有限区间上的扩散算子的惟一性定理

数学物理学报 ›› 2013, Vol. 33 ›› Issue (2): 333-339.

有限区间上的扩散算子的惟一性定理

王於平

南京林业大学应用数学系南京 210037

收稿日期:2011-03-11 修回日期:2012-11-28 出版日期:2013-04-25 发布日期:2013-04-25
基金资助:
国家自然科学基金(11171152)和江苏省自然科学基金 (BK2010489)资助

A Uniqueness Theorem for Diffusion Operators on the Finite Interval

WANG Yu-Ping

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037

Received:2011-03-11 Revised:2012-11-28 Online:2013-04-25 Published:2013-04-25
Supported by:
国家自然科学基金(11171152)和江苏省自然科学基金 (BK2010489)资助

摘要/Abstract

摘要：

讨论了有限区间[0, π]上的扩散算子的逆问题, 对固定的整数n(n∈Z), 证明不同的系数H_k 的扩散算子的第n个特征值的谱集合能够惟一确定势函数 [0, π] 上的(q(x), p(x)) 及边界条件中的系数h.

关键词: 惟一性定理, 逆问题, 扩散算子, 特征值

Abstract:

In this paper, we discuss the inverse problem for diffusion operators on the finite interval [0, π]. For a fixed integer n (n∈Z), the potentials (q(x), p(x)) and coefficient h of the boundary condition can be uniquely determined by the spectral set of the n-th eiginvalues of the diffusion operator for distinct H_k.

Key words: Uniqueness theorem, Inverse problem, Diffusion operator, Eigenvalue

中图分类号:

34A55

王於平. 有限区间上的扩散算子的惟一性定理[J]. 数学物理学报, 2013, 33(2): 333-339.

WANG Yu-Ping. A Uniqueness Theorem for Diffusion Operators on the Finite Interval[J]. Acta mathematica scientia,Series A, 2013, 33(2): 333-339.

参考文献

[1] Ambartsumyan V A. \"{U}ber eine frage der eigenwerttheorie. Zeitschrift f\"{u}r Physik, 1929, 53: 690--695

[2] Borg G. Eine umkehrung der Sturm-Liouvillesehen eigenwertaufgabe. Acta Mathematica, 1945, 78: 1--96

[3] Gesztesy F, Simon B. Inverse spectral analysis with partial information on the potential II: the case of discrete spectrum. Trans Amer Math Soc, 2000, 352(6): 2765--2787

[4] Hald O H. The Sturm-Liouville Problem with symmetric potentials. Acta Math, 1978, 141: 262--291

[5] Levitan B M. On the determination of the Sturm-Liouville operator from one and two spectra. Mathematics of the USSR Izv, 1978, 12}: 179--193

[6] Hochstadt H, Lieberman B. An inverse Sturm-Liouville problem with mixed given data. SIAM Journal of Applied Mathematics, 1978, 34: 676--680

[7] McLaughlin J R, Rundell W. A uniqueness theorem for an inverse Sturm-Liouville problem. Journal of Mathematical Physics, 1987, 28(7): 1471--1472

[8] Koyunbakan H. Inverse spectral problem for some singular differential operators. Tamsui Oxford Journal of Mathematical Sciences, 2009, 25(3): 277--283

[9] Jaulent M, Jean C. The inverse s-wave scattering problem for a class of potentials depending on energy. Comm Math Phys, 1972, 28: 177--220

[10] Gasymov M G, Guseinov G S. Determination of diffusion operators on the spectral data 1. Dokl Akad Nauk Azerb SSR, 1981, 37(2): 19--23

[11] Koyunbakan H, Panakhov E S. Half-inverse problem for diffusion operators on the finite interval. J Math Anal Appl, 2007, 326: 1024--1030

[12] 王於平, 杨传富, 黄振友. Schr\"odinger 算子的二次微分束的半逆问题. 数学物理学报, 2011, 31(6): 1708--1717

[13] Yang C F. Reconstruction of the diffusion operator from nodal data. Z Natureforsch, 2010, 65A: 100--106

[14] Nabiev I M. The inverse quasiperiodic problem for a diffusion operator. Doklady Mathematics, 2007, 76(1): 527--529

[15] Wei G S, Xu H K. On the missing eigenvalue problem for an inverse Sturm-Liouville problem. J Math Pures Appl, 2009, 91: 468--475

[16] Yurko V A. Inverse spectral problems for Sturm-Liouville differential operators on a finite interval. J Inverse and Ill-Posed Problems, 2009, 17: 639--694

[17] 杨传富, 杨孝平. 一般边界条件下Sturm-Liouville 算子的 Ambarzumyan型定理. 数学物理学报, 2010, 30(2): 449--455

[18] Ding X Q, Lou P Z. On Ritz method of an integro-differential equation. Acta Math Scientia, 2009, 29: 687--696

[19] Ding X Q, Lou P Z. Finite element approximatiom of an integro-differential operator. Acta Math Scientia, 2009, 29: 1767--1776

[20] Yurko V A. Method of Spectral Mappings in the Inverse Problem Theory. Utrecht: VSP (Inverse Ill-posed Problems Series), 2002

[21] 刘景麟. 常微分算子谱论. 北京: 科学出版社, 2009

[22] Dunford N, Schwarz J T. Linear Operators (Part II). New York, London: Interscience Publishers, 1963