QUASI-EXACTLY SOLVABLE PROBLEMS AND SU(1,1) GROUP

It is shown that the particular class of one-dimensional quasi-exactly solvable models can be constructed with the help of infinite-dimensional representation of Lie algebra. Hamiltonian of a system is expressed in terms of SU(1,1) generators.

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LIE-ALGEBRAIC APPROACH TO THE PROBLEM OF QUASI-EXACT SOLUBILITY IN QUANTUM MECHANICS

Modern Physics Letters A ◽

10.1142/s0217732390002146 ◽

1990 ◽

Vol 05 (23) ◽

pp. 1891-1899 ◽

Cited By ~ 11

Author(s):

A. G. USHVERIDZE

Keyword(s):

Quantum Mechanics ◽

Lie Algebras ◽

Algebraic Approach ◽

New Method ◽

Exactly Solvable Models ◽

Solvable Models ◽

Simple Lie Algebras ◽

Infinite Dimensional ◽

Exactly Solvable

A new method of constructing quasi-exactly solvable models of quantum mechanics is proposed. This method is based on the use of infinite-dimensional representations of simple and semi-simple Lie algebras.

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Exactly and Quasi-Exactly Solvable Models on the Basis of OSP(2|1)

Modern Physics Letters A ◽

10.1142/s0217732397001680 ◽

1997 ◽

Vol 12 (22) ◽

pp. 1655-1661

Author(s):

A. Shafiekhani ◽

M. Khorrami

Keyword(s):

Symmetry Algebra ◽

Dynamical Symmetry ◽

One Dimension ◽

Exactly Solvable Models ◽

Solvable Models ◽

Exactly Solvable ◽

Solvable Problems

The exactly and quasi-exactly solvable problems for spin one-half in one dimension on the basis of a hidden dynamical symmetry algebra of Hamiltonian are discussed. We take the supergroup, OSP(2|1), as such a symmetry. A number of exactly solvable examples are considered and their spectrum are evaluated explicitly. Also, a class of quasi-exactly solvable problems on the basis of such a symmetry has been obtained.

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QUASI-EXACTLY-SOLVABLE QUANTAL PROBLEMS: ONE-DIMENSIONAL ANALOGUE OF RATIONAL CONFORMAL FIELD THEORIES

International Journal of Modern Physics A ◽

10.1142/s0217751x90000374 ◽

1990 ◽

Vol 05 (04) ◽

pp. 803-832 ◽

Cited By ~ 89

Author(s):

A. YU. MOROZOV ◽

A.M. PERELOMOV ◽

A.A. ROSLY ◽

M.A. SHIFMAN ◽

A.V. TURBINER

Keyword(s):

Mathematical Formulation ◽

Field Theories ◽

Conformal Field Theories ◽

Solvable Models ◽

One Dimensional ◽

Conformal Field ◽

Exactly Solvable ◽

Solvable Problems ◽

Shed Light ◽

Ordinary Quantum

The class of quasi-exactly-solvable problems in ordinary quantum mechanics discovered recently shows remarkable parallels with rational two-dimensional conformal field theories. This fact suggests that investigation of the quasi-exactly-solvable models may shed light on rational conformal field theories. We discuss a relation between these two theoretical schemes and propose a mathematical formulation for the procedure of constructing quasi-exactly solvable systems. This discussion leads us to a kind of generalization of the Sugawara construction.

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INVESTIGATION OF EXCITATION SPECTRA OF EXACTLY SOLVABLE MODELS USING INVERSION RELATIONS

International Journal of Modern Physics B ◽

10.1142/s0217979290000401 ◽

1990 ◽

Vol 04 (05) ◽

pp. 871-893 ◽

Cited By ~ 28

Author(s):

A. KLÜMPER

Keyword(s):

Solution Procedure ◽

Exactly Solvable Models ◽

Solvable Models ◽

Excitation Spectra ◽

One Dimensional ◽

Exactly Solvable ◽

State Models ◽

Set Up ◽

Quantum Antiferromagnet ◽

Inversion Relations

We study the excitations of two-dimensional classical vertex models and related one-dimensional quantum Hamiltonians. The main mathematical means we use are functional equations which are set up and solved for the excitation functions. The method is first applied to the eight-vertex model as a prototype and compared with the traditional solution procedure of the Bethe Ansatz. In the last part we investigate some q-state models, notably the biquadratic spin-1 quantum antiferromagnet and its generalizations, which recently attracted interest due to Haldane’s conjecture.

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An approach to one-dimensional elliptic quasi-exactly solvable models

Pramana ◽

10.1007/s12043-008-0020-5 ◽

2008 ◽

Vol 70 (4) ◽

pp. 575-585

Author(s):

M. A. Fasihi ◽

M. A. Jafarizadeh ◽

M. Rezaei

Keyword(s):

Exactly Solvable Models ◽

Solvable Models ◽

One Dimensional ◽

Exactly Solvable

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A class of exactly solvable models for the Schrödinger equation

Open Physics ◽

10.2478/s11534-013-0301-6 ◽

2013 ◽

Vol 11 (8) ◽

Cited By ~ 4

Author(s):

Charles Downing

Keyword(s):

Differential Equation ◽

Schrödinger Equation ◽

Schrodinger Equation ◽

Mathematical Physics ◽

Transcendental Equation ◽

Exactly Solvable Models ◽

Solvable Models ◽

One Dimensional ◽

Exactly Solvable ◽

The One

AbstractWe present a class of confining potentials which allow one to reduce the one-dimensional Schrödinger equation to a named equation of mathematical physics, namely either Bessel’s or Whittaker’s differential equation. In all cases, we provide closed form expressions for both the symmetric and antisymmetric wavefunction solutions, each along with an associated transcendental equation for allowed eigenvalues. The class of potentials considered contains an example of both cusp-like single wells and a double-well.

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Generalized Generalized Spin Models Associated with Exactly Solvable Models

Progress in Algebraic Combinatorics ◽

10.2969/aspm/02410081 ◽

2018 ◽

Author(s):

Tetsuo Deguchi

Keyword(s):

Exactly Solvable Models ◽

Solvable Models ◽

Spin Models ◽

Exactly Solvable

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Interaction between Different Kinds of Quantum Phase Transitions

Quantum Reports ◽

10.3390/quantum3020015 ◽

2021 ◽

Vol 3 (2) ◽

pp. 253-261

Author(s):

Angel Ricardo Plastino ◽

Gustavo Luis Ferri ◽

Angelo Plastino

Keyword(s):

Phase Transitions ◽

Nuclear Physics ◽

Body Problem ◽

Quantum Phase Transitions ◽

Quantum Phase ◽

Exactly Solvable Models ◽

Solvable Models ◽

Many Body ◽

Pairing Interaction ◽

Exactly Solvable

We employ two different Lipkin-like, exactly solvable models so as to display features of the competition between different fermion–fermion quantum interactions (at finite temperatures). One of our two interactions mimics the pairing interaction responsible for superconductivity. The other interaction is a monopole one that resembles the so-called quadrupole one, much used in nuclear physics as a residual interaction. The pairing versus monopole effects here observed afford for some interesting insights into the intricacies of the quantum many body problem, in particular with regards to so-called quantum phase transitions (strictly, level crossings).

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