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Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws

Douglas F. Watson, Tiziano Valentinuzzi

Abstract

We show that single-valuation exponential kernels, under mild regularity assumptions, converge in the continuum limit to a fourth-order operator with heat asymptotics $Θ(t)\sim t^{-1/4}$ and hence spectral dimension $d_s=\tfrac12$. Independently, a Tauberian analysis implies that any self-adjoint operator with superlinear eigenvalue counting $N(λ)\sim λ\,L(λ)$ must satisfy $Θ(t)\sim t^{-1}L(1/t)$ and therefore has spectral dimension $d_s=2$. Since spectral dimension is invariant under unitary equivalence and compact perturbations, these exponents are incompatible, yielding a structural obstruction that separates single-valuation kernel limits from operators with accelerated spectral growth.

Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws

Abstract

We show that single-valuation exponential kernels, under mild regularity assumptions, converge in the continuum limit to a fourth-order operator with heat asymptotics and hence spectral dimension . Independently, a Tauberian analysis implies that any self-adjoint operator with superlinear eigenvalue counting must satisfy and therefore has spectral dimension . Since spectral dimension is invariant under unitary equivalence and compact perturbations, these exponents are incompatible, yielding a structural obstruction that separates single-valuation kernel limits from operators with accelerated spectral growth.

Paper Structure

This paper contains 62 sections, 18 theorems, 121 equations.

Table of Contents

  1. Statement of the Main Theorems
  2. The Exponential Kernel Principle
  3. Arithmetic Specialization and the Biharmonic Limit
  4. No-go for Local Geometric Operators
  5. Spectral Constraints from the Riemann Zeros
  6. The Double Hilbert--Pólya Obstruction
  7. The Exponential Kernel Principle
  8. Setup and Assumptions
  9. Existence and Uniqueness of the Exponential Form
  10. Lagrangian formulation
  11. Normalization and the partition function
  12. Solving the constraint
  13. Uniqueness of the maximizer
  14. Integrability and Admissibility
  15. General criterion
  16. ...and 47 more sections

Key Result

Theorem 1.1

Under the standing assumptions stated in Section sec:exp-principle, the entropy-maximization problem for each fixed $i\in I$ admits a unique solution $\{T_{ij}\}_{j\in I}$, and this maximizer is necessarily of exponential form: for a uniquely determined parameter $\beta_i>0$ chosen so that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (28)

  • Theorem 1.1: Exponential Kernel Principle
  • Theorem 1.2: Arithmetic biharmonic limit
  • Theorem 1.3: Elliptic spectral dimension
  • Theorem 1.4: Elliptic obstruction for zeta-type spectra
  • Theorem 1.5: Double Hilbert--Pólya Obstruction
  • Proposition 2.1: Exponential maximizer
  • Proposition 2.2: Integrability window
  • Corollary 2.3: Arithmetic exponential kernel
  • Lemma 4.1: Liminf inequality
  • proof
  • ...and 18 more