Spectral Sums Research Articles

The prime geodesic theorem for PSL2(ℤ[i]) and spectral exponential sums

This work addresses the Prime Geodesic Theorem for the Picard manifold $\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}$, which asks for the asymptotic evaluation of a counting function for the closed geodesics on $\mathcal{M}$. Let $E_{\Gamma}(X)$ be the error term in the Prime Geodesic Theorem. We establish that $E_{\Gamma}(X) = O_{\varepsilon}(X^{3/2+\varepsilon})$ on average as well as many pointwise bounds. The second moment bound parallels an analogous result for $\Gamma = \mathrm{PSL}_{2}(\mathbb{Z})$ due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group $\Gamma$ is given. Our numerical experiments visualise in particular that $E_{\Gamma}(X)$ obeys a conjectural bound of size $O_{\epsilon}(X^{1+\varepsilon})$.

Scaling asymptotics of spectral Wigner functions* *SZ was partially supported by NSF Grant DMS-1810747. BH was funded by NSF CAREER Grant DMS-2143754 as well as NSF Grants DMS-1855684, DMS-2133806 and an ONR MURI on Foundations of Deep Learning.

We prove that smooth Wigner–Weyl spectral sums at an energy level E exhibit Airy scaling asymptotics across the classical energy surface Σ E . This was proved earlier by the authors for the isotropic harmonic oscillator and the proof is extended in this article to all quantum Hamiltonians −ℏ 2Δ + V where V is a confining potential with at most quadratic growth at infinity. The main tools are the Herman–Kluk initial value parametrix for the propagator and the Chester–Friedman–Ursell normal form for complex phases with a one-dimensional cubic degeneracy. This gives a rigorous account of Airy scaling asymptotics of spectral Wigner distributions of Berry, Ozorio de Almeida and other physicists.

Almost everywhere convergence of spectral sums for self-adjoint operators

Let L L be a non-negative self-adjoint operator acting on the space L 2 ( X ) L^2(X) , where X X is a positive measure space. Let L = ∫ 0 ∞ λ d E L ( λ ) { L}=\int _0^{\infty } \lambda dE_{ L}({\lambda }) be the spectral resolution of L L and S R ( L ) f = ∫ 0 R d E L ( λ ) f S_R({ L})f=\int _0^R dE_{ L}(\lambda ) f denote the spherical partial sums in terms of the resolution of L { L} . In this article we give a sufficient condition on L L such that lim R → ∞ S R ( L ) f ( x ) = f ( x ) , a.e. \begin{equation*} \lim _{R\rightarrow \infty } S_R({ L})f(x) =f(x),\ \ \text {a.e.} \end{equation*} for any f f such that log ⁡ ( 2 + L ) f ∈ L 2 ( X ) \operatorname {log}(2+L) f\in L^2(X) . These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.

Systematic strong coupling expansion for out-of-equilibrium dynamics in the Lieb-Liniger model

We consider the time evolution of local observables after an interaction quench in the repulsive Lieb-Liniger model. The system is initialized in the ground state for vanishing interaction and then time-evolved with the Lieb-Liniger Hamiltonian for large, finite interacting strength c. We employ the Quench Action approach to express the full time evolution of local observables in terms of sums over energy eigenstates and then derive the leading terms of a 1/c expansion for several one and two-point functions as a function of time t>0 after the quantum quench. We observe delicate cancellations of contributions to the spectral sums that depend on the details of the choice of representative state in the Quench Action approach and our final results are independent of this choice. Our results provide a highly non-trivial confirmation of the typicality assumptions underlying the Quench Action approach.

A physicist’s guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as [Formula: see text]-dimensional balls (with [Formula: see text]), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier–Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review gathers many summation formulas for spectral sums that are dispersed in the literature.

Generalizing Lieb's Concavity Theorem via operator interpolation

We introduce the notion of k-trace and use interpolation of operators to prove the joint concavity of the function (A,B)↦Trk[(Bq2K⁎ApKBq2)s]1k, which generalizes Lieb's concavity theorem and its following results from trace to a class of homogeneous functions Trk[⋅]1k. Here Trk[A] denotes the kth elementary symmetric polynomial of the eigenvalues of A. This result gives an alternative proof for the concavity of A↦Trk[exp⁡(H+log⁡A)]1k that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices.

Heat kernel coefficients on the sphere in any dimension

We derive all heat kernel coefficients for Laplacians acting on scalars, vectors, and tensors on fully symmetric spaces, in any dimension. Final expressions are easy to evaluate and implement, and confirmed independently using spectral sums and the Euler–Maclaurin formula. We also obtain the Green’s function for Laplacians acting on transverse traceless tensors in any dimension, and new integral representations for heat kernels using known eigenvalue spectra of Laplacians. Applications to quantum gravity and the functional renormalisation group, and other, are indicated.

