# Finite-rank approximations of spectral zeta residues

## Abstract

We use the asymptotic expansion of the heat trace to express all residues of spectral zeta functions as regularized sums over the spectrum. The method extends to those spectral zeta functions that are localized by a bounded operator.

## Keywords

Zeta residues Wodzicki residue Dixmier trace Heat kernel Partial spectrum Numerical asymptotics## Mathematics Subject Classification

58J42 58B34 35P20## 1 Introduction

The spectral theory of elliptic operators presents a major connection between functional analysis and differential geometry. It provides a number of interesting relations between the spectrum with multiplicities (which is the complete unitary invariant of self-adjoint operators) and the symbol (which is directly tied to the local expression of pseudodifferential operators). Thereby, it shows how the local structure of such an operator influences its global properties, and vice versa. Of particular interest is the relation between the symbol of an elliptic operator and its spectral asymptotics that is conveyed by the spectral zeta residues.

This note is concerned with expressing the spectral zeta residues as a limit of partial sums of particular functions over the spectrum. Together with the well-known relations between the zeta residues, the symbol, and the heat expansion, this bridges a gap between the continuum set-up of spectral geometry and the finite objects in combinatorial geometry and computer science. For instance, this method allows one to approximate the scalar curvature on a compact Riemannian manifold using only a finite part of its Laplacian spectrum.

We will first provide some background material in Sect. 1.1 and then introduce the main topic and results of this note in Sect. 1.2.

### 1.1 Background: spectral zeta residues and Weyl’s law

*d*-dimensional manifold can be meromorphically extended to the complex plane, with simple poles occurring in the points \(d/2-\mathbb {Z}_{\ge 0} \subset \mathbb {R}\). The residues at these poles (which are proportional to the so-called heat kernel coefficients) relate the spectrum of the Laplace operator, itself an isometry invariant, to other known isometry invariants of the manifold, such as its volume and scalar curvature.

*k*is any nonnegative integer, the residue at \(s=(d-k)/m\) of \(\zeta (\varDelta ,s)\) equals the Wodzicki residue

*m*elliptic pseudodifferential operator on a

*d*-dimensional manifold. As in [10], this yields Weyl’s famous law

The problem of improving on the accuracy in Weyl’s law has attracted much attention over the last century. That is, one wonders whether we can obtain an asymptotic expansion of \(N(\varLambda ) - {{\mathrm{res}}}_{s=d/m} \zeta (\varDelta ,s) \varLambda ^{d/m} \) for specific classes of operators. However, sharp bounds that depend only on the spectral zeta function of \(\varDelta \) have not yet been produced, even in the well-studied case of the Laplacian on flat tori. It would seem natural that the Wiener–Ikehara result could be extended to relate the asymptotic expansion of \(N(\varLambda )\) to the location of the poles of the zeta function. However, this approach is limited by the difficulties of inverse Mellin and Laplace transforms (see e.g. [1]), [2, 9.7.2]. The lower poles of the zeta function can therefore not yet be related to the asymptotics of \(N(\varLambda )\). However, we will explain in the present paper how to relate their residues to the asymptotics of other functionals of the operator spectrum.

### 1.2 Zeta residues as a resummation of the spectrum

*finite*subsets of the operator spectrum. For the first residue, Weyl’s law gives rise to the Dixmier trace

^{1}formula: if \(\lambda _0 \le \lambda _1 \le \cdots \) are the eigenvalues of \(\varDelta \), then

The existence of such asymptotic residue functionals is shown by our Theorem 1, and explicit expressions follow from the conditions of Propositions 1 and 2. Corollary 2 uses the theorem to improve on the convergence of the Dixmier trace formula, and Corollary 3 exhibits the resulting expression for the second pole. Finally, Theorem 2 is a simple extension of our treatment to the localized residue traces \({{\mathrm{res}}}_{s=s_k} {{\mathrm{tr}}}h \varDelta ^{-s}\), for any bounded operator *h*.

## 2 Zeta residues as normal functionals

We can ask how spectral zeta residues relate to finite subsets of the operator spectrum without referring to the original setting of differential geometry. We will henceforth consider the question in such generality, but we will need to restrict ourselves to operators whose zeta functions share an essential property with the Minakshisundaram–Pleijel zeta function.

### Definition 1

A positive, invertible,^{2} self-adjoint, unbounded operator \(\varDelta \) with compact resolvent is said to be *spectrally elliptic* if the ‘heat trace’ \({{\mathrm{tr}}}\text {e}^{- t \varDelta }\) admits an asymptotic expansion \(\sum _{i=0}^\infty c_i t^{- s_i}\) as \(t \rightarrow 0^+\), where the \(s_i\) are decreasingly ordered reals. We say that \(\varPi = \{s_i\}\) is the *heat spectrum* of \(\varDelta \).

