Asymptotic behavior of the quantum defect as a function of an angular momentum and almost normalization of eigenfunctions for a three-dimensional Schrödinger operator with nearly Coulomb potential

Abstract

We study two questions connected to the discrete spectrum of a three-dimensional \linebreak {\sch} operator with Coulomb potential perturbed by a spherically symmetric compactly supported function $q(r)$. It is known that the eigenvalues of this operator can be described by the formula $\lambda^\ell_n=-(n+\ell+\mu^\ell_n)^{-2}$, where $n=1,2,\dots$ is the principal quantum number, $\ell=0,1,2,\dots$ is the azimuthal quantum number (or angular momentum) and $\mu^\ell_n$ is the so-called quantum defect. It is, also, known that for each $\ell$ there exists a limit $\mu^\ell_\infty= {\lim_{n\rightarrow\infty}}\mu^\ell_n$. We assume that $q(r)\geq 0$ and prove the estimate $|\mu^\ell_\infty|\leq C(a e^2/2)^{2\ell+1} (\ell+0.5)^{-(4\ell+3)}$, where $a$ is the radius of the support of $q(r)$ and $C$ is $0.25 \sqrt{\pi}$ times the first moment of $q(r)$ (which is assumed to be finite). It follows from this estimate that $\mu^\ell_\infty$ tends to zero as $\ell\rightarrow \infty$ faster than any negative power of $\ell$. Our second result deals with the eigenfunctions $\{\Phi^\ell_n (r)\}^\infty_{n=1}$ of the radial {\sch} operator corresponding to a fixed value of $\ell$. The natural definition of $\Phi^\ell_n(r)$ which allows one to construct them explicitly, is based on the condition $ {\lim_{r\rightarrow 0}} r^{-\ell-1} \Phi^\ell_n(r)=1$. Such defined eigenfunctions are not normalized in $L^2(0,\infty)$ and the normalization constant $C^\ell_n$ is known only for the pure Coulomb potential. We prove that in the case of a perturbed Coulomb potential the constants $C^\ell_n$ behave as $n^{-3/2}$ when $n\rightarrow\infty$ for any fixed $\ell$ and, therefore, for any $\ell$ the system $\{n^{-3/2} \Phi^\ell_n (r)\}^\infty_{n=1}$ forms an orthogonal Riesz basis in its closed linear hull.