Trace Formula Research Articles

The Selberg trace formula for spin Dirac operators on degenerating hyperbolic surfaces

We investigate the spectrum of the spin Dirac operator on families of hyperbolic surfaces where a set of disjoint simple geodesics shrink to 0 , under the hypothesis that the spin structure is non-trivial along each pinched geodesic. The main tool is a trace formula for the Dirac operator on finite area hyperbolic surfaces. We derive a version of Huber’s theorem and a non-standard small-time heat trace asymptotic expansion for hyperbolic surfaces with cusps. As a corollary we find a simultaneous Weyl law for the eigenvalues of the Dirac operator which is uniform in the degenerating parameter. The main result is the convergence of the Selberg zeta function associated to the Dirac operator on such families of hyperbolic surfaces. A central role is played by a \{\pm 1\} -valued class function \varepsilon determined by the spin structure.

Primes in Arithmetic Progressions to Large Moduli I: Fixed Residue Classes

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than x 1 / 2 x^{1/2} . Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size x 1 / 2 + δ x^{1/2+\delta } with a ‘convenient sized’ factor. As a consequence, the expected asymptotic holds for all but O ( δ Q ) O(\delta Q) moduli q ∼ Q = x 1 / 2 + δ q\sim Q=x^{1/2+\delta } and we get results for moduli as large as x 11 / 21 x^{11/21} . Our proof extends previous techniques of Bombieri, Fouvry, Friedlander and Iwaniec by incorporating new ideas inspired by amplification methods. We combine these with techniques of Zhang and Polymath tailored to our application. In particular, we ultimately rely on exponential sum bounds coming from the spectral theory of automorphic forms (the Kuznetsov trace formula) or from algebraic geometry (Weil and Deligne style estimates).

Trace Formulas for the Magnetic Laplacian and Dirichlet to Neumann Operator—Explicit Expansions

In this paper, we would like to measure how the magnetic effect appears in the heat trace formula associated with the magnetic Laplacian and the magnetic Dirichlet-to-Neumann operator. We propose to the reader an overview of magnetic heat trace formulas through explicit examples. On the way we obtain new formulas and in particular we calculate explicitly some nonlocal terms and logarithmic terms appearing in the Steklov heat trace asymptotics.

N-soliton solutions and asymptotic analysis for the massive Thirring model in the laboratory coordinates via the Riemann-Hilbert approach

Abstract In this paper, the N-soliton solutions for massive Thirring model (MTM) in the laboratory coordinates are analyzed via the Riemann-Hilbert approach. The direct scattering including the analyticity, symmetries and asymptotic behaviors of the Jost solutions as |λ| → ∞ and λ → 0 are given. Considering that the scattering coefficients have simple zeros, the matrix Riemann-Hilbert problem, reconstruction formulas and corresponding trace formulas are also derived. Further, the N-soliton solutions in the reflectionless case are obtained explicitly in the form of determinants. The propagation characteristics of one-soliton solutions and interaction properties of two-soliton solutions are discussed. In particular, the asymptotic expressions of two-soliton solutions as |t| → ∞ are obtained, which show that the velocities and amplitudes of the asymptotic solitons don’t change before and after interaction except the position shifts. In addition, three types of bounded states for two-soliton solutions are presented with certain parametric conditions.

THE REALITY OF SPECTRAL SHIFT FUNCTIONS FOR CONTRACTIONS AND DISSIPATIVE OPERATORS

Recently the authors of this note solved a famous problem that remained open during many years and proved that for arbitrary contractions on Hilbert space with trace class difference there exists an integrable spectral shift function, for which an analogue of the Lifshits—Krein trace formula holds. Similar results were also obtained for pairs of dissipative operators. It turns out that in contrast with the case of self-adjoint or unitary operators, it can happen that there is no real-valued integrable spectral shift function. In this note we state results that give sufficient conditions for the existence of a real-valued integrable spectral shift function for pairs of contractions and pairs of dissipative operators.