The definition implies that there is some \(s_0 \in \mathbb {R}\) such that \(\varGamma (s) {{\mathrm{tr}}}\varDelta ^{-s}\) converges for \(\mathfrak {R}s > s_0\) and can be analytically continued to a meromorphic function \(\varGamma (s) \zeta (\varDelta ,s)\) whose poles are all simple and located in \(\varPi \subset (-\,\infty ,s_0]\).

This particular terminology was chosen because the motivating example of such operators are the positive elliptic differential^{3} operators, cf. [7, 11]. Indeed, for a positive elliptic differential operator of order *m* on a *n*-dimensional manifold, the heat spectrum is contained in \(\frac{n}{m},\frac{n-1}{m},\cdots ,\frac{1}{m}\).

Definition 1 suggests that we will in fact be technically concerned with the asymptotic heat trace coefficients \(c_i\), and the reader who is more familiar with that terminology may rest assured that they are indeed what we are talking about. However, because zeta residues are mathematically the more general notion, the text will mainly refer to the coefficients by that name.

Our main theorem shows how the zeta residues of a spectrally elliptic operator are related to restrictions of its spectrum. We will provide the proof in Sect. 2.1.

The precise asymptotic formula for the residue \({{\mathrm{res}}}_{s=s_k} \zeta (\varDelta ,s)\) at some pole \(s_k \in \varPi \) depends only on the location of the ‘previous’ poles, that is, on the set \(\varPi \cap [s_k,\infty )\). The Dixmier trace, for instance, requires knowledge of \(s_0\) and of the fact that no poles \(s_{-1} > s_0\) exist. For a given finite set \(\{s_i\}\) of such poles, all operators with zeta residues contained in this set can be considered simultaneously, hence the following definition.

### Definition 2

For any finite set \(\{s_i\}_{i=0}^k\) of decreasingly ordered reals, let \(\mathscr {D}(\{s_i\}_{i=0}^k)\) be the set of all spectrally elliptic operators \(\varDelta \) whose heat spectrum \(\varPi \) satisfies \(\varPi \cap [s_k,\infty ) \subset \{s_i\}\), i.e. whose heat trace admits an asymptotic expansion \(\sum _{i=0}^k c_i t^{-s_i} + O(t^{-s_{k+1}})\) as \(t \rightarrow 0^+\), for some \(s_{k+1} < s_k\).

*F*is any Borel measurable function, we will denote the summation of

*F*over the strictly positive eigenvalues smaller than \(\varLambda \) by

### Theorem 1

*F*such that, for all \(\varDelta \in \mathscr {D}(\{s_i\}_{i=0}^k)\),

The problem posed in Introduction is then trivially resolved by the following corollary, together with an explicit choice of *F*.

### Corollary 1

### Remark 1

A uniform bound over \(\mathscr {D}(\{s_i\}_{i=0}^k)\) on the rate of convergence is too much to ask: it depends on the complete spectrum of \(\varDelta \). However, the convergence rate will be shown to always be \(O(\varLambda ^{-1})\) if the next pole is at \(s_k - 1\). For examples of arbitrary error for given \(\varLambda \), see, for instance, [9, 12]. For comparison, see Remark 4 on page 10.

### Remark 2

The exclusion of logarithmic terms in the heat expansion from Definition 1 corresponds to an exclusion of higher-order poles of \(\varGamma (s) \zeta (\varDelta ,s)\). However, Theorem 1 is unchanged if higher-order poles are allowed, as long as they lie below \(s_k\). Moreover, the approach can probably be modified to accommodate such logarithmic terms, at the expense of brevity and asymptotic rate of convergence.

### 2.1 Asymptotic series for zeta function residues

### Proof of Theorem 1

*f*one has

*F*to rewrite the integral as a trace

*F*decays as

The following asymptotic estimate is completely straightforward.

### Proposition 1

*f*be a piecewise continuous function supported in \((1,\infty )\) that is \(O(t^{-m})\) for all \(m \in \mathbb {R}\) towards \(\infty \). Let \(\varDelta \in \mathscr {D}(\{s_i\}_{i=0}^k)\). Then, the Mellin convolution integral of

*f*with \({{\mathrm{tr}}}\text {e}^{-t \varDelta }\) has an asymptotic expansion

Thus, if we simply choose *f* so that its moments \(\int _0^\infty t^{-s_i} f(t) \text {d}t \) vanish for \(i \ne k\) and normalize it so that \(\int _0^\infty t^{-s_k} f(t) \text {d}t = 1\), we gain access to the coefficient \(c_k\).

