Least Favourable Spectral Density Matrices

• The paper presents explicit polynomial bounds for spectral density functions over complex group rings, linking them to Novikov-Shubin invariants.
• It employs recursive univariate reduction and measure-theoretic techniques to derive sharp estimates for matrices with minimal width.
• The framework offers practical tools for analyzing worst-case spectral behavior in free abelian groups, influencing studies in L2-invariants and topology.

Least favourable spectral density matrices arise in the paper of matrices over the complex group ring of free abelian groups, notably $\mathbb{C}[\mathbb{Z}^d]$, and are intrinsically connected to the estimation of spectral density functions and Novikov-Shubin invariants. These matrices are characterized by achieving the minimal possible spectral decay near zero, subject to algebraic constraints imposed by their minors, and are central to understanding the "worst-case" behavior for the accumulation of small eigenvalues of such operators.

1. Spectral Density Functions Over $\mathbb{C}[\mathbb{Z}^d]$

Given an $(m, n)$-matrix $A$ with entries in the complex group ring $\mathbb{C}[\mathbb{Z}^d]$, one considers the operator $r_A^{(2)}$ acting on the Hilbert space $l^2(\mathbb{Z}^d)^n$. The spectral density function $F(r_A^{(2)})(\lambda)$, for $\lambda \geq 0$, is defined as the trace of the spectral projection onto the subspace corresponding to spectrum below $\lambda$. For the case $m = n = 1$, with $p \in \mathbb{C}[\mathbb{Z}^d]$, this function is given by

$$F(r_p^{(2)})(\lambda) = \mu_{\mathbb{T}^d}\left(\{ (z_1,\dots, z_d) \in \mathbb{T}^d \mid |p(z_1,\dots, z_d)| \leq \lambda \}\right),$$

where $\mu_{\mathbb{T}^d}$ denotes the normalized Haar measure on the torus $\mathbb{T}^d$.

2. Polynomial Bounds and Key Algebraic Invariants

Explicit polynomial upper bounds for the spectral density function are governed by combinatorial invariants derived from the matrix's maximal nontrivial minors. For a maximal square submatrix $B$ of $A$ with The determinant $p = \det_{\mathbb{C}[\mathbb{Z}^d]}(B)$ and $k =$ size of $B$, the relevant invariants are:

• Width $w_d(p)$: Defined recursively, this captures the maximal vanishing order (when viewing $p$ as a polynomial in each variable) across all coordinate projections.
• Leading coefficient $\operatorname{lead}(p)$: The coefficient corresponding to the leading term (multidegree) of $p$.
• Norm $|B|_1$: The $l^1$-norm (sum of absolute values of coefficients) of $B$.

The precise estimate, valid for $w_d(p) \geq 1$ and all $\lambda \geq 0$, is

$$F(r_A^{(2)})(\lambda) - F(r_A^{(2)})(0) \leq 8.13 \sqrt{47}\ d\,w_d(p) k 2^{k-2} (|B|_1)^{k-1} \frac{1}{|\operatorname{lead}(p)|} \lambda^{1/(dw_d(p))}.$$

If $w_d(p) = 0$, the spectral density function is a step function: it is zero for $\lambda < |\operatorname{lead}(p)|$ and one otherwise.

3. Least Favourable (Extremal) Spectral Density Matrices

The "least favourable", or extremal, spectral density matrices correspond to those for which the width $w_d(p)$ is minimized, making the associated polynomial as close to a monomial or as highly degenerate as possible. For such matrices:

• The accumulation of small eigenvalues—and thus the measure of the near-zero spectrum—cannot be faster than the rate dictated by the width and the ambient dimension, up to explicit constants.
• The bound above is sharp up to those constants, in the sense that there exist canonical examples where the spectral density function saturates this bound.

A plausible implication is that among all possible matrices respecting a given value of $w_d(p)$, those with minimal width realize the "worst-case" scenario for spectral concentration near zero; conversely, increasing the width improves the decay rate.

4. Novikov-Shubin Invariants and Spectral Asymptotics

The Novikov-Shubin invariant $\alpha_{r_A^{(2)}}$ quantifies the asymptotic order of vanishing of the spectral density function near zero: $$\alpha_{r_A^{(2)}} := \liminf_{\lambda \to 0^+} \frac{\log(F(r_A^{(2)})(\lambda) - F(r_A^{(2)})(0))}{\log \lambda}.$$ The bounds above yield an explicit lower bound

$\alpha_{r_A^{(2)}} \geq d\,w_d(p),$

ensuring that the invariant is strictly positive for any nontrivial matrix over $\mathbb{C}[\mathbb{Z}^d]$. In particular, the sharpness of this estimate confirms previous conjectures and results that for matrices over group rings of finitely generated free abelian groups, the Novikov-Shubin invariant is always strictly positive. For settings outside virtually abelian or free groups, the invariant may be zero, demonstrating the result's optimality in this algebraic context.

5. Derivation Techniques and Recursion

The polynomial bounds are derived using recursive, induction-based techniques on the spatial dimension $d$ and the group ring element structure. The analysis involves:

• Univariate reduction: Sequentially fixing variables and analyzing roots and vanishing orders to control the width.
• Explicit measure-theoretic estimates: Calculating the size of the sub-level set $\{|p(z)| \leq \lambda\}$ in the torus via Fourier analysis and volume bounds.
• Matrix reduction: Relating spectral density functions for the full matrix to those of its maximal minors using a spectral-minor comparison lemma. This translates results for single polynomials to general matrix cases, modulo explicit multiplicative constants.

6. Broader Context and Applications

This framework generalizes previous results for integral group rings to the complex coefficient case. The explicit algebraic control over spectral density functions and Novikov-Shubin invariants is foundational for the paper of $L^2$-invariants and their applications. It provides constructive tools for topological, geometric, and global analytic investigations involving the spectrum of group ring operators, especially in the context of free abelian or virtually abelian groups. The results offer insight into spectral behavior in related settings, such as twisted $L^2$-invariants and their use in geometry and topology, strengthening previously conjectured and partially established positivity properties.

7. Summary of Principal Formulas

Quantity	Formula	Variables
Spectral density (scalar case)	$$F(r_p^{(2)})(\lambda) = \mu_{\mathbb{T}^d}(\{|p(z)| \leq \lambda\})$$	$p \in \mathbb{C}[\mathbb{Z}^d]$, $\lambda \geq 0$
Polynomial bound (matrix case)	$$F(r_A^{(2)})(\lambda) - F(r_A^{(2)})(0) \leq C \lambda^{1/(dw_d(p))}$$	$C$ as above, $w_d(p)$, $k=|B|$, $\lambda$
Novikov-Shubin lower bound	$\alpha_{r_A^{(2)}} \geq d\,w_d(p)$	$A$ as above

The main contribution is the provision of explicit, invariant-controlled bounds on small eigenvalue densities for this class of operators, with direct implications for spectral asymptotics and the topology of free abelian coverings. The analysis precisely identifies and constrains the least favourable behavior, extending previous algebraic and analytic results to a broader and more computable setting.