Integration of the Korteweg–de Vries type equations with a loaded term in the class of periodic functions

In this paper, we study the integrability of a Korteweg–de Vries type equation with a loaded term in the class of periodic functions using direct and inverse spectral problems posed for the Sturm–Liouville operator on the entire axis. We present some information about the Sturm–Liouville operator with a periodic coefficient and its application to solving a loaded Korteweg–de Vries type equation. We show that the function constructed using the obtained system of Dubrovin differential equations and trace formulas is a solution to the problem posed. We present important consequences about the period of the solution with respect to $x$ and about the analyticity of the solution with respect to $x$ for a Korteweg–de Vries type equation with a loaded term.

Mixed single, double, and triple poles solutions for the space-time shifted nonlocal DNLS equation with nonzero boundary conditions via Riemann–Hilbert approach

In this paper, we investigate the space-time shifted nonlocal derivative nonlinear Schrödinger (DNLS) equation under nonzero boundary conditions using the Riemann–Hilbert (RH) approach for the first time. To begin with, in the direct scattering problem, we analyze the analyticity, symmetries, and asymptotic behaviors of the Jost eigenfunctions and scattering matrix functions. Subsequently, we examine the coexistence of N-single, N-double, and N-triple poles in the inverse scattering problem. The corresponding residue conditions, trace formulae, θ condition, and symmetry relations of the norming constants are obtained. Moreover, we derive the exact expression for the mixed single, double, and triple poles solutions with the reflectionless potentials by solving the relevant RH problem associated with the space-time shifted nonlocal DNLS equation. Furthermore, to further explore the remarkable characteristics of soliton solutions, we graphically illustrate the dynamic behaviors of several representative solutions, such as three-soliton, two-breather, and soliton-breather solutions. Finally, we analyze the effects of shift parameters through graphical simulations.

On the Laplace equation on bounded subanalytic manifolds

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Bethe vectors and recurrence relations for twisted Yangian based models

We study Olshanski twisted Yangian based models, known as one-dimensional “soliton non-preserving” open spin chains, by means of algebraic Bethe Ansatz. The even case, when the bulk symmetry is \mathfrak{gl}_{2n}𝔤𝔩2n and the boundary symmetry is \mathfrak{sp}_{2n}𝔰𝔭2n or \mathfrak{so}_{2n}𝔰𝔬2n, was studied in [Ann. Henri Poincaré 20, 339 (2018)]. In the present work, we focus on the odd case, when the bulk symmetry is \mathfrak{gl}_{2n+1}𝔤𝔩2n+1 and the boundary symmetry is \mathfrak{so}_{2n+1}𝔰𝔬2n+1. We explicitly construct Bethe vectors and present a more symmetric form of the trace formula. We use the composite model approach and Y(\mathfrak{gl}_n)Y(𝔤𝔩n)-type recurrence relations to obtain recurrence relations for twisted Yangian based Bethe vectors, for both even and odd cases.

Trace regularization problem for a fourth‐order differential operator on separable Banach space

In this study, we employ new results on the semi‐inner product, the adjoint operator, and the Schatten class of operators on a separable Banach space, which allow us to investigate a differential operator with unbounded operator‐valued coefficients. Specifically, we derive an asymptotic formula for the second regularized trace based on the asymptotic and spectral properties of the extended operator. For this purpose, we use the theory of continuous dense embeddings and known results regarding the regularized trace in Hilbert space. We conclude our paper by providing examples that support our findings.

The Petersson/Kuznetsov trace formula with prescribed local ramifications

The Petersson/Kuznetsov trace formula with prescribed local ramifications

Trace formulas revisited and a new representation of KdV solutions with short-range initial data

Trace formulas revisited and a new representation of KdV solutions with short-range initial data

Bessel functions on GL(n), I

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on GL(n) as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on GL(n) and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on GL(n) with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of GL(4) Bessel functions.

