54 research outputs found
From Quantum to Trigonometric Model: Space-of-Orbits View
A number of affine-Weyl-invariant integrable and exactly-solvable quantum
models with trigonometric potentials is considered in the space of invariants
(the space of orbits). These models are completely-integrable and admit extra
particular integrals. All of them are characterized by (i) a number of
polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for
exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii)
a rational form of the potential and the polynomial entries of the metric in
the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants
(the same holds for rational models when polynomial invariants are used instead
of exponential ones), they admit (iv) an algebraic form of the gauge-rotated
Hamiltonian in the exponential invariants (in the space of orbits) and (v) a
hidden algebraic structure. A hidden algebraic structure for
(A-B-C{-D)-models, both rational and trigonometric, is related to the
universal enveloping algebra . For the exceptional -models,
new, infinite-dimensional, finitely-generated algebras of differential
operators occur. Special attention is given to the one-dimensional model with
symmetry. In particular, the origin
of the so-called TTW model is revealed. This has led to a new quasi-exactly
solvable model on the plane with the hidden algebra .Comment: arXiv admin note: substantial text overlap with arXiv:1106.501
Particular Integrability and (Quasi)-exact-solvability
A notion of a particular integrability is introduced when two operators
commute on a subspace of the space where they act. Particular integrals for
one-dimensional (quasi)-exactly-solvable Schroedinger operators and
Calogero-Sutherland Hamiltonians for all roots are found. In the classical case
some special trajectories for which the corresponding particular constants of
motion appear are indicated.Comment: 13 pages, typos correcte
From Quantum (Calogero) to (Rational) Model
A brief and incomplete review of known integrable and
(quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian
coordinates) potentials is given. All of them are characterized by (i) a
discrete symmetry of the Hamiltonian, (ii) a number of polynomial
eigenfunctions, (iii) a factorization property for eigenfunctions, and admit
(iv) the separation of the radial coordinate and, hence, the existence of the
2nd order integral, (v) an algebraic form in invariants of a discrete symmetry
group (in space of orbits).Comment: dedicated to Willard Miller; 20 page
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