The Asymptotic Formula For The Eigenvalues Of The Second Order Partial Differential Equation With Operator Coefficient

Kevser Ozden Koklu Yildiz Technical University

In this study, it is shown that the spectrum is pure discrete and obtained the asymptotic expression for number eigenvalues of the operator generated with the differential expression

− ∑_{i=1}^{3} \frac{∂}{∂x_{i}} (p(x)\frac{∂u}{∂x_{i}} ) + Q(x)u

and the boundary condition

u|_{x_{3}=0} = 0

in E_{3}^{+} = {x = (x_{1}, x_{2}, x_{3}) : x_{3} ≥ 0}, a semi space of space E_{3}. Here p(x) is a continuously differentiable positive scalar function, Q(x) is a self-adjoint positive operator defined in separable Hilbert space for every x ∈ E_{3}^{+} whose inverse is completely continuous.

Joint work with Mehmet Bayramoglu and Oya Baykal.