# Asymptotic Heat Kernel Expansion in the Semi-Classical Limit

@article{Br2008AsymptoticHK, title={Asymptotic Heat Kernel Expansion in the Semi-Classical Limit}, author={Christian B{\"a}r and Frank Pf{\"a}ffle}, journal={Communications in Mathematical Physics}, year={2008}, volume={294}, pages={731-744} }

Let $${H_\hbar = \hbar^{2}L +V}$$, where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of $${H_\hbar}$$ as $${\hbar \searrow 0}$$. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum…

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## References

SHOWING 1-10 OF 21 REFERENCES

### Trace ideals and their applications

- Mathematics
- 1979

Preliminaries Calkin's theory of operator ideals and symmetrically normed ideals convergence theorems for $\mathcal J_P$ Trace, determinant, and Lidskii's theorem $f(x)g(-i\nabla)$ Fredholm theory…

### Heat Kernels and Dirac Operators

- Mathematics
- 1992

The past few years have seen the emergence of new insights into the Atiyah-Singer Index Theorem for Dirac operators. In this book, elementary proofs of this theorem, and some of its more recent…

### Small ħ asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials

- Physics
- 1984

To a gauge field on a principalG-bundleP→M is associated a sequence of quantum mechanical Hamiltonians, as Planck's constant ħ→0 and a sequence of representations πn ofG is taken. This paper studies…

### LOWER BOUNDS FOR THE HELMHOLTZ FUNCTION

- Mathematics
- 1965

Abstract : A mathematical theorem is established for traces of products of bounded Hermitian and definiteoperators. This theorem is applied to the equilibrium partition function by exploiting an…

### Inequality with Applications in Statistical Mechanics

- Mathematics
- 1965

We prove for Hermitian matrices (or more generally for completely continuous self‐adjoint linear operators in Hilbert space)A and B that Tr (eA+B ) ≤ Tr (eAeB ). The inequality is shown to be sharper…

### Riemannian Geometry: A Modern Introduction

- Mathematics
- 1994

1. Riemannian manifolds 2. Riemannian curvature 3. Riemannian volume 4. Riemannian coverings 5. Surfaces 6. Isoperimetric inequalities (constant curvature) 7. The kinetic density 8. Isoperimetric…

### Proof and Refinements of an Inequality of Feynman

- Mathematics
- 1965

A lower bound given by Feynman for the quantum mechanical free energy of an oscillator is proved, refined, and generalized. The method, application of a classical inequality to path integrals, also…

### Lectures on Differential Geometry

- Mathematics, Geology
- 1994

In 1984, the authors gave a series of lectures on differential geometry in the Institute for Advanced Studies in Princeton, USA. These lectures are published in this volume, which describes the major…

### Functional integration and quantum physics

- Physics
- 1979

Introduction The basic processes Bound state problems Inequalities Magnetic fields and stochastic integrals Asymptotics Other topics References Index Bibliographic supplement Bibliography.