Neumann Trace Research Articles

To quantum averages through asymptotic expansion of classical averages on infinite-dimensional space

We study asymptotic expansions of Gaussian integrals of analytic functionals on infinite-dimensional spaces (Hilbert and nuclear Frechet). We obtain an asymptotic equality coupling the Gaussian integral and the trace of the composition of scaling of the covariation operator of a Gaussian measure and the second (Frechet) derivative of a functional. In this way we couple classical average (given by an infinite-dimensional Gaussian integral) and quantum average (given by the von Neumann trace formula). We can interpret this mathematical construction as a procedure of “dequantization” of quantum mechanics. We represent quantum mechanics as an asymptotic projection of classical statistical mechanics with infinite-dimensional phase space. This space can be represented as the space of classical fields, so quantum mechanics is represented as a projection of “prequantum classical statistical field theory.”

Miniworkshop: L2-Spectral Invariants and the Integrated Density of States

Both the study of L^2 -spectral invariants in geometry and the investigation of the integrated density of states in mathematical physics have attracted much attention in recent years. While the two topics are strongly related, the corresponding communities are rather unaware of each others work and methods. The main aim of this mini-workshop was to bring together people from both fields and provide a basis for interaction. Accordingly, the first two days of the conference were spent with survey talks solicited by the organizers to highlight concepts and methods. There were 9 such talks with durations between 60 and 90 minutes. The second half of the conference was devoted to more detailed investigations. Most participants used the opportunity to present their current research in the area of the meeting. There were 13 such talks. The results presented in those talks contained significant contributions e.g. to the Atiyah conjecture about integrality of L^2 -Betti numbers for a completely new class of groups by Peter Linnell, a mathematically rigorous derivation using von Neumann traces of the asymptotics of the specific heat near absolute zero by Mikhael Shubin, and approximation results for the integrated density of states in various new contexts. Altogether the conference was attended by 17 participants.

A pre-quantum classical statistical model with infinite-dimensional phase space

We show that quantum mechanics can be represented as an asymptotic projection of statistical mechanics of classical fields. Thus our approach does not contradict to a rather common opinion that quantum mechanics could not be reduced to statistical mechanics of classical particles. Notions of a system and causality can be reestablished on the prequantum level, but the price is sufficiently high -- the infinite dimension of the phase space. In our approach quantum observables, symmetric operators in the Hilbert space, are obtained as derivatives of the second order of functionals of classical fields. Statistical states are given by Gaussian ensembles of classical fields with zero mean value (so these are vacuum fluctuations) and dispersion $\alpha$ which plays the role of a small parameter of the model (so these are small vacuum fluctuations). Our approach might be called {\it Prequantum Classical Statistical Field Theory} - PCSFT. Our model is well established on the mathematical level. However, to obtain concrete experimental predictions -- deviations of real experimental averages from averages given by the von Neumann trace formula - we should find the energy scale $\alpha$ of prequantum classical fields.

Trace formulas for non-self-adjoint periodic Schrödinger operators and some applications

Recently, a trace formula for non-self-adjoint periodic Schrödinger operators in L 2 ( R ) associated with Dirichlet eigenvalues was proved in [Differential Integral Equations 14 (2001) 671–700]. Here we prove a corresponding trace formula associated with Neumann eigenvalues. In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas, we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential K e 2 i x , where K ∈ C .

Approximating spectral invariants of Harper operators on graphs II

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation speed.

Optimality of the trace of a product of matrices

A simple and short proof of the optimality conditions in the John von Neumann trace inequality for singular values is shown. Possible generalizations and special cases are also considered.

A sharp trace result on a thermo-elastic plate equation with coupled hinged/Neumann boundary conditions

We consider a thermo-elastic plate equation with rotational forces [Lagnese.1]and with coupled hinged mechanical/Neumann thermal boundary conditions (B.C.).We give a sharp result on the Neumann trace of the mechanical velocity, which is '$\frac{1}{2}$' sharper in the space variable than the result than one would obtain by a formal application of trace theory on the optimal interior regularity. Two proofs byenergy methods are given: one which reduces the analysis to sharp wave equation'sregularity theory; and one which analyzes directly the corresponding Kirchoff elasticequation. Important implications of this result are noted.

A supplement to the von Neumann trace inequality for singular values

The von Neumann trace inequality for singular values is reexamined by determining the possible values for the diagonal elements contributing to the trace. The possible values for the eigenvalues contributing to the trace are also found.

A trace inequality with a subtracted term

For fixed real or complex matrices A and B, the well-known von Neumann trace inequality identifies the maximum of |tr( UAVB)z.sfnc;, as U and V range over the unitary group, the maximum being a bilinear expression in the singular values of A and B. This paper establishes the analogue of this inequality for real matrices A and B when U and V range over the proper (real) orthogonal group. The maximum is again a bilinear expression in the singular values, but there is a subtracted term when A and B have determinants of opposite sign.

Spectral flow, eta invariants, and von Neumann algebras

In this paper, we prove that the spectral flow of a path of tangential signature operators on a closed odd dimensional manifold parametrized by unitary representations of the fundamental group is a homotopy invariant of the manifold and the path in the space of unitary representations, for a large class of fundamental groups with the help of results due to Weinberger ( Proc. Nat. Acad. Sci. U.S.A. 85 (1988), 5362–5363). We also prove that an eta invariant for normal covering spaces introduced by Cheeger and Gromov ( J. Differential Geom. 21 (1985), 1–34) and defined using the von Neumann trace, is a homotopy invariant for Bieberbach groups. Finally, we compute this invariant for closed locally symmetric spaces, and obtain the surprising result that the eta invariant for such spaces is a differential invariant.