Stability and uniqueness properties of Taylor approximations of matrix functions

We consider Taylor approximations of matrix functions and establish bounds controlling the size of a perturbation in terms of the size of the Taylor remainder or, equivalently, the associated spectral shift function(s). This can be viewed as a stability property for the inverse problem of recovering the perturbation from the spectral shift function(s). From our bounds we derive a uniqueness result for spectral sums: if the trace of the Taylor remainder of order n≥2 equals zero for a finite number of monomials, then the perturbation is zero. This uniqueness result is in contrast with nonuniqueness of the first order Taylor remainder. As an application we characterize equality of two graphs in terms of self-returning walks.

A spectral reciprocity formula and non-vanishing for L-functions on GL(4)×GL(2)

We introduce a new type of summation formula for central values of GL(4)×GL(2)L-functions, when varied over Maaß forms. By rewriting such a sum in terms of GL(4)×GL(1)L-functions and applying a new “balanced” Voronoi formula, the sum can be shown to be equal to a differently-weighted average of the same quantities. By controlling the support of the spectral weighting functions on both sides, this reciprocity formula gives estimates on spectral sums that were previously obtainable only for lower rank groups. The “balanced” Voronoi formula has Kloosterman sums on both sides, and can be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL(4) Hecke eigenvalues. As an application we show that for any self-dual cusp form Π for SL(4,Z), there exist infinitely many Maaß forms π for SL(2,Z) such that L(1/2,Π×π)≠0.

Averaging over Heegner Points in the Hyperbolic Circle Problem

For $\Gamma={\hbox{PSL}_2( {\mathbb Z})}$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $\Gamma z$ inside the hyperbolic disc centered at $z$ with radius $\cosh^{-1}(X/2)$. We show that, by averaging over Heegner points $z$ of discriminant $D$, Selberg's error term estimate can be improved, if $D$ is large enough. The proof uses bounds on spectral exponential sums, and results towards the sup-norm conjecture of eigenfunctions, and the Lindel\of conjecture for twists of the $L$-functions attached to Maa{\ss} cusp forms.

Approximating Spectral Sums of Large-Scale Matrices using Stochastic Chebyshev Approximations

Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis, and computational biology (e.g., protein folding), just to name a few application areas. We propose a linear-time randomized algorithm for approximating the trace of matrix functions of large symmetric matrices. Our algorithm is based on coupling function approximation using Chebyshev interpolation with stochastic trace estimators (Hutchinson's method), and as such requires only implicit access to the matrix, in the form of a function that maps a vector to the product of the matrix and the vector. We provide rigorous approximation error in terms of the extremal eigenvalue of the input matrix, and the Bernstein ellipse that corresponds to the function at hand. Based on our general scheme, we provide algorithms with provable guarantees for important matrix computations, including log-determinant, trace of matrix inverse, Estrada index, Schatten $p$-norm, and testing positive definiteness. We experimentally evaluate our algorithm and demonstrate its effectiveness on matrices with tens of millions dimensions.

Local average in hyperbolic lattice point counting, with an Appendix by Niko Laaksonen

The hyperbolic lattice point problem asks to estimate the size of the orbit $$\Gamma z$$ inside a hyperbolic disk of radius $$\cosh ^{-1}(X/2)$$ for $$\Gamma $$ a discrete subgroup of $${\hbox {PSL}_2( {{\mathbb {R}}})} $$ . Selberg proved the estimate $$O(X^{2/3})$$ for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for $${\hbox {PSL}_2( {{\mathbb {Z}}})} $$ . The result is that the error term can be improved to $$O(X^{7/12+{\varepsilon }})$$ . The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maaß cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound $$O(X^{1/2+{\varepsilon }})$$ . In the appendix the relevant exponential sum over the spectral parameters is investigated.

Spectral sum rules for confining large-N theories

We consider asymptotically-free four-dimensional large-$N$ gauge theories with massive fermionic and bosonic adjoint matter fields, compactified on squashed three-spheres, and examine their regularized large-$N$ confined-phase spectral sums. The analysis is done in the limit of vanishing 't Hooft coupling, which is justified by taking the size of the compactification manifold to be small compared to the inverse strong scale $\Lambda^{-1}$. Our results motivate us to conjecture some universal spectral sum rules for these large $N$ gauge theories.

Form factor relocalisation and interpolating renormalisation group flows from the staircase model

We investigate the staircase model, introduced by Aliosha Zamolodchikov through an analytic continuation of the sinh-Gordon S-matrix to describe interpolating flows between minimal models of conformal field theory in two dimensions. Applying the form factor expansion and the c-theorem, we show that the resulting c-function has the same physical content as that found by Zamolodchikov from the thermodynamic Bethe Ansatz. This turns out to be a consequence of a nontrivial underlying mechanism, which leads to an interesting localisation pattern for the spectral integrals giving the multi-particle contributions. We demonstrate several aspects of this form factor relocalisation, which suggests a novel approach to the construction of form factors and spectral sums in integrable renormalisation group flows with non-diagonal scattering.

