Quadratic Forms, Real Zeros and Echoes of the Spectral Action
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866914173925982208 |
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| author | Connes, Alain van Suijlekom, Walter D. |
| author_facet | Connes, Alain van Suijlekom, Walter D. |
| contents | 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 $λ$ is a simple, isolated eigenvalue with even eigenfunction $ξ$, then all the zeros of the entire function $\widehat ξ(z)$, the Fourier transform of $ξ$, lie on the real line. The proof proceeds in five steps. (1) We give a C*-algebraic proof of a corollary of Carathéodory-Fejér'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 $ξ\in \ker T$, then the polynomial $P(z) = \sum ξ_j z^j$ 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 Carathéodory-Fejér'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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quadratic Forms, Real Zeros and Echoes of the Spectral Action Connes, Alain van Suijlekom, Walter D. Operator Algebras Number Theory 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 $λ$ is a simple, isolated eigenvalue with even eigenfunction $ξ$, then all the zeros of the entire function $\widehat ξ(z)$, the Fourier transform of $ξ$, lie on the real line. The proof proceeds in five steps. (1) We give a C*-algebraic proof of a corollary of Carathéodory-Fejér'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 $ξ\in \ker T$, then the polynomial $P(z) = \sum ξ_j z^j$ 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 Carathéodory-Fejér'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. |
| title | Quadratic Forms, Real Zeros and Echoes of the Spectral Action |
| topic | Operator Algebras Number Theory |
| url | https://arxiv.org/abs/2511.23257 |