Dispersion estimates for the discrete Hermite operator
It is well known that the classical Hermite orthogonal polynomials in one variable, \(H_{n}\left(x\right) \), play a crucial role in the exact solutions of the one-dimensional Schrödinger equation modeling the {quantum harmonic oscillator}: \[ -\frac{\hbar ^{2}}{2m}\frac{d^{2}\psi (x)}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}\psi (x)=E\psi (x). \] In this framework, ``exact solutions'' means that the whole energy spectrum is analytically computed: \[ E_{n}=\hbar \omega \left( n+\frac{1}{2}\right) ,\quad n=0,1,2,\ldots , \] as well as the corresponding set of wave functions (also known as \textit{Hermite functions}) \[ \psi _{n}(x)=\frac{1}{\sqrt{2^{n}n!}}\left( \frac{m\omega }{\pi \hbar }\right) ^{1/4}e^{-\frac{m\omega x^{2}}{2\hbar }}H_{n}\left( \sqrt{\frac{m\omega }{\hbar }}x\right) ,\quad n=0,1,2,\ldots . \] In this nice and very technical paper, the authors study the one-dimensional discrete Schrödinger equation associated with the self-adjoint Jacobi operator \[ H_{d}=\left( \begin{array}{cccccc} 0 & \sqrt{2} & 0 & 0 & 0 & \cdots \\ \sqrt{2} & 0 & \sqrt{4} & 0 & 0 & \cdots \\ 0 & \sqrt{4} & 0 & \sqrt{6} & 0 & \cdots \\ 0 & 0 & \sqrt{6} & 0 & \sqrt{8} & \cdots \\ 0 & 0 & 0 & \sqrt{8} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \right), \] whose generalize eigenfunctions are precisely the Hermite functions. They obtain a sharp \(l^{\infty }\)\ estimate of the kernel \(a_{n,m}(t)\), for \(m=0,1\), \(m=n\), and \(t\in \lbrack 1,\infty ]\) for the \textit{propagator} \(e^{itH_{d}}\), associated with the Hermite functions. They also obtain an interesting recurrence relation for \(a_{n,m}(t)\), and they conjecture that the aforesaid \(l^{\infty }\) estimate holds true for general values of the parameter \(m\).
scientific article; zbMATH DE number 7453740
scientific article