Trace Formula for p-Adics

We give a p-adic analogue of Selberg's trace formula relating the trace of a semigroup generated by a natural elliptic operator with a sum over contributions coming from closed geodesics. The construction uses probabilistic methods to define the generator.     

Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems

In this paper, we give a survey on the Hill-type formula and its applications.Moreover, we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions, which include the standard Neumann, Dirichlet and periodic boundary conditions. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Further, based on the Hill-type formula, we derive the Krein-type trace formula. As applications, we give nontrivial estimations for the eigenvalue problem and the relative Morse index.     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, [(Z)\tilde]\tilde{Z} , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

Trace Formula for Sturm--Liouville Operators with Singular Potentials

Mathematical Notes - Suppose that u(x) is a function of bounded variation on the closed interval [0,π], continuous at the endpoints of this interval. Then the Sturm—Liouville operator...     

On a Trace Formula for Functions of Noncommuting Operators

Mathematical Notes - The main result of the paper is that the Lifshits-Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we...     

Trace Formula for Integral Points on the Three-Dimensional Sphere

Doklady Mathematics - Average values over integral points on a three-dimensional sphere with an arbitrary smooth weight function are studied. For them, an expansion of the mean product of two...     

On the Trace Formula of Perturbation Theory. II

In a previous paper for a class of pairs of operators in a Hilbert space with nuclear difference or under a more general nuclearity condition there was introduced a spectral shift functional. Here we consider the local integrability of this spectral shift functional, i.e., the problem when this functional can locally be represented by an integrable function. The general results are then applied to pairs of definitizable and locally definitizable unitary operators in a Krein space.     

Generalized Trace Formula for Polynomials Orthogonal in Continuous-Discrete Sobolev Spaces

Functional Analysis and Its Applications - In continuous-discrete Sobolev spaces a generalized trace formula for orthogonal polynomials $$\{\widehat{q}_n\}_{n=0}^\infty$$ is obtained. The proof of...     

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.     

Determinant Formula for the Trace Class Perturbation of Heisenberg Commutation Relation

In this paper we establish a determinant formula for a pair of unbounded self-adjoint operators u and v satisfying a commutation relation where D is in the trace class, and some other condition. The functions f and g are rational functions satisfying certain condition.     

Spectral Theory for Riemannian Manifolds with Cusps and a Related Trace Formula

A new orthogonality relation for normed linear spaces is introduced using a concept of area of a parallelogram given by E. Silverman. Comparisons are drawn between this relation and an earlier relation used by G. Birkhoff. In addition, this new relation is utilized to obtain new characterizations of inner-product spaces.     

The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions

We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel and provide small-t asymptotics for the trace of the heat kernel. Submitted: May 20, 2008., Accepted: January 6, 2009.