We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
The distributions of some transmission eigenvalues in one dimension.
- Authors
YALIN ZHANG; GUOLIANG SHI
- Abstract
This work focuses on two main topics about the transmission eigenvalue problem defined on the unit interval [0, 1]: (i) the distributions of transmission eigenvalues in a region Σε which is symmetric about the axes; (ii) the existence of real transmission eigenvalues under some conditions on acoustic profiles n1 (x) and n2 (x). These subjects are central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Specifically, in the case where n1(x) = n2(x) for x = 0, 1, we show how to locate the transmission eigenvalues in Σ∈. The existence of infinitely many real transmission eigenvalues is proven in the following cases: (i) n2 (0) /n1 (0) ≠ n2 (1) /n1 (1) and n1 (0) n1 (1) ≠ n2 (0) n2 (1) ; (ii) n1 (0) = n2 (0), n1 (1) = n2 (1) and n'1 (1) -- n'2 (1) = n'1 (0) -- n'2 (0). The asymptotics of the real eigenvalues, as the byproduct of their existence, is obtained in both cases. All the results are obtained under the assumptions on the values of ni (x) and n'i (x) for i = 1, 2, x = 0, 1. There are no more restrictions on the values of n1 (x), n2 (x) in the interval (0, 1).
- Subjects
EIGENVALUES; DIFFERENTIAL equations; ASYMPTOTIC theory of algebraic ideals; EIGENFREQUENCIES; EIGENANALYSIS; EXPONENTIAL stability
- Publication
IMA Journal of Applied Mathematics, 2017, Vol 82, Issue 3, p601
- ISSN
0272-4960
- Publication type
Article
- DOI
10.1093/imamat/hxx005