Robert Milson

An extended class of orthogonal polynomials defined by a Sturm–Liouville problem

Abstract We present two infinite sequences of polynomial eigenfunctions of a Sturm–Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X 1 -Jacobi and X 1 -Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [ − 1 , 1 ] or the half-line [ 0 , ∞ ) , respectively, and they are a basis of the corresponding L 2 Hilbert spaces. Moreover, we prove a converse statement similar to Bochners theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions { p i } i = 1 ∞ , then it must be either the X 1 -Jacobi or the X 1 -Laguerre Sturm–Liouville problem. A Rodrigues-type formula can be derived for both of the X 1 polynomial sequences.

Classification of the Weyl tensor in higher dimensions

PCT No. PCT/GB88/00981 Sec. 371 Date May 1, 1990 Sec. 102(e) Date May 1, 1990 PCT Filed Nov. 14, 1988 PCT Pub. No. WO89/04245 PCT Pub. Date May 18, 1989.Flexible reinforced polymeric material (30), typically a tape, comprises two elongate flexible support layers (31, 39) and a plurality of lengthwise extending cords (36) secured thereto. The respective edges of the support layers (31, 39) are transversely offset to result in a pair of edge regions and some cords (36) are sandwiched between the support layers while at least one cord (32, 33, 40) and (37, 38, 41) is secured to one of the edge regions. Preferably the two edge regions have corresponding cord arrangements (32, 33, 40) and (37, 38, 41) provided such that the cords (32, 33, 40) of one edge region may be caused to interlock with those (37, 38, 41) at the other edge region.

Exceptional orthogonal polynomials and the Darboux transformation

We adapt the notion of the Darboux transformation to the context of polynomial Sturm–Liouville problems. As an application, we characterize the recently described Xm Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux–Crum transformation.

Two-step Darboux transformations and exceptional Laguerre polynomials

It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these parameters, the classical Laguerre and the type II X(l)-Laguerre polynomials are recovered.

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 λ 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 λ. Explicit expressions for such recurrence relations are given.

Vanishing scalar invariant spacetimes in higher dimensions

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.

Bianchi identities in higher dimensions

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 or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers–Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.

Alignment and algebraically special tensors in Lorentzian geometry

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. First, we show that the alignment condition is equivalent to the principal null direction equation. In 4 dimensions this recovers the usual Petrov types are recovered. For higher dimensions we prove that, in general, a Weyl tensor does not possess any aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.

All spacetimes with vanishing curvature invariants

All Lorentzian spacetimes with vanishing invariants constructed from the Riemann tensor and its covariant derivatives are determined. A subclass of the Kundt spacetimes results and we display the corresponding metrics in local coordinates. Some potential applications of these spacetimes are discussed.

A Conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm–Liouville problems, but without the assumption that an eigenpolynomial 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 Cariñena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux–Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPSs. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.