Isospectral Operators
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Isospectral Operators Mu-Fa Chen · Xu Zhang
Received: 7 March 2014 / Revised: 19 March 2014 / Accepted: 24 March 2014 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2014
Abstract For a large class of integral operators or second-order differential operators, their isospectral (or cospectral) operators are constructed explicitly in terms of htransform (duality). This provides us a simple way to extend the known knowledge on the spectrum (or the estimation of the principal eigenvalue) from a smaller class of operators to a much larger one. In particular, an open problem about the positivity of the principal eigenvalue for birth–death processes is solved in the paper. Keywords
Isospectral · Harmonic function · Integral operator · Differential operator
Mathematics Subject Classification (2010)
58J53 · 37A30
1 Introduction Let us consider the elliptic operators L=
ai j (x)∂i2j +
i, j
L=
i, j
bi (x)∂i + c(x),
i
a˜ i j (x)∂i2j +
b˜i (x)∂i ,
i
M.-F. Chen (B) School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, People’s Republic of China e-mail: [email protected] http://math.bnu.edu.cn/chenmf/main_eng.htm X. Zhang College of Applied Sciences, Beijing University of Technology, Beijing 100022, People’s Republic of China e-mail: [email protected]
123
M.-F. Chen, X. Zhang
on L 2 (μ) and L 2 (μ) ˜ (real), respectively, where μ˜ = h 2 μ for a given measure μ and some h = 0. Their main difference is that c(x) ≡ 0. We are interested in when the operators L and L are L 2 -isospectral in the following sense: L f˜, f˜ μ˜ , for every f˜ := f / h, f ∈ D(L). (L f, f )μ = Here is one of our typical results in the note (cf. Theorems 3.1 and 3.6 in Sect. 3). Theorem 1.1 (1) Given L on L 2 (μ) having domain D(L), let h = 0, μ-a.e. be LL: harmonic: Lh = 0, μ-a.e., then, L is L 2 -isospectral to L = { f : f h ∈ D(L)}, L = L 0 + 2h −1 a∇h, ∇, D where L 0 = L − c. ˜ having domain D L , then for each h = 0, μ-a.e., L is L 2 (2) Given L on L 2 (μ) isospectral to L: 1 2 2 ˜ ∇ + 2 a∇h, L = L − a∇h, ˜ ∇h − Lh , h h h
D(L) = f : f / h ∈ D L , where ·, · denotes the Euclidean inner product. As a typical application of Theorem 1.1, we obtain the next result. To state it, we need to explain the meaning of eigenvalue in different sense. We say that λ is an eigenvalue of L in the ordinary sense if Lg = λg for some g = 0. It is called a L 2 -eigenvalue if additionally, g ∈ L 2 (μ). Corollary 1.2 For each h ∈ C 2 (R), h = 0, a.e., the operator
h 2 h 1 d2 h d h + − x+ +x − L = 2 dx 2 h dx h h 2h h
has L 2 -eigenvalues λn L h = −n with eigenfunctions gn (x) = (−1)n h(x)e x
2
dn −x 2 e , n 0, dx n
respectively. A particular class of L h is the following: Lb =
1 1 d2 d 2 2 + b(x) − b(x) − b (x) − x + 1 , b ∈ C 1 (R). 2 dx 2 dx 2
Proof Noting that t
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