Narrow Results

We realize the number of bound states of a Schrödinger operator on ${ℝ}^n$ as an index pairing in all dimensions. Expanding on ideas of Guillopé and others, we use high-energy corrections to find representatives of the $K$-theory class of the scattering operator. These representatives allow us to compute the number of bound states using an integral formula involving heat kernel coefficients.

We consider the selfadjoint operator H = H₀ + V, where H₀ is the free semi-classical Dirac operator on ℝ³. We suppose that the smooth matrix-valued potential V = O(〈x〉^-δ), δ > 0, has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator H by complex distortions of ℝ³. We establish an upper bound O(h^-3) for the number of resonances in any compact domain. For δ > 3, a representation of the derivative of the spectral shift function ξ(λ,h) related to the semi-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h^-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit–Wigner approximation formula for the derivative of the spectral shift function.

Let G/ℚ be an inner form of Sp₄/ℚ which does not split over ℝ. Consequently, it is not quasi-split. In this paper we determine completely the automorphic cohomology of G. That is, we describe the Eisenstein and the cuspidal cohomology of congruence subgroups Γ of G with respect to arbitrary coefficient systems. In particular we establish precise nonvanishing results for cuspidal cohomology. In addition, we calculate the residual spectrum of G.

We derive a trace formula for Green's functions of position-dependent (effective) mass Schrödinger equations that are defined on a real, finite interval and connected by a Darboux transformation of arbitrary order. Our findings generalize former results (J. Phys. A37, 10287 (2004)) on constant mass Schrödinger equations to the effective mass case.

Following the semiclassical formalism of Strutinsky et al.,¹¹ we have obtained the complete eigenvalue spectrum for a particle enclosed in an infinitely high spheroidal cavity. Our spheroidal trace formula also reproduces the results of a spherical billiard in the limit η → 1.0. Inclusion of repetition of each family of the orbits with reference to the largest one significantly improves the eigenvalues of the sphere and an exact comparison with the quantum mechanical results is observed up to the second decimal place for kR₀ ≥ 7. The contributions of the equatorial, planar (in the axis of symmetry plane) and non-planar (three-dimensional) orbits are obtained from the same trace formula by using the appropriate conditions. The resulting eigenvalues compare very well with the quantum mechanical eigenvalues at normal deformation. It is interesting that the partial sum of equatorial orbits leads to eigenvalues with maximum angular momentum projection, while the summing of planar orbits leads to eigenvalues with L_z = 0 except for L = 1. The remaining quantum mechanical eigenvalues are observed to arise from the three-dimensional (3D) orbits. Very few spurious eigenvalues arise in these partial sums. This result establishes the important role of 3D orbits even at normal deformations.

We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here, we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szegö kernel along the diagonal.

For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Apéry numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr_2k(Γ₀(N), n) of the Hecke operator T_n acting on the space of cusp forms S_2k(Γ₀(N)). We show that for (n, N) = 1, these traces interpolate p-adically in the weight aspect.

Let F be a number field or a p-adic field of odd residual characteristic. Let E be a quadratic extension of F, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (respectively, admissible) representations from the unitary group U(3, E/F) to the unitary group U(3, F' E/F'). As a consequence, we classify, up to certain restrictions, the packets of U(3, F' E/F') which contain irreducible automorphic (respectively, admissible) representations invariant under the action of the Galois group Gal(F' E/E). We also determine the invariance of individual representations. This work is the first study of base change into an algebraic group whose packets are not all singletons, and which does not satisfy the rigidity, or "strong multiplicity one", theorem. Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (respectively, admissible) member is Galois-invariant. The restriction that the residual characteristic of the local fields be odd may be removed once the multiplicity one theorem for U(3) is proved to hold unconditionally without restriction on the dyadic places.

Assuming the irreducibility of characteristic polynomials of Hecke operators $T_2$, we show that two normalized Hecke eigenforms of level one are distinguished by their second Fourier coefficients.

In this paper, we investigate the average

Ngô Bao Châu has been awarded a Fields Medal for his proof of the fundamental lemma. I shall try to describe the role of the fundamental lemma in the theory of automorphic forms. I hope that this will make it clear why the result will be a cornerstone of the subject. I will also try to give some sense of Ngô's proof. It is a profound and beautiful argument, built on insights mathematicians have contributed for over thirty years.

The trace formula is a major tool in the theory of automorphic forms. It was conceived by Selberg and extensively developed by Arthur. Among other things it is applicable to the study of spectral asymptotics as well as to (special cases of) Langlands functoriality conjectures. An important variant invented by Jacquet – the relative trace formula – is used to study period integrals and invariant functionals.

This note is a brief survey of our results on the stability criteria of S-periodic orbits in Hamiltonian system, by using the conditional Fredholm determinant and the trace formula. Most results appear in the papers.^9,10