Self-adjoint representations of a certain sequence of spectral differential equations

For {dollar}alpha>-1{dollar} and M {dollar}>{dollar} 0 the Laguerre-type polynomials (L{dollar}rmsbsp{lcub}n{rcub}{lcub}alpha,M{rcub}{dollar}) form a complete orthogonal sequence in a Lebesgue-Hilbert space with inner product {dollar}rm(f,g) {lcub}:={rcub} {lcub}1overGamma(alpha+1){rcub}intsbsp{lcub}0{rcub}{lcub}infty{rcub}f(t) overline{lcub}g(t){rcub}tspalpha {lcub}e{rcub}sp{lcub}-t{rcub}dt+Mf(0)overline{lcub}g(0){rcub}.{dollar}; This Hilbert space is called the right-definite space. In the case {dollar}alphain{lcub}0,1,2,cdots{rcub}{dollar} and M {dollar}>{dollar} 0 the polynomials satisfy a {dollar}rm(2alpha+4)sp{lcub}th{rcub}{dollar}-order spectral differential equation called the Laguerre-type equation of order ({dollar}alpha{dollar} +): {dollar}rmellsbalpha f(t) =lambda f(t).{dollar}; This spectral differential equation has a unique self-adjoint representation in the right-definite space, which has the Laguerre-type polynomials as a complete set of eigenfunctions. In the 4{dollar}sp{lcub}rm th{rcub}{dollar}- and {dollar}6sp{lcub}rm th{rcub}{dollar}-order cases no boundary conditions are needed to produce the self-adjoint representation of {dollar}ellsbalpha.{dollar} In all of the higher order cases only one boundary condition is needed; indeed it is given by {dollar}rm limsb{lcub}tdownarrow0{rcub}f(t)=f(0).{dollar}; The Laguerre-type expression is made formally self-adjoint when multiplied by the classical weight function {dollar}rmomegasbalpha(t) {lcub}:={rcub} tspalpha esp{lcub}-t{rcub}/Gamma(alpha+1){dollar}:; {dollar}rmomegasbalpha(t)ellsbalpha f(t)=M(alpha+1)sumsbsp{lcub}k =1{rcub}{lcub}alpha+2{rcub}((-1)sp{lcub}k{rcub}{lcub}sumsbsp{lcub}j=0{rcub}{lcub}alpha+2-k{rcub}{lcub}tsp{lcub}2k-2+ j{rcub}esp{lcub}-t{rcub}over k!(k-1)!j!{rcub}{rcub}fsp{lcub}k{rcub}(t))sp{lcub}(k){rcub}-{lcub}(tsp{lcub}alpha +1{rcub}fspprime(t))spprimeoveralpha!{rcub}.{dollar}; The differential expression {dollar}ellsbalpha{dollar} generates a unique self-adjoint differential operator in the left-definite space, which is a Sobolev space with inner product; {dollar}rm M(alpha+1)sumsbsp{lcub}k=1{rcub}{lcub}alpha+2{rcub}intsbsp{lcub}0{rcub}{lcub}infty {rcub}{lcub}sumsbsp{lcub}j=0{rcub}{lcub}alpha+2-k{rcub}{lcub}tsp{lcub}2k-2+j{rcub}esp{lcub}-t{rcub}over k!(k-1)!j!{rcub}{rcub}fsp{lcub}(k){rcub}(t)overline{lcub}gsp{lcub}(k){rcub}(t){rcub}dt+ {lcub}1overalpha!{rcub}intsbsp{lcub}0{rcub}{lcub}infty{rcub}(tsp{lcub}alpha+ 1{rcub}fspprime(t)overline{lcub}gspprime(t){rcub}+Ktspalpha f(t)overline{lcub}g(t){rcub})esp{lcub}-t{rcub}dt+KMf(0)overline{lcub}g(0){rcub}.{dollar}; No boundary conditions are required to generate the left-definite self-adjoint spectral representation for any of the Laguerre-type differential expressions. To study this left-definite problem, the spectral theorem plays a fundamental role.