The regularized trace formula for a fourth order differential operator given in a finite interval

Serpil Karayel,Yonca Sezer

doi:10.1186/s13660-015-0823-0

Serpil Karayel, Yonca Sezer

Open Access

https://doi.org/10.1186/s13660-015-0823-0

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Abstract

In this work, a regularized trace formula for a differential operator of fourth order with bounded operator coefficient is found.

Highlights

Investigations into the regularized trace formulas of scalar differential operators started with the work [ ] firstly
We find the following regularized trace formula for a self-adjoint differential operator L of fourth order with bounded operator coefficient:
The integral on the right-hand side of the last equality can be written as tr λ QR λ j

Summary

Introduction

Investigations into the regularized trace formulas of scalar differential operators started with the work [ ] firstly. We find the following regularized trace formula for a self-adjoint differential operator L of fourth order with bounded operator coefficient:. We denote the norms by · H and · and inner products by (·, ·)H and (·, ·) in H and H , respectively, and we denote the sum of eigenvalues of a kernel operator A by tr A = trace A. Every point of this set is an eigenvalue of L which has infinite multiplicity. The integral on the right-hand side of the last equality can be written as tr λ QR λ j

Let ε be a constant satisfying the condition ε

Considering for m

From this relation we obtain