14,177 research outputs found
Quantum transfer matrices for discrete and continuous quasi-exactly solvable problems
We clarify the algebraic structure of continuous and discrete quasi-exactly
solvable spectral problems by embedding them into the framework of the quantum
inverse scattering method. The quasi-exactly solvable hamiltonians in one
dimension are identified with traces of quantum monodromy matrices for specific
integrable systems with non-periodic boundary conditions. Applications to the
Azbel-Hofstadter problem are outlined.Comment: 15 pages, standard LaTe
Integrability of one degree of freedom symplectic maps with polar singularities
In this paper, we treat symplectic difference equations with one degree of
freedom. For such cases, we resolve the relation between that the dynamics on
the two dimensional phase space is reduced to on one dimensional level sets by
a conserved quantity and that the dynamics is integrable, under some
assumptions. The process which we introduce is related to interval exchange
transformations.Comment: 10 pages, 2 figure
New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum
quasi-Yang-Baxter algebras being simple but non-trivial deformations of
ordinary algebras of monodromy matrices realize a new type of quantum dynamical
symmetries and find an unexpected and remarkable applications in quantum
inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter
algebras the standard procedure of QISM one obtains new wide classes of quantum
models which, being integrable (i.e. having enough number of commuting
integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic
Bethe ansatz solution for arbitrarily large but limited parts of the spectrum).
These quasi-exactly solvable models naturally arise as deformations of known
exactly solvable ones. A general theory of such deformations is proposed. The
correspondence ``Yangian --- quasi-Yangian'' and `` spin models ---
quasi- spin models'' is discussed in detail. We also construct the
classical conterparts of quasi-Yang-Baxter algebras and show that they
naturally lead to new classes of classical integrable models. We conjecture
that these models are quasi-exactly solvable in the sense of classical inverse
scattering method, i.e. admit only partial construction of action-angle
variables.Comment: 49 pages, LaTe
Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
Inozemtsev models are classically integrable multi-particle dynamical systems
related to Calogero-Moser models. Because of the additional q^6 (rational
models) or sin^2(2q) (trigonometric models) potentials, their quantum versions
are not exactly solvable in contrast with Calogero-Moser models. We show that
quantum Inozemtsev models can be deformed to be a widest class of partly
solvable (or quasi-exactly solvable) multi-particle dynamical systems. They
posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A
new method for identifying and solving quasi-exactly solvable systems, the
method of pre-superpotential, is presented.Comment: LaTeX2e 28 pages, no figure
Exactly-Solvable Models Derived from a Generalized Gaudin Algebra
We introduce a generalized Gaudin Lie algebra and a complete set of mutually
commuting quantum invariants allowing the derivation of several families of
exactly solvable Hamiltonians. Different Hamiltonians correspond to different
representations of the generators of the algebra. The derived exactly-solvable
generalized Gaudin models include the Bardeen-Cooper-Schrieffer,
Suhl-Matthias-Walker, the Lipkin-Meshkov-Glick, generalized Dicke, the Nuclear
Interacting Boson Model, a new exactly-solvable Kondo-like impurity model, and
many more that have not been exploited in the physics literature yet
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