Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.     

Stable local trace formula

Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg’s trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.     

A relative Kuznietsov trace formula on G2

In this paper, we set up the general formulation to study distinguished residual representations of a reductive group G by the relative trace formula approach. This approach simplifies the argument of [JR], which deals with this type of relative trace formula for a special symmetric pair (GL(2n), Sp(2n)) and also works for non-symmetric, spherical pairs. To illustrate our idea and method, we complete our relative trace formula (both the geometric side identity and the spectral side identity) for the case (G ₂, SL(3)). Received: 6 February 1999     

Kreĭn–Saakyan and Trace Formulas for a Pair of Canonical Differential Operators

An analog of the Kreĭn–Saakyan formula is derived for any pair of relatively prime self-adjoint extensions of a minimal symmetric canonical differential operator. This allows us to deduce a trace formula in the matrix case. I am grateful to Sh. Saakyan for his interest in this work and lively discussion. Received: December 8, 2006. Accepted: December 30, 2006.     

The Spectrum and Trace Formula for Bounded Perturbations of Differential Operators

Spectrum properties and a method for deriving a regularized trace formula for perturbations of operators with discrete spectra in a separable Hilbert space are studied. A trace formula for a local perturbation of a two-dimensional harmonic oscillator in a strip is obtained based on this method.     

The Classical Formula for the Regularized Trace of a Multidimensional Harmonic Oscillator

In this article we investigate the spectrum of a finite perturbation of a two-dimensional harmonic oscillator and obtain the classical formula of the first regularized trace. Bibliography: 12 titles.     

Singularities of the wave trace for the Friedlander model

There has recently been renewed interest in the trace formula–in particular, that of the initial case of GL(2)–due to counting applications in the function field case. For these applications, one needs a very precise form of the trace formula, with all terms computed explicitly. Our aim in this work is to compute the trace formula for GL(2) over a number field in as full detail as was done for the function field case and to give an accessible exposition, being motivated by these applications to counting, but also by pure curiosity as to the optimal form of this plastic formula. We also explain a correction argument in our context here of GL(2). The idea is to introduce a global summand which does not change the formula globally but changes the local weighted orbital integrals at the hyperbolic terms, so that their limit at the identity becomes a unipotent contribution to the trace formula. This gives a harmonious and pleasing form to the formula. Finally, we put the trace formula in an invariant form; thus all its terms are distributions whose value at a test function f ^y (x) = f(y ^?1 xy) is independent of y in GL(2,A).     

On the Carey–Helton–Howe–Pincus trace formula

In this article, we give a new proof of the Carey–Helton–Howe–Pincus trace formula using Kato's theory of “relatively-smooth” operators and Krein's trace formula.     

Track formulas and derivations in simple jordan pairs

In finite dimensional Lie algebras, Jordan algebras, and other algebraic structures the study of derivations has been facilitated by having a nontrivial trace formula on hand (see for example [?]) . Tuere is no common pattern in proving these trace formulas, they all depend on the underlying structures. In this note we derive such a trace formula for finite dimensional central simple Jordan pairs. We use it to determine all derivations the Killing form and the dimensions of the derivation algebras of the Jordan pairs. Dur primary tool is a Trace Reduction Formula.     

N维向量Sturm-Liouville算子的正则迹及其在反谱问题中的应用

研究了赋予一般分离型边界条件的N维向量Sturm-Liouville方程的特征值问题,获得了该系统的一个正则迹公式.迹公式不仅形式美观,而且它在反谱理论中具有重要的作用.     

Trace formula for almost Lie algebra of operators

In this present paper, the almost Lie algebra of operators is introduced. By a natural homomorphism, this almost Lie algebra of operators is mapped to a Lie algebra. By choosing a basis in this Lie algebra, a bilinear functional on the enveloping algebra of this Lie algebra is defined through the trace of some operators which are related to the communtators. A general trace formula is obtained by means of the partial derivatives and symmetric operation in the enveloping algebra. More concrete formula is also obtained in terms of a linear functional on the commutators in some special cases.     

