Asymptotyc behavior of the eigenvalue numbers of the operator Δ^{2} + Q(x)

Mehmet Bayramoglu Yildiz Technical University, Turkey

Let H is a separable Hilbert Space and E_{n} is an n-dimensional Euclidean space. We denoted the Hilbert space of all H-values functions defined on E_{n} which are measurable and satisfy the condition

∫_{0}^{∞} ∥f(x)∥_{H}^{2} dx < ∞,

with H_{1} = L_{2}(E_{n}, H). In this study, we obtain asymptotic behaviour of the numbers of the eigenvalues ≤ λ (λ > 0) when λ ⟶ +∞ of the operator generated by the expression

(1)  Δ^{2} + Q(x),

in H_{1}. In (1), ∀x = (x_{1}, x_{2}, ⋯, x_{n}) ∈ E_{n}, Q(x) = Q^{∗}(x) ≥ I, ( I is an identity operator in H),Q^{−1}(x) is a compact operator in H and Δ^{2} is a biharmonic operator. Joint work with Oya Baykal and Kevser Koklu.