impurity

1998 ◽  
Vol 12 (23) ◽  
pp. 2435-2446
Author(s):  
Shu Chen ◽  
Yupeng Wang ◽  
Fan Yang ◽  
Fu-Cho Pu
Keyword(s):  
Bethe Ansatz ◽  
Coupling Coefficients ◽  
Ansatz Method ◽  
Fermion Model ◽  
One Dimensional ◽  
Consistent Condition ◽  
Self Consistent ◽  
Special Relations ◽  
Bethe Ansatz Solution ◽  
Special Case

We investigate a 1D fermion model with boundary impurity. When the boundary impurity coupling coefficients fullfil some special relations, this model is integrable. The integrable condition can be determined by the self-consistent condition. Furthermore, the eigenvalue and the Bethe ansatz equation are also obtained by using the coordinate Bethe ansatz method, thus the ground state properties is discussed in some special case.

scholarly journals QUASI-EXACTLY SOLVABLE SCHRÖDINGER EQUATIONS, SYMMETRIC POLYNOMIALS AND FUNCTIONAL BETHE ANSATZ METHOD

2018 ◽  
Vol 58 (2) ◽  
pp. 118 ◽  
Author(s):  
Christiane Quesne
Keyword(s):  
Bethe Ansatz ◽  
Linear Equations ◽  
Polynomial Solution ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Symmetric Polynomials ◽  
Ansatz Method ◽  
Exactly Solvable ◽  
Solvable Extension ◽  
Elementary Symmetric Polynomials

For applications to quasi-exactly solvable Schrödinger equations in quantum mechanics, we consider the general conditions that have to be satisfied by the coefficients of a second-order differential equation with at most <em>k </em>+ 1 singular points in order that this equation has particular solutions that are <em>n</em>th-degree polynomials. In a first approach, we show that such conditions involve <em>k </em>- 2 integration constants, which satisfy a system of linear equations whose coefficients can be written in terms of elementary symmetric polynomials in the polynomial solution roots whenver such roots are all real and distinct. In a second approach, we consider the functional Bethe ansatz method in its most general form under the same assumption. Comparing the two approaches, we prove that the above-mentioned <em>k </em>- 2 integration constants can be expressed as linear combinations of monomial symmetric polynomials in the roots, associated with partitions into no more than two parts. We illustrate these results by considering a quasi-exactly solvable extension of the Mathews-Lakshmanan nonlinear oscillator corresponding to <em>k </em>= 4.

scholarly journals Target space entanglement in quantum mechanics of fermions and matrices

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Sotaro Sugishita
Keyword(s):  
Mutual Information ◽  
Ground State ◽  
Entanglement Entropy ◽  
Algebraic Approach ◽  
Upper Bounds ◽  
Wave Functions ◽  
Target Space ◽  
Single Interval ◽  
Area Law ◽  
Formula Expression

Abstract We consider entanglement of first-quantized identical particles by adopting an algebraic approach. In particular, we investigate fermions whose wave functions are given by the Slater determinants, as for singlet sectors of one-matrix models. We show that the upper bounds of the general Rényi entropies are N log 2 for N particles or an N × N matrix. We compute the target space entanglement entropy and the mutual information in a free one-matrix model. We confirm the area law: the single-interval entropy for the ground state scales as $$ \frac{1}{3} $$ 1 3 log N in the large N model. We obtain an analytical $$ \mathcal{O}\left({N}^0\right) $$ O N 0 expression of the mutual information for two intervals in the large N expansion.

scholarly journals Integrable extended Hubbard models with boundary Kondo impurities

2001 ◽  
Vol 64 (3) ◽  
pp. 445-467
Author(s):  
Anthony J. Bracken ◽  
Xiang-Yu Ge ◽  
Mark D. Gould ◽  
Huan-Qiang Zhou
Keyword(s):  
Hilbert Space ◽  
Bethe Ansatz ◽  
Inverse Scattering ◽  
Magnetic Moments ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Ansatz Method ◽  
One Dimensional ◽  
Hubbard Models ◽  
Kondo Impurities

Three kinds of integrable Kondo impurity additions to one-dimensional q-deformed extended Hubbard models are studied by means of the boundary Z2-graded quantum inverse scattering method. The boundary K matrices depending on the local magnetic moments of the impurities are presented as nontrivial realisations of the reflection equation algebras in an impurity Hilbert space. The models are solved by using the algebraic Bethe ansatz method, and the Bethe ansatz equations are obtained.

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