# The spectral function of an elliptic operator

@article{Hrmander1968TheSF, title={The spectral function of an elliptic operator}, author={Lars H{\"o}rmander}, journal={Acta Mathematica}, year={1968}, volume={121}, pages={193-218} }

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 achieved by means of a complete description of the singularities of the Fourier transform of the spectral function for low frequencies.

## 747 Citations

The spectral function of a singular differential operator of order 2m

- Mathematics
- 2010

We study the spectral function of a self-adjoint semibounded below differential operator on a Hilbert space and obtain the formulae for the spectral function of the operator with general boundary…

A COMPLETE ASYMPTOTIC EXPANSION OF THE SPECTRAL FUNCTION OF SECOND ORDER ELLIPTIC OPERATORS IN

- Mathematics
- 1985

A complete asymptotic expansion as , , ( arbitrary) is obtained for the spectral function of second order elliptic operators in satisfying the condition of not being "trapped", i.e. the requirement…

PARAMETRIX AND ASYMPTOTICS OF THE SPECTRAL FUNCTION OF DIFFERENTIAL OPERATORS IN

- Mathematics
- 1987

A new formula is obtained for a global parametrix of the Cauchy problem for hyperbolic equations and systems which is an analogue of the representation of the Green function in terms of the spectral…

The eigenvalue distribution of elliptic operators with Hölder continuous coefficients

- Mathematics
- 1991

This is the continuation of the previous paper [11], in which we attempted the improvement of the remainder estimate for the eigenvalue distribution of the elliptic operator of order 2m with Holder…

Localization of the Spectral Expansions Associated with the Partial Differential Operators

- MathematicsNonlinear Systems and Complexity
- 2018

In this paper we discuss precise conditions of the summability and localization of the spectral expansions associated with various partial differential operators. In this we study the problems in the…

Distribution of eigenvalues of an elliptic operator in a bounded region

- Mathematics
- 1990

Estimates of the remainder in the classical asymptotic expressions for the distribution of the eigenvalues of an elliptic differential operator defined in a bounded region are studied.

Asymptotics near the boundary of spectral functions of elliptic self-adjoint boundary problems

- Mathematics
- 1975

We obtain the asymptotic behavior of the spectral function of an elliptic self-adjoint boundary problem in a closed domain and give a uniform estimate for the remainder. The estimate for the…

Asymptotics of the spectral function of an elliptic differential operator of second order

- Mathematics
- 1982

In the work the Riesz means of the spectral function of a boundary-value problem for an elliptic differential operator of second order are studied for the case of a geodesically concave boundary.…

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

- Mathematics
- 1999

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach…

A remark on convergence almost-everywhere of eigenfunction expansions of elliptic operators.

- Mathematics
- 2019

In this paper it is proposed a very simple method for estimating the maximal operator in $L_1$. Using this method one can considerably improve the existing theorems on convergence almost-everywhere…

## References

SHOWING 1-8 OF 8 REFERENCES

On the asymptotic behavior of spectral functions and resolvent kernels of elliptic operators

- Mathematics
- 1967

Asymptotic formulas with remainder estimates are derived for spectral functions of general elliptic operators. The estimates are based on asymptotic expansion of resolvent kernels in the complex…

Pseudo-Differential Operators

- Mathematics
- 1965

Contents: Second Order Elliptic Operators.- Pseudo-Differential Operators.- Elliptic Operators on a Compact Manifold without Boundary.- Boundary Problems for Elliptic Differential Operators.-…

Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

- MathematicsCanadian Journal of Mathematics
- 1949

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (x i ), i = 1, 2, … Ν, the line-element dr is given by…

Zeta Functions on the Unitary Sphere

- MathematicsCanadian Journal of Mathematics
- 1952

In an earlier paper [5], the author defined a zeta function on the real sphere , whereas in the present paper it is proposed to define one on the unitary sphere where x i's are complex numbers and…