A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

AbstractWe show that any self-adjoint operator A (bounded or unbounded) in a Hilbert space H=(V,(·,·)) that is bounded below generates a continuum of Hilbert spaces {Hr}r>0 and a continuum of self-adjoint operators {Ar}r>0. For reasons originating in the theory of differential operators, we call each Hr the rth left-definite space and each Ar the rth left-definite operator associated with (H,A). Each space Hr can be seen as the closure of the domain D(Ar) of the self-adjoint operator Ar in the topology generated from the inner product (Arx,y) (x,y∈D(Ar)). Furthermore, each Ar is a unique self-adjoint restriction of A in Hr. We show that the spectrum of each Ar agrees with the spectrum of A and the domain of each Ar is characterized in terms of another left-definite space. The Hilbert space spectral theorem plays a fundamental role in these constructions. We apply these results to two examples, including the classical Laguerre differential expression ℓ[·] in which we explicitly find the left-definite spaces and left-definite operators associated with A, the self-adjoint operator generated by ℓ[·] in L2((0,∞);tαe−t) having the Laguerre polynomials as eigenfunctions

A construction of real weight functions for certain orthogonal polynomials in two variables

AbstractH.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are(x2+y)uxx+2xyuxy+y2uyy+gxux+g(y−1)uy=λu, andx2uxx+2xyuxy+(y2−y)uyy+g(x−1)ux+g(y−γ)uy=λu. Even though they showed that these equations have a sequence of weak orthogonal polynomial solutions, they were unable to show that these polynomials were, in fact, orthogonal. The orthogonality of these two polynomial sequences was recently established by Kwon, Littlejohn, and Lee solving an open problem from 1967.In this paper, we construct explicit weight functions for these two orthogonal polynomial sequences, using a method first developed by Littlejohn and then further developed by Han, Kim, and Kwon. Moreover, two additional partial differential equations were found by Kwon, Littlejohn, and Lee that have sequences of orthogonal polynomial solutions. These equations are given by(x2−x)uxx+2xyuxy+y2uyy+(dx+e)ux+(dy+h)uy=λu,xuxx+2yuxy+(dx+e)ux+(dy+h)uy=λu. In each of these examples, we also produce explicit orthogonalizing weight functions

Differential equations having orthogonal polynomial solutions

AbstractNecessary and sufficient conditions for an orthogonal polynomial system (OPS) to satisfy a differential equation with polynomial coefficients of the form (∗) LN[y] = ∑i=1Nli(x)y(i)(x) = λny(x) were found by H.L. Krall. Here, we find new necessary conditions for the equation (∗) to have an OPS of solutions as well as some other interesting applications. In particular, we obtain necessary and sufficient conditions for a distribution w(x) to be an orthogonalizing weight for such an OPS and investigate the structure of w(x). We also show that if the equation (∗) has an OPS of solutions, which is orthogonal relative to a distribution w(x), then the differential operator LN[·] in (∗) must be symmetrizable under certain conditions on w(x)

Askey-Wilson Type Functions, With Bound States

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of $q$-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction

Legendre polynomials, Legendre-Stirling numbers, and the left-definite spectral analysis of the Legendre differential expression

AbstractIn this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A(k) in L2(−1,1), generated from the classical second-order Legendre differential equationℓL,k[y](t)=−((1−t2)y′)′+ky=λy(t∈(−1,1)),that has the Legendre polynomials {Pm(t)}m=0∞ as eigenfunctions; here, k is a fixed, nonnegative constant. More specifically, for k>0, we explicitly determine the unique left-definite Hilbert–Sobolev space Wn(k) and its associated inner product (·,·)n,k for each n∈N. Moreover, for each n∈N, we determine the corresponding unique left-definite self-adjoint operator An(k) in Wn(k) and characterize its domain in terms of another left-definite space. The key to determining these spaces and inner products is in finding the explicit Lagrangian symmetric form of the integral composite powers of ℓL,k[·]. In turn, the key to determining these powers is a remarkable new identity involving a double sequence of numbers which we call Legendre–Stirling numbers