Study of the asymptotic eigenvalue distribution and trace formula of a second order operator-differential equation
- PDF / 531,784 Bytes
- 22 Pages / 595.276 x 793.701 pts Page_size
- 85 Downloads / 254 Views
RESEARCH
Open Access
Study of the asymptotic eigenvalue distribution and trace formula of a second order operatordifferential equation Nigar Mahar Aslanova1,2 Correspondence: nigar. [email protected] 1 Department of Differential Equation, Institute of Mathematics and Mechanics-Azerbaijan National Academy of Science, 9, F. Agayev Street, Baku AZ1141, Azerbaijan Full list of author information is available at the end of the article
Abstract The purpose of writing this article is to show some spectral properties of the Bessel operator equation, with spectral parameter-dependent boundary condition. This problem arises upon separation of variables in heat or wave equations, when one of the boundary conditions contains partial derivative with respect to time. To illustrate the problem and the proof in detail, as a first step, the corresponding operator’s discreteness of the spectrum is proved. Then, the nature of the eigenvalue distribution is established. Finally, based on these results, a regularized trace formula for the eigenvalues is obtained. MSC: 34B05; 34G20; 34L20; 34L05; 47A05; 47A10. Keywords: Hilbert space, discrete spectrum, regularized trace
Introduction Let L2 = L2 (H, [0, 1]) ⊕ H, where H is a separable Hilbert space with a scalar product (·, ·) and a norm ||·|| inside of it. By definition, a scalar product in L2 is 1 (y(t), z(t)) dt −
(Y, Z)L2 =
1 (y1 , z1 ), h
h < 0,
(1)
0
where Y = {y (t), y1}, Z = {z (t), z1} and y(t), z(t) Î L2 (H, [0, 1]) for which L2 (H, [0, 1]) is a space of vector functions y(t) such that 01 y (t)2 dt < ∞. Now, consider the equation: l[y] ≡ −y (t) +
ν 2 − 14 y (t) + Ay (t) + q(t)y (t) = λy(t), t2
y (1) − hy (1) = λy (1)
t ∈ (0, 1), ν ≥ 1,
(2) (3)
in L2 (H, [0, 1]), where A is a self-adjoint positive-definite operator in H which has a compact inverse operator. Further, suppose the operator-valued function q(t) is weakly measurable, and ||q(t)|| is bounded on [0, 1] with the following properties: 1. q(t) has a second-order weak derivative on [0, 1], and q(l) (t) (l = 0, 1, 2) are selfadjoint operators in H for each t Î [0, 1], [q(l) (t)]* = q(l) (t), q(l) (t) Î s1(H). Here © 2011 Aslanova; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Aslanova Boundary Value Problems 2011, 2011:7 http://www.boundaryvalueproblems.com/content/2011/1/7
Page 2 of 22
s1(H) is a trace class, i.e., a class of compact operators in separable Hilbert space H, whose singular values form a convergent series (denoting the compact operator 1
by B, then its singular values are the eigenvalues of (BB∗ ) 2). If {n } is a basis |(Bϕn , ϕn )|. For formed by the orthonormal eigenvectors of B, then Bσ1 (H) = simplicity, denote the norm in s1(H) by ||·||1. 2. The functions ||q(l) (t)||1 (l = 0, 1, 2) are bounded on [0, 1]. 3. The relat
Data Loading...
