arXiv subjects
Publications and source records attributed to M. Malamud.
We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two- and three-dimensional Schrödinger operators with point interactions on a finite set. In particular, we prove that any realization has purely absolutely continuous non-negative spectrum.
The main object under consideration is a class $Φ_n\backslashΦ_{n+1}$ of radial positive definite functions on $\R^n$ which do not admit \emph{radial positive definite continuation} on $\R^{n+1}$. We find certain necessary and sufficient conditions for the Schoenberg representation measure $ν_n$ of $f\in Φ_n$ in order that the inclusion $f\in Φ_{n+k}$, $k\in\N$, holds. We show that the class $Φ_n\backslashΦ_{n+k}$ is rich enough by giving a number of examples. In particular, we give a direct proof of $Ω_n\inΦ_n\backslashΦ_{n+1}$, which avoids Schoenberg's theorem, $Ω_n$ is the Schoenberg kernel. We show that $Ω_n(a\cdot)Ω_n(b\cdot)\inΦ_n\backslashΦ_{n+1}$, for $a\not=b$. Moreover, for the square of this function we prove surprisingly much stronger result: $Ω_n^2(a\cdot)\inΦ_{2n-1}\backslashΦ_{2n}$. We also show that any $f\inΦ_n\backslashΦ_{n+1}$, $n\ge2$, has infinitely many negative squares. The latter means that for an arbitrary positive integer $N$ there is a finite Schoenberg matrix $\kS_X(f) := \|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m}$, $X := \{x_j\}_{j=1}^m \subset\R^{n+1}$, which has at least $N$ negative eigenvalues.
Given a function $f$ on the positive half-line $\R_+$ and a sequence (finite or infinite) of points $X=\{x_k\}_{k=1}^ω$ in $\R^n$, we define and study matrices $\kS_X(f)=\|f(|x_i-x_j|)\|_{i,j=1}^ω$ called Schoenberg's matrices. We are primarily interested in those matrices which generate bounded and invertible linear operators $S_X(f)$ on $\ell^2(\N)$. We provide conditions on $X$ and $f$ for the latter to hold. If $f$ is an $\ell^2$-positive definite function, such conditions are given in terms of the Schoenberg measure $σ(f)$. We also approach Schoenberg's matrices from the viewpoint of harmonic analysis on $\R^n$, wherein the notion of the strong $X$-positive definiteness plays a key role. In particular, we prove that \emph{each radial $\ell^2$-positive definite function is strongly $X$-positive definite} whenever $X$ is separated. We also implement a "grammization" procedure for certain positive definite Schoenberg's matrices. This leads to Riesz--Fischer and Riesz sequences (Riesz bases in their linear span) of the form $\kF_X(f)=\{f(x-x_j)\}_{x_j\in X}$ for certain radial functions $f\in L^2(\R^n)$. Examples of Schoenberg's operators with various spectral properties are presented.