The spectral resolution of Laguerre operators in right definite and left definite spaces.

摘要

The connection between the singular Sturm-Liouville problem and the differential operator generated by the Laguerre differential equation {dollar}xyprimeprime + (1 + alpha - x)yprime + lambda y = 0{dollar} is considered for {dollar}all alpha{dollar} in both the right and left definite cases.; First, the boundary value Sturm-Liouville problem is considered on an interval, {dollar}(a,b),{dollar} where both {dollar}a{dollar} and {dollar}b{dollar} are singular points. We use the Weyl {dollar}m(lambda){dollar} function to find {dollar}Lsp2(a, b; w){dollar} solutions near a and b. Using these we define abstract boundary conditions. These, in turn, assist us to define a self-adjoint differential operator and to derive its resolvent. We then discuss the spectral functions and the generalized Fourier transform of an arbitrary function in {dollar}Lsp2(a, b; w).{dollar}; Second, in the right definite case, a shifted Laguerre equation when {dollar}alpha {lcub}-{rcub}1{dollar} are introduced. We show that both Laguerre operators given by {dollar}ly = lbrack {lcub}-{rcub}(xsp{lcub}alpha+1{rcub}esp{lcub}-x{rcub}yprime)primerbrack /xspalpha esp{lcub}-x{rcub},{dollar} when {dollar}alpha {lcub}-{rcub}1,{dollar} respectively, are self-adjoint under the norm {dollar}(y, z) = intsbsp{lcub}0{rcub}{lcub}infty{rcub} xspalpha esp{lcub}-x{rcub}y bar z dx.{dollar} It is also shown that the spectral resolution of the Laguerre differential operator for all {dollar}alpha {lcub}-{rcub}1.{dollar}; Finally, in the left definite case, the Laguerre operators given by {dollar}ly = lbrack {lcub}-{rcub}(xsp{lcub}alpha+1{rcub} esp{lcub}-x{rcub} yprime)prime + xspalpha esp{lcub}-x{rcub} yrbrack /xspalpha esp{lcub}-x{rcub},{dollar} when {dollar}alpha {lcub}-{rcub}1,{dollar} respectively, are shown to be self-adjoint under the energy norm {dollar}langle y, zrangle = intsbsp{lcub}0{rcub}{lcub}infty{rcub} lbrack (xsp{lcub}alpha+1{rcub} esp{lcub}-x{rcub})yprimebar zprime + xspalpha esp{lcub}-x{rcub} y bar zrbrack dx.{dollar} The spectra for both the shifted, {dollar}alpha {lcub}-{rcub}1,{dollar} Laguerre equations are discrete, {dollar}{lcub}n - alpha + 1{rcub}sbsp{lcub}n=0{rcub}{lcub}infty{rcub}{dollar} and {dollar}{lcub}n + 1{rcub}sbsp{lcub}n=0{rcub}{lcub}infty{rcub},{dollar} with eigenfunctions {dollar}{lcub}xsp{lcub}-alpha{rcub}Lsbsp{lcub}n{rcub}{lcub}(-alpha){rcub}(x){rcub}sbsp{lcub}n=0{rcub}{lcub}infty{rcub}{dollar} and {dollar}{lcub}Lsbsp{lcub}n{rcub}{lcub}(alpha){rcub}(x){rcub}sbsp{lcub}n=0{rcub}{lcub}infty{rcub}{dollar}, respectively. Their spectral resolutions in {dollar}Lsp2(0, infty; xspalpha esp{lcub}-x{rcub}){dollar} hold in the new space {dollar}Hsp1(0, infty; xsp{lcub}alpha+1{rcub} esp{lcub}-x{rcub}, xspalpha esp{lcub}-x{rcub}){dollar} as well.