Weyl's theory in anL2-basis Gauss quadrature of the spectral density

Abstract

When a complete orthonormal square-integrable real basis set $\left\{{\chi }_n\right\}$ can be found in which a model one-dimensional Hamiltonian $H_0$ is represented by an infinite symmetric tridiagonal matrix, analysis of the spectrum of $H_0$ leads to a close analogy of Weyl's theory of the Schrödinger equation. With $H_0$ tridiagonal, the Schrödinger equation becomes a three-term recursion relation in $n$ instead of a differential equation. The Sturm-sequence polynomials $p_n\mathrm{}\left(E\right)\mathrm{}$ form the solution regular at $n=0\mathrm{}$, while a second, Weyl's solution $q_n\mathrm{}\left(E\right)\mathrm{}$, is irregular. Then the resolvent ${(E-H_0)}^{-1\mathrm{}}$ is proportional to ${p_n}_{<}{q_n}_{>}$. Furthermore, completeness implies the orthogonality of the $p_n$, and hence, that truncating to a finite basis generates a Gauss quadrature of the spectral density with abscissas at the eigen-values of the truncated $H_0$, and just as in Weyl's theory, approximating the spectral density as a Stieltjes integral. By truncating the representation of any additional potential to a finite matrix, the theory can be extended in analogy to $R$-matrix theory to potential and even multichannel scattering, yielding an explicit construction of the Fredholm determinant, whose zeros locate the resonances and bound states. In addition, such an analysis reveals, at least in the basis sets in which $H_0$ is tridiagonal, how other $L^2$-function methods such as stabilization, Stieltjes imaging, and coordinate rotation work and how accurate they are.