Abstract:We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in R 2 . Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amount and the curvature of the boundary is unbounded. By transforming t… Show more
“…In dimension two, there are better bounds of R B (µ). The bound O(µ 2/3 ) was given in Kuznecov and Fedosov [12] and Colin de Verdière [3], which was improved a little bit in [4] and recently to O(µ 131/208 (log µ) 18627/8320 ) in [5]. We guess that O (µ 1/2+ ) may be its true size.…”
Section: Introductionmentioning
confidence: 80%
“…Proof. When ν is a nonnegative integer, a proof is provided in the appendix of [5]. For real positive ν we recall the integral representation of J ν (x) for x > 0 ([1, p.360])…”
Section: Zeros Of the Bessel Function J ν (X)mentioning
confidence: 99%
“…Our domain D here is simply a special case of it (with r = 0 and R = 1 there). To prove this lemma, we just need to repeat the computation of N u D 1 (µ) in the proof of [5,Theorem 4.1]. We give a sketch in the appendix.…”
Section: Lemma 31 There Exists a Constantmentioning
confidence: 99%
“…For results on other interesting (planar) domains, see [10] for ellipses, [5] for annuli, [11] for domains of separable variables type, etc.…”
Let B = {x ∈ R d : |x| < R} (d ≥ 3) be a ball. We consider the Dirichlet Laplacian associated with B and prove that its eigenvalue counting function has an asymptotics18627 8320 ä as µ → ∞.
“…In dimension two, there are better bounds of R B (µ). The bound O(µ 2/3 ) was given in Kuznecov and Fedosov [12] and Colin de Verdière [3], which was improved a little bit in [4] and recently to O(µ 131/208 (log µ) 18627/8320 ) in [5]. We guess that O (µ 1/2+ ) may be its true size.…”
Section: Introductionmentioning
confidence: 80%
“…Proof. When ν is a nonnegative integer, a proof is provided in the appendix of [5]. For real positive ν we recall the integral representation of J ν (x) for x > 0 ([1, p.360])…”
Section: Zeros Of the Bessel Function J ν (X)mentioning
confidence: 99%
“…Our domain D here is simply a special case of it (with r = 0 and R = 1 there). To prove this lemma, we just need to repeat the computation of N u D 1 (µ) in the proof of [5,Theorem 4.1]. We give a sketch in the appendix.…”
Section: Lemma 31 There Exists a Constantmentioning
confidence: 99%
“…For results on other interesting (planar) domains, see [10] for ellipses, [5] for annuli, [11] for domains of separable variables type, etc.…”
Let B = {x ∈ R d : |x| < R} (d ≥ 3) be a ball. We consider the Dirichlet Laplacian associated with B and prove that its eigenvalue counting function has an asymptotics18627 8320 ä as µ → ∞.
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