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A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in n − 4 spacelike dimensions… (More)

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. In the first instance , we show that the alignment condition is equivalent to the 4-dimensional PND equation, and thereby recover the usual Petrov types. For the higher dimensional case, we prove… (More)

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 reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to… (More)

- D Gómez-Ullate, N Kamran, R Milson
- 2008

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solv-able quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to… (More)

A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by dropping the assumption that the first element of the orthogonal polynomial sequence be a constant. This approach gives rise to new families of complete orthogonal polynomial systems… (More)

- Robert Milson
- 1998

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 counterexample is given based on the trigonometric… (More)

We describe a theory of tensor classification based on the notion of aligned null directions. Applications include the classification problem for the Weyl and Ricci tensors. In the first instance, we show that the alignment condition is equivalent to the 4-dimensional PND equation, and thereby recover the usual Petrov types. For the higher dimensional case,… (More)

Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunc-tions of Sturm–Liouville problems, but without the assumption that an eigenpolyno-mial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by… (More)

Our goal in this paper is to extend the theory of quasi-exactly solvable Schrödinger operators beyond the Lie-algebraic class. Let Pn be the space of n-th degree polynomials in one variable. We first analyze exceptional polynomial subspaces M ⊂ Pn, which are those proper subspaces of Pn invariant under second order differential operators which do not… (More)