Cyclic base change of cuspidal automorphic representations over function fields

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

Exact quantization conditions and full transseries structures for PT symmetric anharmonic oscillators

We study exact Wentzel–Kramers–Brillouin analysis (EWKB) for a PT symmetric quantum mechanics (QM) defined by the potential VPT(x)=ω2x2+gx2K(ix)ϵ with ω∈R≥0, g∈R>0 and K,ϵ∈N to clarify its perturbative/nonperturbative structure. In our analysis, we mainly consider the massless cases, i.e., ω=0, and derive the exact quantization conditions (QCs) for arbitrary (K,ϵ) including all perturbative/nonperturbative corrections. From the exact QCs, we clarify full transseries structure of the energy spectra with respect to the inverse energy level expansion, and then formulate the Gutzwiller trace formula, the spectral summation form, and the Euclidean path integral. For the massive cases, i.e., ω>0, we show the fact that, by requiring the existence of the solution of the exact QCs, the path of analytic continuation in EWKB is uniquely determined for a given N=2K+ϵ, and in consequence the exact QCs, the energy spectra, and the three formulas are all perturbative. Similarities to Hermitian QMs and resurgence are also discussed as additional remarks. Published by the American Physical Society 2024

Continuity and minimization of spectrum related with the two-component Novikov equation

This article is concerned with the spectral problem{dϕ1(x)=ϕ2(x)dh(x),dϕ2(x)=ϕ3(x)dg(x),dϕ3(x)=−λϕ1(x)dx, x∈(−1,1),ϕ2(−1)=0, ϕ3(−1)=0, ϕ3(1)=0,which is associated with the two-component Novikov equation. The coefficients g and h are functions of bounded variation. The first aim is to prove the continuous dependence of the n-th eigenvalue on the coefficients g and h with different topologies; in addition, we discuss the Fréchet differentiability of the algebraically simple eigenvalues on the coefficients g, h with respect to the norm topology of total variations. Secondly, utilizing the theory of oscillatory (non-symmetric) kernels, conditions on g, h under which the spectrum is nonnegative and algebraically simple are obtained. Finally, we solve the extremal problems of the lowest nonzero eigenvalue, when the norm ‖⋅‖V of the coefficients g, h is given. The latter can be seen as an application of our first two main results and the trace formulas.

Trace formula for differential operators with frozen argument

We derive the formula for the regularized trace for rank-one perturbations of self-adjoint operators and then justify its applicability to a wide class of differential operators with frozen argument.

On the cubic Shimura lift to $\operatorname{PGL}(3)$: The fundamental lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of \mathrm{SL}_{2} to automorphic representations on \mathrm{PGL}_{2} . Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of \mathrm{SL}_{3} to \mathrm{PGL}_{3} , and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation \Theta_{\mathrm{SO}_8} for split \mathrm{SO}_{8}(\mathbb{A}) , consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on \mathrm{PGL}_{3}(\mathbb{A}) involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of \mathrm{SL}_{3}(\mathbb{A}) . We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.

Trace formula and Levinson's theorem as an index pairing in the presence of resonances

Trace formula and Levinson's theorem as an index pairing in the presence of resonances

Estimates and higher-order spectral shift measures in several variables

In recent years, higher-order trace formulas of operator functions have attracted considerable attention to a large part of the perturbation theory community. In this direction, we prove estimates for traces of higher-order derivatives of multivariate operator functions with associated scalar functions arising from multivariate analytic function space and, as a consequence, derive higher-order spectral shift measures for pairs of tuples of commuting contractions under Hilbert-Schmidt perturbations. These results substantially extend the main results of [26], where the estimates were proved for traces of first and second-order derivatives of multivariate operator functions. In the context of the existence of higher-order spectral shift measures, our results extend the relative results of [6,20] from a single-variable to a multivariate case under Hilbert-Schmidt perturbations. Our results rely crucially on heavy uses of explicit expressions of higher-order derivatives of operator functions and estimates of the divided difference of multivariate analytic functions, which are developed in this paper, along with the spectral theorem of tuple of commuting normal operators. In conclusion, we explore the significance of our results and provide relevant examples.