Quadratic Forms, Real Zeros and Echoes of the Spectral Action

TL;DR

This paper proves that under certain spectral conditions, the Fourier transform of the eigenfunction associated with a quadratic form's lowest eigenvalue has all zeros on the real line, extending classical Toeplitz matrix results.

Contribution

It extends Carathéodory-Fejér's theorem to continuous convolution operators and spectral actions, linking spectral properties to zero localization of Fourier transforms.

Findings

Zeros of Fourier transforms lie on the real line under specified spectral conditions.

Extension of Toeplitz matrix zero localization to continuous convolution operators.

Connection between spectral action structures and zero distribution of eigenfunction transforms.

Abstract

For a real distribution $\mathcal{D}$ on the interval $[0,L]$ with $\tilde{\mathcal{ D}}$ the associated even distribution on the interval $[-L, L]$, we prove that if the associated quadratic form with Schwartz kernel $\tilde{\mathcal{D}}(x - y)$ defines a lower-bounded selfadjoint operator on $L^2([-\frac{L}{2}, \frac{L}{2}])$, whose lowest spectral value $\lambda$ is a simple, isolated eigenvalue with even eigenfunction $\xi$, then all the zeros of the entire function $\widehat \xi(z)$, the Fourier transform of $\xi$, lie on the real line. The proof proceeds in five steps. (1) We give a C*-algebraic proof of a corollary of Carath\'eodory-Fej\'er's 1911 structure Theorem for Toeplitz matrices: if $T \in M_n(\mathbb{C})$ is a Hermitian, positive semidefinite Toeplitz matrix of rank $n - 1$, and $\xi \in \ker T$, then the polynomial $P(z) = \sum \xi_j z^j$ has all its zeros on the unit circle. (2) We formulate and prove a continuous analogue of this result, replacing the Toeplitz matrix with a convolution operator with continuous kernel $h(x - y)$, and the polynomial $P(z)$ with the Fourier transform of the eigenfunction corresponding to the largest eigenvalue. (3) We analyze finite-dimensional truncations of the quadratic forms defined by real, even distributions $\mathcal{D}$ on $[-L, L]$, and observe that the resulting matrices exhibit a structure previously encountered in perturbative expansions of the spectral action. (4) We establish an analogue of Carath\'eodory-Fej\'er's corollary for matrices of this specific structure, thereby extending the zero localization result beyond the classical Toeplitz setting. (5) Finally, we apply a classical theorem of Hurwitz concerning the zeros of uniform limits of holomorphic functions to deduce the general result stated above.