Properties of the operator domains of the fourth-order Legendre-type differential expressions
Legendre-type polynomials are obtained when the Gram-Schmidt process is applied to \(\{x^ n: n\in \mathbb{N}_ 0\}\) in \(L^ 2([-1,1]; \mu)\), where the measure \(\mu\) is generated by \[ \widehat\mu(x)= \begin{cases} -1-M, & x\in (-\infty,-1],\\ x, & x\in (-1,1),\\ -1+ N, & x\in [1,\infty),\end{cases} \] and \(M\), \(N\) are non-negative constants. In each of the five possible cases the polynomials satisfy a formally symmetric spectral differential equation: one of these is of second order (the classical case \(M= N=0\)), three of fourth order (discovered by H. L. Krall) and one of sixth order (discovered by L. L. Littlejohn). This paper deals with the three cases for which the differential equations are of fourth order. The fourth order differential expressions generate self-adjoint operators in \(L^ 2([-1,1]; \mu)\) with discrete spectra and whose eigenfunctions are the appropriate Legendre-type polynomials. Properties of the domains of these operators are determined.
formally symmetric spectral differential equation
scientific article; zbMATH DE number 597490
scientific article