Geometry of ends, spectrum of Laplacian, scattering, and inverse problem

• Project/Area Number: 21540215
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Shizuoka University
• Principal Investigator: KUMURA Hironori 静岡大学, 理学部・数学科, 准教授 (30283336)
• Co-Investigator(Renkei-kenkyūsha): KASUE Atsushi 金沢大学, 理学部, 教授 (40152657) ; AKUTAGAWA Kazuo 東北大学, 情報学部, 教授 (80192920)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: ラプラス作用素 / スペクトラム / 極限吸収原理 / 絶対連続性 / 平均曲率 / 極小部分多様体 / ラプラス-ベルトラミ作用素 / 固有値の非存在 / リーマン部分多様体 / ノン・コンパクト・リーマン多様体 / ラプラス・ベルトラミ作用素 / スペクトル / エンド / 放射曲率 / ラプラシアン

Research Abstract

(1) I obtained a sharp criterion of the curvature which shows infinitely many discrete spectrum of the Laplace-Beltrami operator.
(2) I obtained a sharp result which clarifies the relation between the measure growth rate and the absence of embedded eigenvalues of drift Laplacians.
(3) I proved that the limiting absorption principle holds on Riemannian manifolds having ends with various measure growth rates.
(4) I provided the explicitly calculated radii of geodesic balls of a complete noncompact Riemannian submanifold which must exit from the cylindrically bounded domain U, in case its mean curvature is sufficiently small.

Research Products

• [Journal Article] The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II (2011)
  • Author(s): 久村裕憲
  • Journal Title: Bull. London Math. Soc. Volume: 43 Pages: 985-1003
  • Peer Reviewed
• [Journal Article] The lower bound of the Ricci curvature which yields the infinite number of the discrete spectrum of the Laplacian (2011)
  • Author(s): 久村裕憲
  • Journal Title: Annales de l' institut Fourier Volume: 61 Pages: 1557-1572
  • Peer Reviewed
• [Journal Article] The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II (2011)
  • Author(s): 久村裕憲
  • Journal Title: Bull.London Math.Soc. Volume: 43 Pages: 985-1003
  • Peer Reviewed
• [Journal Article] The lower bound of the Ricci curvature which yields the infinite number of the discrete spectrum of the Laplacian (2011)
  • Author(s): 久村裕憲
  • Journal Title: Annales de l'institut Fourier Volume: 61 Pages: 1557-1572
  • Peer Reviewed
• [Journal Article] The radial curvature of an end that makes eigenvalues vanish in the essential spectrum I (2010)
  • Author(s): 久村裕憲
  • Journal Title: Math. Ann Volume: 346 Pages: 795-828
  • NAID: 120001941020
  • Peer Reviewed
• [Journal Article] The radial curvature of an end that makes eigenvalues vanish in the essential spectrum I (2010)
  • Author(s): Hironori Kumura
  • Journal Title: Mathematische Annalen Volume: 346 Pages: 795-828
  • NAID: 120001941020
  • Peer Reviewed
• [Journal Article] The radial curvature of an end that makes eigenvalues vanish in the essential spectrum I (2010)
  • Author(s): Hironori Kumura
  • Journal Title: Mathematische Annalen 346 Pages: 795-828
  • NAID: 120001941020
  • Peer Reviewed
• [Journal Article] Perturbation of a warped product metric of an end and the growth property of solutions to eigenvalue equation
  • Author(s): 久村裕憲
  • Journal Title: J. Math.
  • Peer Reviewed
• [Presentation] ノンコンパクト多様体上のシュレーディンガー作用素 (2010)
  • Author(s): 久村裕憲
  • Organizer: 東北大学数学教室幾何セミナー
  • Place of Presentation: 数学棟208
  • Year and Date: 2010-10-12
• [Presentation] ノンコンパクト多様体上のシュレーディンガー作用素 (2010)
  • Author(s): 久村裕憲
  • Organizer: 東北大学数学教室幾何セミナー
  • Place of Presentation: 東北大学理学部(宮城県)
  • Year and Date: 2010-10-12
• [Presentation] Absence of eigenvalues and convergence of radial curvatures to zero (2010)
  • Author(s): 久村裕憲
  • Organizer: 日本数学会春季総合分科会
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-25
• [Presentation] Ricci曲率と離散スペクトルの無限性・有限性 (2010)
  • Author(s): 久村裕憲
  • Organizer: 日本数学会春季総合分科会
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-25
• [Presentation] Ricci 曲率と離散スペクトルの無限性・有限性 (2010)
  • Author(s): 久村裕憲
  • Organizer: 日本数学会幾何学分科会
  • Place of Presentation: 慶応大学理工学部
  • Year and Date: 2010-03-25
• [Presentation] Absence of eigenvalues and convergence of radial curvatures to zero (2010)
  • Author(s): 久村裕憲
  • Organizer: 日本数学会幾何学分科会
  • Place of Presentation: 慶応大学理工学部
  • Year and Date: 2010-03-25
• [Presentation] On the discrete spectrum of Schr odinger operator on noncompact Riemannian manifolds (2009)
  • Author(s): 久村裕憲
  • Organizer: 日本数学会春季総合分科会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-28