*f*towards \(\infty \), it has absolutely convergent Laplace transform

*F*. As \(\left| f(t) {{\mathrm{tr}}}\text {e}^{- \varepsilon t \varDelta } \right| \) is absolutely integrable as well, we can apply the Fubini–Tonelli theorem to obtain

### Proposition 2

*f*and \(\varDelta \) are as in Proposition 1 and additionally

*F*satisfies

### Proof

*t*, we see that

This completes the proof of Theorem 1 that started on page 5.

### 2.2 Explicit formulas for the first two poles

One reason to look for series that converge to zeta residues is to obtain geometric information from finite-dimensional approximations of the Laplacian on a Riemannian manifold. For instance, if \(\varDelta \) is the Laplacian, the first two residues are proportional to the volume and the total scalar curvature, respectively.

The first pole is a classical object of study. Its residue is expressed by Dixmier’s singular trace and is used, for instance, for the zeta regularization of divergent series. It is connected to counting asymptotics by the Wiener–Ikehara theorem, and for a Laplace operator on a compact Riemannian manifold, it is proportional to the volume.

Our Theorem 1 provides the following formula for the first residue.

### Corollary 2

### Proof

Use \(f = [t \ge 1] \text {e}^{-t} \), with Laplace transform \(F(s) = \text {e}^{-1-s}/(1+s)\), in Proposition 1. Then, divide by \(\int _0^\infty t^{-s_0} f(t) \text {d}t\) to satisfy the conditions of Proposition 2. \(\square \)

### Remark 3

The present series converges faster in general than the logarithmic trace suggested by Weyl’s asymptotic formula; the remainder is \(O(\varepsilon ^{s_0 - s_1}) = O(\varLambda ^{s_1-s_0} (\log \varLambda )^{s_0 - s_{1}})\), whereas for e.g. the Laplacian on the circle the logarithmic trace of \(\varDelta ^{-1/2}\) convergences only as \(\sum _{n=1}^N \frac{1}{n \log N} - 1 = \frac{\gamma + O(N^{-1})}{\log N}\).

Now, the second pole is of particular interest because it is the first pole for which no asymptotic residue formula was previously known and because it provides a way to calculate the total scalar curvature of a Riemannian manifold from partial spectra of the Laplacian.

### Corollary 3

### Proof

Use \(f = [t \ge 1] \text {e}^{-t} - 2^{s_0 - 1} [t \ge 2] \text {e}^{-t/2}\), which clearly satisfies the conditions of Proposition 1 and has vanishing moment \(\int _0^\infty t^{-s_0} f(t) \text {d}t\). Therefore, if we divide by \(\int _0^\infty t^{1-s_0} f(t) \text {d}t\), the result will be as in Proposition 2. The Laplace transform of *f* is \(F(s) = \text {e}^{-1-s}/(1+s) - 2^{s_0} \text {e}^{-1-2s}/(1+2s)\). \(\square \)

### Example 1

*n*th eigenvalue of the Laplacian on the sphere and let \(p_n = 2n+1\) be its multiplicity. As before, write \(\varepsilon _m = 2 q_m^{-1} \log q_m\). We will use the Euler–Maclaurin formula to estimate the series from Corollary 3,

Because \(F(s) = O(s^{-1} \text {e}^{-s})\) as \(s \rightarrow \infty \), the order zero boundary term is just \(\frac{1}{2} + \frac{1}{2} p_m F(\varepsilon _m q_m) = \frac{1}{2} + O(p_m q_m^{-2} / \log q_m)\).

*t*, one can reproduce the asymptotics as \(t \rightarrow 0\) afterwards. In the present formulation, however, we need only take a single limit in

*m*and have expressed the residue as a single universal series in the eigenvalues, with an asymptotic bound on the error over all of \(\mathscr {D}(s_0,s_0-1)\).

## 3 Localization

If \(\varDelta \) is spectrally elliptic, Theorem 1 allows us to calculate its zeta residues as a series in its eigenvalues. However, the classical theory of elliptic pseudodifferential operators assigns to them not just the total zeta residues, but rather a set of zeta *densities*. To be precise, the somewhat simpler situation for elliptic differential operators is as follows.

*M*be a Riemannian manifold of dimension

*n*equipped with a smooth Hermitian vector bundle

*E*, and let \(\varDelta \) be a positive, self-adjoint, elliptic differential operator of order \(m>0\), acting inside the Hilbert space of square-integrable sections of

*E*. Then, the following localized asymptotics are available.

