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Register a new userInclusion of Diffraction Effects in the Gutzwiller Trace Formula
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The Gutzwiller trace formula is extended to include diffraction effects. The new trace formula involves periodic rays which have non-geometrical segments as a result of diffraction on the surfaces and edges of the scatter.
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We study the effect of edge diffraction on the semiclassical analysis of two dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is used to study the cardioid billiard, which has a single vertex. The formula works well for most of the short orbits we analyzed but fails for a few diffractive orbits due to a breakdown in the formalism for certain geometries. We extend the symbolic dynamics to account for diffractive orbits and use it to show that in the presence of parity symmetry the trace formula decomposes in an elegant manner such that for the cardioid billiard the diffractive orbits have no effect on the odd spectrum. Including diffractive orbits helps resolve peaks in the density of even states but does not appear to affect their positions. An analysis of the level statistics shows no significant difference between spectra with and without diffraction.
We stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about the genuine $L^2$-automorphic spectrum, into a linear combination of stable trace formulas of products of split odd orthogonal groups via endoscopic transfer. By adapting the strategies of Arthur and Moeglin-Waldspurger from the linear case, the proof is built on a long induction process that mixes up local and global, geometric and spectral data. As a by-product, we also stabilize the local trace formula for metaplectic groups over any local field of characteristic zero.
We found that if $u$ and $v$ are any two unitaries in a unital $C^*$-algebra with $|uv-vu|<2$ such that $uvu^*v^*$ commutes with $u$ and $v,$ then the SCA, $A_{u,v}$ generated by $u$ and $v$ is isomorphic to a quotient of the rotation algebra $A_theta$ provided that $A_{u,v}$ has a unique tracial state. We also found that the Exel trace formula holds in any unital $C^*$-algebra. Let $thetain (-1/2, 1/2)$ be a rational number. We prove the following: For any $ep>0,$ there exists $dt>0$ satisfying the following: if $u$ and $v$ are two unitary matrices such that $$ |uv-e^{2pi itheta}vu|<dtandeqn {1over{2pi i}}tau(log(uvu^*v^*))=theta, $$ then there exists a pair of unitary matrices $tilde{u}$ and $tilde{v}$ such that $$ tilde{u}tilde{v}=e^{2pi itheta} tilde{v}tilde{u},,, |u-tilde{u}|<epandeqn |v-tilde{v}|<ep. $$ Furthermore, a generalization of this for all real $theta$ is obtained for unitaries in unital infinite dimensional simple $C^*$-algebras of tracial rank zero.
Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C*-algebras of G and H. As an application, we exploit the role of group C*-algebras as recipients of higher indices of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.
A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrodinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.
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