Quadratic Forms, Real Zeros and Echoes of the Spectral Action

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.