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, $m=0,1,2,...$, of deformations exists for each family of shape-invariant potentials. We prove that the $m$-th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules $\mathcal{P}^{(m)}_m\subset\mathcal{P}^{(m)}_{m+1}\subset...$, where $\mathcal{P}^{(m)}_n$ is a codimension $m$ subspace of $$. In particular, we prove that the first ($m=1$) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules $\mathcal{P}^{(1)}_n = < 1,z^2,...,z^n>$. By construction, these algebraically deformed Hamiltonians do not have an $\mathfrak{sl}(2)$ 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

Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials

In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux-Poschl-Teller potential.Comment: 28 pages, 6 figure

An experimental study of perceptions of lectures attuned to different learning styles.

An experimental study into the attunement of lectures to students' learning styles, in which analysis was undertaken on 77 students, from the degree disciplines of Physiotherapy, Statistics, Nursing and Psychology. The aim of the study was to discover if students perceived that they had learnt more effectively in lectures attuned to their learning styles, as predicted on the basis of Kolb (1984). The students were presented with four lectures each attuned to a different learning style, at the end of each lecture they were asked to assess their perceptions of the lecture and learning within the lecture via a questionnaire. The results indicated that there were no significant differences in the students' perceived learning within attuned lectures when compared to the non-attuned lectures. This contradicts the connection Kolb claims between approach and learning style.Close examination of Kolb's learning style theory revealed a number of serious anomalies and internal inconsistencies within his work (Claimed negative correlations between dialectic pairs, mixed learning styles and questionnable support for learning styles from split brain research). These theoretical anomalies were suplemented by experimental results that indicated that learning styles were not stable over time. The implications of this analysis are discussed in detail. Finally speculative further investigations were carried out to in an attempt to provide a more appropriate interpretation of Kolb's work. This further work yielded interesting results that are reported and would merit further study