# Second term of the spectral asymptotic expansion of the Laplace - Beltrami operator on manifolds with boundary

@article{Ivrii1980SecondTO, title={Second term of the spectral asymptotic expansion of the Laplace - Beltrami operator on manifolds with boundary}, author={Victor Ivrii}, journal={Functional Analysis and Its Applications}, year={1980}, volume={14}, pages={98-106} }

A weld head control and guidance system utilizing non-contact proximity sensors which generate a signal based on the electrical conductivity of a workpiece. The system includes electrical controls which receive the signal generated by the proximity sensors and control movement of a movable torch mount to maintain a welding torch in a proper orientation and position with respect to a workpiece.

#### 164 Citations

The Trace of the Heat Kernel in Domains with Nonsmooth Boundaries

- Mathematics
- 1992

In this note, I would like to describe some recent work that considers the relationship between the smoothness of the boundary of a domain in R n and the spectral properties of the Laplacian in the… Expand

Periods of multiple reflecting geodesics and inverse spectral results

- Mathematics
- 1987

where AD is a self-adjoint operator in L2(Q) corresponding to the laplacian - A with Dirichlet boundary condition on aQ. The singularities of aD(t) are connected with the periods of all periodic… Expand

Asymptotic expansion of the trace of the heat kernel associated to the Dirichlet-to-Neumann operator

- Mathematics
- 2015

Abstract For a given bounded domain Ω with smooth boundary in a smooth Riemannian manifold ( M , g ) , by decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian… Expand

Semi-classical analysis of the Laplace operator with Robin boundary conditions

- Mathematics
- 2012

We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in… Expand

C-infinity Scaling Asymptotics for the Spectral Function of the Laplacian

- Mathematics, Physics
- 2016

This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near… Expand

Spectral Geometry: An Introduction and Background Material for this Volume

- Mathematics
- 1997

Inverse spectral geometry (ISG) has for the last couple of decades exhibited a very strong dynamics. A good deal of this dynamics stems from the fact that ISG is a melting pot for ideas and results… Expand

One can hear the area and curvature of boundary of a domain by hearing the Steklov eigenvalues

- Mathematics
- 2013

For a given bounded domain $\Omega$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we show that the Poisson type upper-estimate of the heat kernel associated to the… Expand

Microlocal analysis of global hyperbolic propagators on manifolds without boundary

- Mathematics
- 2020

The main goal of this thesis is to construct explicitly, modulo smooth terms, propagators for physically meaningful hyperbolic partial differential equations (PDEs) and systems of PDEs on closed… Expand

The trace of the heat kernel in Lipschitz domains

- Mathematics
- 1993

We establish the existence of an asymptotic expansion as t —► 0+ for the trace of the heat kernel for the Neumann Laplacian in a bounded Lip- schitz domain. The proof of an asymptotic expansion for… Expand

$$C^\infty $$C∞ Scaling Asymptotics for the Spectral Projector of the Laplacian

- Mathematics
- 2018

This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near… Expand

#### References

SHOWING 1-9 OF 9 REFERENCES

The spectral function of an elliptic operator

- Mathematics
- 1968

In this paper we shall obtain the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator. This is… Expand

The spectrum of positive elliptic operators and periodic bicharacteristics

- Mathematics
- 1975

Let X be a compact boundaryless C ∞ manifold and let P be a positive elliptic self-adjoint pseudodifferential operator of order m>0 on X. For technical reasons we will assume that P operates on… Expand

Fourier integral operators. I

- Mathematics
- 1971

Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, but their value… Expand