Dynamical mass generation of vector mesons from QCD trace anomaly

Mass formulas for the light-vector mesons written in terms of the gluon condensate i.e., the trace anomaly in quantum chromodynamics (QCD), are derived on the basis of finite energy QCD sum rules. We utilize sum rules with $s^n$ and $s^{n+1/2}$ weights, which relate the energy-weighted spectral sums to the vacuum expectation values of certain commutation relations. After evaluating the commutation relations, the sum rules with $s^n$ weights are reduced to the familiar ones obtained from the operator product expansion (OPE). On the other hand, the sum rules with $s^{n+1/2}$ weights cannot be derived from OPE. They give new relations between the spectral sums and QCD vacuum fluctuations. To derive simple mass formula, we adopt the pole + continuum Ansatz for the spectral function, and solve coupled equations given by the sum rules with $s^{0,1}$ weights and the new sum rule with $s^{1/2}$ weight. Application of our approach to the axial-vector meson is also discussed.

Form factor perturbation theory from finite volume

Using a regularization by putting the system in finite volume, we develop a novel approach to form factor perturbation theory for non-integrable models described as perturbations of integrable ones. This permits to go beyond first order in form factor perturbation theory and in principle works to any order. The procedure is carried out in detail for double sine-Gordon theory, where the vacuum energy density and breather mass correction is evaluated at second order. The results agree with those obtained from the truncated conformal space approach. The regularization procedure can also be used to compute other spectral sums involving disconnected pieces of form factors such as those that occur e.g. in finite temperature correlators.

Relation between chiral symmetry breaking and confinement in YM-theories

Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov loop are thoroughly investigated. The approach by Gattringer is generalized to mode sums which reconstruct the Polyakov loop locally. This opens the possibility to study the mode sum approximation to the Polyakov loop correlator. The approach is rederived for the ab initio continuum formulation of Yang-Mills theories, and the convergence of the mode sum is studied in detail. The mode sums are then explicitly calculated for the Schwinger model and SU(2) gauge theory in a homogeneous background field. Using SU(2) lattice gauge theory, the IR dominated mode sums are considered and the mode sum approximation to the static quark antiquark potential is obtained numerically. We find a good agreement between the mode sum approximation and the static potential at large distances for the confinement and the high temperature plasma phase.

Finite element synthesis of structural or acoustic receptances in view of practical design applications

The present study addresses the modal synthesis of dynamic responses or receptances, which can be considered as a major issue in vibratory or acoustic design. A new method termed “method of orthocomplement” offers to increase the accuracy of modal superposition. Two apparently independent domains are considered: first, the numerical control of round-off errors in finite-dimensional models; second, the regularization of modal boundary discontinuities in infinite-dimensional formulations. Concerning the finite-dimensional systems, using the proposed method in conjunction with pseudo-inversion theorems, proves that “inertia relief corrections” remedy modal truncation problems in singular floating systems, just as static corrections do for simply supported systems. The proof is based on hitherto unpublished expressions of inertial contributions in terms of orthogonal projectors. It is worth noting that such expressions present by themselves some technical interest. Modal formulae being delivered in closed form, a thorough analysis of errors becomes possible. Special “spectrograms” are introduced to appraise the convergence of ordinary or modified modal sums. When applied to infinite-dimensional modal boundary discontinuities, the method of orthocomplement eliminates the Gibbs oscillations that affect the ordinary spectral sums, and produces the same corrected formulae as it did for finite-dimensional examples. Effective solutions for the computation of vibratory or acoustic boundary receptances, are developed in that way. The paper once again illustrates how orthogonal projectors can contribute to the determination of static corrections. Future applications to coupling problems, model reduction and modal synthesis are briefly presented at the end of the text.

Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov loop

We investigate and compute spectral sums of the Wilson lattice Dirac operator for quenched $SU(3)$ gauge theory. It is demonstrated that there exist sums which serve as order parameters for the confinement-deconfinement phase transition and get their main contribution from the IR end of the spectrum. They are approximately proportional to the Polyakov loop. In contrast to earlier studied spectral sums, some of them are expected to have a well-defined continuum limit.

Complete spectra of the Dirac operator and their relation to confinement

Abstract We compute complete spectra of the staggered lattice Dirac operator for quenched SU ( 3 ) gauge configurations below and above the critical temperature. The confined and the deconfined phases are characterized by a different response of the Dirac eigenvalues to a change of the fermionic boundary conditions. We analyze the role of the eigenvalues in recently developed spectral sums representing the Polyakov loop. We show that the Polyakov loop gets its main contributions from the UV end of the spectrum.