Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space

Abstract

We consider the Friedrichs self-adjoint extension for a differential operator $A$ of the form $A=A_0+q\left(x\right)\cdot$, which is defined on a bounded domain $\Omega \subset {ℝ}^n$, $n\ge 1$ (for $n=1$ we assume that $\Omega =\left(a,b\right)$ is a finite interval). Here $A_0=A_0\left(x,D\right)$ is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth coefficients and a potential $q\left(x\right)$ is a real-valued integrable function satisfying the generalized Kato condition. Under these assumptions for the coefficients of $A$ and for positive $\lambda$ large enough we obtain the existence of Green's function for the operator $A+\lambda I$ and its estimates up to the boundary of $\Omega$. These estimates allow us to prove the absolute and uniform convergence up to the boundary of $\Omega$ of Fourier series in eigenfunctions of this operator. In particular, these results can be applied for the basis of the Fourier method which is usually used in practice for solving some equations of mathematical physics.