94 research outputs found
The Darboux transformation and algebraic deformations of shape-invariant potentials
We investigate the backward Darboux transformations (addition of a lowest
bound state) of shape-invariant potentials on the line, and classify the
subclass of algebraic deformations, those for which the potential and the bound
states are simple elementary functions. A countable family, , of
deformations exists for each family of shape-invariant potentials. We prove
that the -th deformation is exactly solvable by polynomials, meaning that it
leaves invariant an infinite flag of polynomial modules
, where
is a codimension subspace of . In
particular, we prove that the first () algebraic deformation of the
shape-invariant class is precisely the class of operators preserving the
infinite flag of exceptional monomial modules . By construction, these algebraically deformed Hamiltonians do
not have an hidden symmetry algebra structure.Comment: 18 pages, 3 figures. Paper has been considerably extended and
revised. References adde
Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces
In this paper we derive structure theorems that characterize the spaces of
linear and non-linear differential operators that preserve finite dimensional
subspaces generated by polynomials in one or several variables. By means of the
useful concept of deficiency, we can write explicit basis for these spaces of
differential operators. In the case of linear operators, these results apply to
the theory of quasi-exact solvability in quantum mechanics, specially in the
multivariate case where the Lie algebraic approach is harder to apply. In the
case of non-linear operators, the structure theorems in this paper can be
applied to the method of finding special solutions of non-linear evolution
equations by nonlinear separation of variables.Comment: 23 pages, typed in AMS-LaTe
Quasi-exact solvability in a general polynomial setting
Our goal in this paper is to extend the theory of quasi-exactly solvable
Schrodinger operators beyond the Lie-algebraic class. Let \cP_n be the space
of n-th degree polynomials in one variable. We first analyze "exceptional
polynomial subspaces" which are those proper subspaces of \cP_n invariant
under second order differential operators which do not preserve \cP_n. We
characterize the only possible exceptional subspaces of codimension one and we
describe the space of second order differential operators that leave these
subspaces invariant. We then use equivalence under changes of variable and
gauge transformations to achieve a complete classification of these new,
non-Lie algebraic Schrodinger operators. As an example, we discuss a finite gap
elliptic potential which does not belong to the Treibich-Verdier class.Comment: 29 pages, 10 figures, typed in AMS-Te
Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
We prove that every rational extension of the quantum harmonic oscillator
that is exactly solvable by polynomials is monodromy free, and therefore can be
obtained by applying a finite number of state-deleting Darboux transformations
on the harmonic oscillator. Equivalently, every exceptional orthogonal
polynomial system of Hermite type can be obtained by applying a Darboux-Crum
transformation to the classical Hermite polynomials. Exceptional Hermite
polynomial systems only exist for even codimension 2m, and they are indexed by
the partitions \lambda of m. We provide explicit expressions for their
corresponding orthogonality weights and differential operators and a separate
proof of their completeness. Exceptional Hermite polynomials satisfy a 2l+3
recurrence relation where l is the length of the partition \lambda. Explicit
expressions for such recurrence relations are given.Comment: 25 pages, typed in AMSTe
Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials
Using the method of point canonical transformation, we derive some exactly
solvable rationally extended quantum Hamiltonians which are non-Hermitian in
nature and whose bound state wave functions are associated with Laguerre- or
Jacobi-type exceptional orthogonal polynomials. These Hamiltonians are
shown, with the help of imaginary shift of co-ordinate: , to be both quasi and pseudo-Hermitian. It turns
out that the corresponding energy spectra is entirely real
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