arXiv subjects

Robert Milson

Publications and source records attributed to Robert Milson.

48 records · Page 3Linked to original sources

arXiv2006

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.

nlin.SI↗

arXiv2006

Quasi-exact solvability beyond the SL(2) algebraization

We present evidence to suggest that the study of one dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual $\sla(2)$ approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the $\sla(2)$ Lie algebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic hamiltonian cannot be expressed as a polynomial in the generators of $\sla(2)$. We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie-algebraic approach.

nlin.SI↗

arXiv2006

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.

nlin.SI↗

arXiv2005

Adapting CBPP platforms for instructional use

Commons based peer-production (CBPP) is the de-centralized, net-based approach to the creation and dissemination of information resources. Underlying every CBPP system is a virtual community brought together by an internet tool (such as a web site) and structured by a specific collaboration protocol. In this talk we will argue that the value of such platforms can be leveraged by adapting them for pedagogical purposes. We report on one such recent adaptation. The Noosphere system is a web-based collaboration environment that underlies the popular Planetmath website, a collaboratively written encyclopedia of mathematics licensed under the GNU Free Documentation License (FDL). Recently, the system was used to host a graduate-level mathematics course at Dalhousie University, in Halifax, Canada. The course consisted of regular lectures and assignment problems. The students in the course collaborated on a set of course notes, encapsulating the lecture content and giving solutions of assigned problems. The successful outcome of this experiment demonstrated that a dedicated Noosphere system is well suited for classroom applications. We argue that this ``proof of concept'' experience also strongly suggests that every successful CBPP platform possesses latent pedagogical value.

cs.DL↗

arXiv2004

Alignment and the classification of Lorentz-signature tensors

We define the notion of an aligned null direction, a Lorentz-signature analogue of the eigenvector concept that is valid for arbitrary tensor types. The set of aligned null directions is described by a a system of alignment polynomials whose coefficients are derived from the components of the tensor. The algebraic properties of the alignment polynomials can be used to classify the corresponding tensors and to put them into normal form. The alignment classification paradigm is illustrated with a discussion of bivectors and of Weyl-type tensors. Note: an earlier version of this manuscript was published in the proceedings of SPT 2004. The present version has been expanded to include a discussion of complexified alignment. Section 4 also corrects errors contained in the earlier manuscript.

gr-qc↗

arXiv2003

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 $<1,z,...,z^n>$. 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.

quant-ph↗

arXiv2002

Reflection quotients in Riemannian Geometry. A Geometric Converse to Chevalley's Theorem

Chevalley's theorem and it's converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely that a freely-generated polynomial subring is closed with respect to the gradient product.

math.DG↗

arXiv2000

Spectral residues of second-order differential equations: towards a new class of summation identities and inversion formulas

The present article deals with differential equations with spectral parameter from the point of view of formal power series. The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residue. The article focuses on second-order, self-adjoint problems. In such a setting every potential function determines a sequence of spectral residues. This correspondence is invertible, and gives rise to a combinatorial inversion formula. Other interesting combinatorial consequences are obtained by considering spectral residues of exactly-solvable potentials of 1-dimensional quantum mechanics. It is also shown that the Darboux transformation of 1-dimensional potentials corresponds to a simple negation of the corresponding spectral residues. This fact leads to another combinatorial inversion formula. Finally, there is a brief discussion of applications. The topics considered are enumeration problems and integrable systems.

math-ph↗

arXiv2000

Composition sum identities related to the distribution of coordinate values in a discrete simplex

Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first and second order, ordinary, linear differential equations. Regarding the first class, the corresponding identities amount to a proof of the exponential formula of labelled counting. The identities in the second class can be used to establish certain geometric properties of the simplex of bounded, ordered, integer tuples. We present three theorems that support the conclusion that the inner dimensions of such an order simplex are, in a certain sense, more ample than the outer dimensions. As well, we give an algebraic proof of a bijection between two families of subsets in the order simplex, and inquire as to the possibility of establishing this bijection by combinatorial, rather than by algebraic methods.

math.CO↗

arXiv1999

Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems

We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a re duction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasi-exactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization o f the annihilating differential operators for spaces of analytic functions are presented.

solv-int↗

arXiv1998

Imprimitively generated Lie-algebraic Hamiltonians and separation of variables

Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the A_2 root system.

solv-int↗

arXiv1997

On the Liouville transformation and exactly-solvable Schrodinger equations

The present article discusses the connection between exactly-solvable Schrodinger equations and the Liouville transformation. This transformation yields a large class of exactly-solvable potentials, including the exactly-solvable potentials introduced by Natanzon. As well, this class is shown to contain two new families of exactly solvable potentials.

solv-int↗