- 1.The operator \(\text {e}^{-t \varDelta }\) is given by a smooth kernel \(k(x,y,t): E_y \rightarrow E_x\), and as \(t \rightarrow 0^+\), its restriction to the diagonal admits an asymptotic expansion in smooth sections \(k_j\) of \({{\mathrm{End}}}E\),$$\begin{aligned} k(x,x,t) \sim \sum _{j=0}^{\infty } k_j(x) t^{(j-n)/m}. \end{aligned}$$
- 2.For any continuous section
*h*of \({{\mathrm{End}}}E\) and any \(j \ge 0\), the following residues exist and satisfy$$\begin{aligned} {{\mathrm{res}}}_{s=(j-n)/m} \varGamma (s) {{\mathrm{tr}}}h \varDelta ^{-s} = \int _M {{\mathrm{tr}}}(h(x) k_j(x)) \text {d} {{\mathrm{vol}}}_M(x). \end{aligned}$$

*h*

*localizes*the residue trace, and we are interested in expressing the localized residues in a fashion similar to Theorem 1. We will solve this localization problem in a slightly more general setting, along the lines of the previous treatment of the zeta function residues.

*F*be the Laplace transform of a piecewise continuous function

*f*supported in \([1,\infty )\), which is \(O(t^{-m})\) for all \( m \in \mathbb {R}\) towards \( \infty \), satisfies \(\int _0^\infty t^{-s_i} f(t) \text {d}t = 0\) for all \(i < k\), and is normalized to satisfy \(\int _0^\infty t^{-s_k} f(t) \text {d}t = 1\).

### Theorem 2

### Proof

*f*. That is,

## 4 Final remarks and suggestions

The original motivation behind this note was to confirm the point of view that finite-rank cut-offs of spectral triples can carry geometric information in noncommutative geometry. The following remarks all proceed in that direction.

### Remark 4

Our Theorem 1 provides a partial counterweight to a classical result by Colin de Verdière [12]. On the one hand, he showed that a finite set of Laplace eigenvalues carries no information on the metric if no pointwise bounds on the sectional curvature are imposed. On the other hand, we now see that for each metric and each desired accuracy, there is a bound \(\varLambda \) such that the set of eigenvalues smaller than \(\varLambda \), with multiplicity, allow computation of all zeta residues (and hence, of the associated global invariants of the metric) up to that accuracy. The local version, using Theorem 2, of this statement is that for each metric and each \(\varepsilon \) there is a finite-rank projection \(P_\varLambda = [0,\varLambda ](\varDelta _g)\) such that the cut-off matrix \(P_\varLambda \varDelta _g P_\varLambda \) together with the cut-off \(P_\varLambda C(M) P_\varLambda \) of the function algebra yields the residues of \({{\mathrm{tr}}}a \varDelta ^{-s}\) up to an error of \(\Vert a\Vert \varepsilon \).

### Remark 5

*finite-dimensional*noncommutative geometry. Let (

*A*,

*H*,

*D*) be a spectral triple such that \(D^2\) is spectrally elliptic with heat spectrum \(\{(j-n)/2 \}_{j \in \mathbb {Z}_{>0}}\). In [5], the scalar curvature functional of such a spectral triple was defined to be the map

*H*is finite-dimensional but a dimension spectrum of

*D*is specified in advance (e.g. by modelling considerations), this residue always vanishes and Theorem 2 suggests it should perhaps be replaced by

*H*and

*F*is as in Corollary 3. Mutatis mutandis, the same applies to the volume and other spectral invariants.

### Remark 6

The calculation of the residues of localized zeta functions, Theorem 2, can be combined with the local index formula of Connes and Moscovici [6] for the Chern character of the Fredholm module associated to a spectral triple (*A*, *H*, *D*), in order to estimate some \({\text {KK}}\)-theoretic index pairings numerically. If the square of the operator *D* is spectrally elliptic, the index pairings can be expressed as a series in the spectrum of \(D^2\) and the coefficients \({{\mathrm{tr}}}\pi _{\lambda } a_0 [D,a_1]^{(k_1)} \cdots [D,a_n]^{(k_n)} \pi _{\lambda }\), where \(\pi _\lambda \) projects onto the eigenspace associated with the eigenvalue \(\lambda \), and the commutators are defined recursively as \([D,a]^{(k+1)} \overset{\underset{\mathrm {def}}{}}{=}[D^2,[D,a]^{(k)}]\) and \(a_0,\cdots ,a_n \in A\). A similar statement holds in the presence of a grading.

## Footnotes

- 1.
As restricted to the ideal on which this formula converges. For a treatment of the Dixmier trace on the larger ideal \(L^{(1,\infty )}\), see [3].

- 2.
If not, just restrict to the complement of the kernel.

- 3.
The classical elliptic

*pseudo*differential operators may have logarithmic terms in the heat expansion, leading to double poles of \(\varGamma (z) \zeta (z)\) (e.g. at negative integers), and are thus excluded in general. However, see Remark 2.

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