This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article... more
This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article of Aomoto and Forrester. We consider first the one-degree of freedom case and compute the Birkhoff series defined near each of its stationary points. In general, the analysis of the system motivates finding some expression for the inverses of a rational map introduced by Aomoto and Forrester. We derive here some diagrammatic expansion series for these inverses.
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Several explicit examples of quasi exactly solvable ‘discrete ’ quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the... more
Several explicit examples of quasi exactly solvable ‘discrete ’ quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/sin 2 x potential deformed by a cos 2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions. 1
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Ruijsenaars–Schneider models associated with An−1 root system with a discrete coupling constant are studied. The eigenvalues of the Hamiltonian are given in terms of the Bethe ansatz formulas. Taking the “nonrelativistic” limit, we obtain... more
Ruijsenaars–Schneider models associated with An−1 root system with a discrete coupling constant are studied. The eigenvalues of the Hamiltonian are given in terms of the Bethe ansatz formulas. Taking the “nonrelativistic” limit, we obtain the spectrum of the corresponding Calogero–Moser systems in the third formulas of Felder et al.
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Several explicit examples of quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogs of the well-known... more
Several explicit examples of quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogs of the well-known quasiexactly solvable systems, the harmonic oscillator (with∕without the centrifugal potential) deformed by a sextic potential, and the 1∕sin2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.
Research Interests: Mathematics, Mathematical Physics, Physics, Quantum Mechanics, Quantum, and 10 moreMathematical Sciences, Physical sciences, Wave Equation, Quantum Harmonic Oscillator, Bound States, Difference equation, Eigenvalues and Eigenvectors, Eigenvalues and Eigenfunctions, Degree of Freedom, and eigenfunction
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Research Interests: Number Theory, Mathematical Physics, Probability Theory, Quantum Theory, Quantum Mechanics, and 11 moreMatrix Theory, Mathematical Sciences, Physical sciences, Brownian Motion, Markov chain, Markov Process, Orthogonal Polynomial, Similarity Transformation, Matrix Algebra, Birth and Death Process, and transition probability
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Research Interests: Mathematics, Mathematical Physics, Physics, Quantum Theory, Quantum Mechanics, and 12 moreMathematical Sciences, Physical sciences, Wave Equation, Schrodinger equation, Bound States, Difference equation, Unified Theory, Polynomials, Eigenvalues and Eigenfunctions, Orthogonal Polynomial, eigenfunction, and Schrödinger equation
ABSTRACT
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This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article... more
This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article of Aomoto and Forrester. We consider first the one-degree of freedom case and compute the Birkhoff series defined near each of its stationary points. In general, the analysis of the system motivates finding some expression for the inverses of a rational map introduced by Aomoto and Forrester. We derive here some diagrammatic expansion series for these inverses.
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Crum's theorem in one-dimensional quantum mechanics asserts the existence of an associated Hamiltonian system for any given Hamiltonian with the complete set of eigenvalues and eigenfunctions. The associated system is iso-spectral to... more
Crum's theorem in one-dimensional quantum mechanics asserts the existence of an associated Hamiltonian system for any given Hamiltonian with the complete set of eigenvalues and eigenfunctions. The associated system is iso-spectral to the original one except for the ...
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This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article... more
This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article of Aomoto and Forrester. We consider first the one-degree of freedom case and compute the Birkhoff series defined near each of its stationary points. In general, the analysis of the system motivates finding some expression for the inverses of a rational map introduced by Aomoto and Forrester. We derive here some diagrammatic expansion series for these inverses.