Koplienko Trace Formula

Koplienko (Sib Math J 25(5): 735–743, 1984) gave a trace formula for perturbations of self-adjoint operators by operators of Hilbert–Schmidt class ${\mathcal{B}_2(\mathcal{H})}$ . Recently Gesztesy et?al. (Basics Z Mat Fiz Anal Geom 4(1):63–107, 2008) gave an alternative proof of the trace formula when the operators involved are bounded. In this article, we give a still another proof and extend the formula for unbounded case by reducing the problem to a finite dimensional one as in the proof of Krein trace formula by Voiculescu (On a Trace Formula of M. G. Krein. Operator Theory: Advances and Applications, vol. 24, pp. 329–332. Birkhauser, Basel, 1987), Sinha and Mohapatra (Proc Indian Acad Sci (Math Sci) 104(4):819–853, 1994).     

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a h-pseudo-differential operator on whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of “real principal type”. The new contribution to the trace formula is valid for all time in a compact subset of but the result is modest since we have restrictions on the dimension.     

Applications of Connes'' Geodesic Flow to Trace Formulas in Noncommutative Geometry

The “trace formula” of Chazarain, Duistermaat, and Guillemin expresses that the singularities of the distribution trace of the wave group on a compact Riemannian manifoldXis included in the set of periods of the geodesic flow restricted toS*X. Most of the objects involved in this trace formula have analogues in Connes' Noncommutative Geometry. This paper shows, on several significant examples of Noncommutative Geometry, that Connes' definition of geodesic flow leads to statements analogous to the classical trace formula of Chazarain, Duistermaat, and Guillemin.     

On dimension formulas for Jacobi forms

In this paper, we discuss the dimension formula for Jacobi forms via the Selberg trace formula, an explicit dimension formula for the space of Jacobi cusp forms of degree 2 is given as an application.     

On the Euler characteristic of the discrete spectrum

This paper, which is largely expository in nature, seeks to illustrate some of the advances that have been made on the trace formula in the past 15 years. We review the basic theory of the trace formula, then introduce some ideas of Arthur and Kottwitz that allow one to calculate the Euler characteristic of the S-cohomology of the discrete spectrum. This Euler characteristic is first expressed as a trace of a certain test function on the space of automorphic forms, and then, by the stable trace formula, is converted into a sum of orbital integrals. A result on global measures allows us to calculate these integrals in terms of the values of certain Artin L-functions at negative integers.Our intention is to show how advances in the theory have allowed one to render such calculations completely explicit. As a byproduct of this calculation, we obtain the existence of automorphic representations with certain local behavior at the places in S.     

The trace formula in Banach spaces

A classical result of Grothendieck and Lidskii says that the trace formula (that the trace of a nuclear operator is the sum of its eigenvalues provided the sequence of eigenvalues is absolutely summable) holds in Hilbert spaces. In 1988, Pisier proved that weak Hilbert spaces satisfy the trace formula. We exhibit a much larger class of Banach spaces, called Γ-spaces, that satisfy the trace formula. A natural class of asymptotically Hilbertian spaces, including some spaces that are ?₂ sums of finite-dimensional spaces, are Γ-spaces. One consequence is that the direct sum of two Γ-spaces need not be a Γ-space.     

Gutzwiller’s semiclassical trace formula and Maslov-type index theory for symplectic paths

Gutzwiller’s famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with a view toward the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley–Zehnder index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley–Zehnder’s index theory in which we will recall all essential ingredients is the Maslov-type index theory for symplectic paths developed by Long and his collaborators. It would shed new light on the computations and understandings of the semiclassical trace formula. The insights in Gutzwiller’s work also seems plausible for the studies of Hamiltonian systems.     

Prolate spheroidal wave functions,Sonine spaces,and the Riemann zeta function

In this paper we study underlying spaces associated with A. Connes? trace formula (see Connes (1999) [3], Li (2010) [14]). In particular the explicit formula in the theory of prime numbers is expressed as the trace of an operator acting on a Hilbert space, which is a direct sum of a Sonine space, the space of prolate spheroidal wave functions, and a variant of the space of prolate spheroidal wave functions. A formula is obtained for the orthogonal projection of the Hilbert space onto the Sonine space. A base is given for the variant space of the space of prolate spheroidal wave functions.     

A Hilbert-Schmidt integral operator and the Weil distribution

In this paper, a positive operator is given. It is shown that the product of this positive operator and the convolution operator is a trace class Hilbert-Schmidt integral operator and has nonnegative eigenvalues. A formula is given for the trace of this product operator. It seems that this product operator is the closest trace class integral operator which has nonnegative eigenvalues and is related to the Weil distribution in the context of Connes’ program for the Riemann hypothesis. A relation is given between the trace of the product operator and the Weil distribution.