Darboux transformation of the Laguerre operator
For \(\alpha>-1\), let \(L_n(x,\alpha)\), \(n\) a non-negative integer, denote the classical Laguerre polynomials orthogonal in \(L^2(\mathbb R_+,w_\alpha)\), where \(w_\alpha(x)=x^\alpha \cdot e^{-x}\). For \(\alpha>-1\), the classical Laguerre operator \(L_\alpha\) is defined on \(L^2(\mathbb R_+,w_\alpha)\) by \[ L_\alpha(y):= xy^{\prime\prime}+(\alpha+1-x)y^\prime. \] This operator is self-adjoint and the classical Laguerre polynomials are its eigenvalues. If \(\alpha<-1\) and not equal to a negative integer, the operator \(L_\alpha\) is called a non-classical Laguerre operator and its eigenvalues \(L_n(x,\alpha)\), called the non-classical Laguerre polynomials, are orthogonal in the Pontryagin space \(\Pi(\alpha)\). The author uses the following factorization operators \[ L=\left(x\frac{d}{dx}+\left(\alpha +1\right)\right)\quad \text{and}\quad R=\left(\frac{d}{dx}-1\right), \] and defines the Darboux transformation \(D^{\alpha+}\) of the operator \(L_\alpha\) as the operator \[ L_{\alpha+1}(y)=RL(y)+(\alpha+1)y. \] The author shows that, if \(\alpha>-1\), then the Darboux transformation \(D^{\alpha+}\) of \(L_{\alpha}\) is the classical Laguerre operator on \(L^2(\mathbb R_{+},w_{\alpha+1})\); if \(\alpha<-2\) (not equal to a negative integer), then the Darboux transformation \(D^{\alpha+}\) of \(L_{\alpha}\) is the non-classical Laguerre operator on \(\Pi(\alpha+1)\) and in the case \(\alpha\in(-2,-1)\), the Darboux transformation \(D^{\alpha+}\) of the non-classical Laguerre operator \(L_\alpha\) on \(\Pi(\alpha)\) is the classical Laguerre operator on \(L^2(\mathbb R_+,w_{\alpha+1})\). Additionally, for \(\alpha\) not equal to a negative integer, if \(L_{\alpha+1}\) is the Darboux transformation \(D^{\alpha+}\) of \(L_\alpha\), then the eigenvalues are transformed by the Christoffel formula. The author obtains analogous results for the Darboux transformation \(D^{\alpha-}\) defined by the factorization operators \[ {\mathcal L}=\left(\frac{d}{dx}-1\right)\quad \text{and}\quad {\mathcal R}=\left(x\frac{d}{dx}+\alpha\right). \]
scientific article; zbMATH DE number 6860224
scientific article