Asymptotic expression of negative eigenvalue number of an operator differential equation with even order according to small parameter

having weighted function in eigenvalue

Inci Albayrak

Yildiz Technical University Mathematical Engineering Davutpasa Istanbul Turkey

Let H be seperable Hilbert space and while ρ(x) > 0 is a continuous function in [0, ∞). Let H_{1} a Hilbert space of H-valued measurable function f(x), (0 ≤ x < ∞) that satisfies ∫_{0}^{∞}ρ(x)∥f(x)∥^{2}dx < ∞ condition.

Let L_{h}, an operator generated by differential expression

ℓ_{h}(y) = \frac{1}{ρ(x)}\{(−1)^{n}hy^{(2n)} − Q(x)y\}, (0 ≤ x < ∞)

and boundary conditions

y^{(j)}(0) = 0, j = 0, 1, ..., n − 1

in H_{1} space. Q(x) in ℓ_{h}(y) is an operator function which satisfied Q^{∗}(x) = Q(x) ≥ 0, Q(x) ∈

σ_{∞}, lim_{x→∞} ∥Q(x)∥ = 0 and h > 0 is small parameter. Under this conditions, negative spectrum of L_{h} operator consists of eigenvalues.

Let N_{h}(ϵ) denoted numbers of negative eigenvalues of L_{h} that smaller than −ϵ, (ϵ > 0).

When h → 0

N_{ϵ}(h) ∼ \frac{1}{π \sqrt[2n]{h}} ∑_{j} ∫_{α_{j}(x)>ϵ} ρ(x) \sqrt[2n]{α_{j}(x) − ϵ}dx

is obtained. Here, α_{1}(x) ≥ α_{2}(x) ≥ ... are eigenvalues of Q(x). While n = 1, asymptotic expression of N_{h}(ϵ) is found in [1].

REFERENCE

1. M.Bayramoğlu,I.Albayrak, Second International Symposium on Mathematical and Computational Applications, Baku, September 1-3,1999.

Joint work with Oya Baykal and Mehmet Bayramoğlu.