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计算数学  2018, Vol. 40 Issue (1): 63-84    DOI:
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重访听音辩鼓问题
刘小会1,2, 曹建文1, 张娅1
1. 中国科学院软件研究所, 北京 100190;
2. 中国科学院大学, 北京 100190
THE ISO-SPECTRAL PROBLEM, REVISITED IN PLANAR CASE
Liu Xiaohui1,2, Cao Jianwen1, Zhang Ya1
1. Institute of Software, Chinese Academy of Science, Beijing 100190, China;
2. University of Chinese Academy of Science, Beijing 100190, China
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摘要 听音辨鼓这个反问题发展至今已经半个世纪,许多数学和物理学家都做出了很多有益的贡献.这个挑战性问题由美国数学家M.Kac 1966年正式提出,用数学语言描述为欧几里得空间中,是否可以找到两个(或更多)非等距单连通区域是等谱的?C.Gordon等人1992年在二维平面上给出一对等谱区域,首次对Kac的问题说"No".问题发展至今,只有17类平面等谱区域.它们都遵循一系列镜像反演规则,成对等谱,保持反演规则不变,改变基本构建块的形状,可以形成无穷多同类的等谱对.本文重访17类等谱区域,探究构建块之间的镜像反演规则.通过折叠方法,建立17类等谱区域特征函数之间的迁移映射关系.结合符号计算,列出17类等谱区域移植矩阵的通解.此外,利用Bernstein-Bézier多项式,计算等谱区域的广义特征值.
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关键词17类等谱区域   镜像反演规则   折叠方法   移植矩阵     
Abstract: "Can one hear the shape of a drum?" was proposed by Kac in 1966. In mathematical word, this question deduces to "is it possible to find two (or more) non-isometric Euclidean simply connected domains for which the sets of eigenvalues are identical?". It wasn't until 1992 that C. Gordon et al. answered negatively by finding a pair of non-isometric planar domains with the same Laplace spectrum. Still, there are essentially only 17 families of examples that say no to Kac's question. This paper revisits the serials of reflection rule inherent in the 17 families of iso-spectral domains. By the method of paper-folding, we construct transplantation mapping of the 17 iso-spectral pairs. With the technique of symbolic calculation, we also obtain the transplantation matrix of the 17 iso-spectral pairs. Moreover, after utilizing the Bernstein-Bézier patches, the approximated eigenvalues of some iso-spectral pairs are calculated.
Key words17 families of iso-spectral pairs   Reflection rule   Paper-folding method   Transplantation matrix   
收稿日期: 2017-01-22;
基金资助:

国家自然科学基金~(91430214,91230109,11301507)资助项目.

引用本文:   
. 重访听音辩鼓问题[J]. 计算数学, 2018, 40(1): 63-84.
. THE ISO-SPECTRAL PROBLEM, REVISITED IN PLANAR CASE[J]. Mathematica Numerica Sinica, 2018, 40(1): 63-84.
 
[1] Kac M. Can one hear the shape of a drum[J]. Amer. Math. Monthly, 1966, 73:1-23.
[2] Milnor J. Eigenvalues of Laplace operator on certain manifolds[J]. Proc. Nat. Acad. Sci. USA, 1964, 51:542.
[3] Ikeda A. On lens spaces which are isospectral but not isometric[J]. Ann. Sci. Ecole Normale Super., 1980, 13:303-315.
[4] Vigneras M F. Riemannienes isospectrales et non isometrigues[J]. Ann. Math., 1980, 11221-32.
[5] Urakawa H. Bounded domains which are isospectral but not congruent[J]. Ann. Sci. Ecole Norm. Sup., 1982, 441-456.
[6] Melrose R. Isospectral sets of drumheads are compact in C[R]. 1983, No. 48-83.
[7] Protter M H. Can one hear the shape of a drum? Revisited[J]. SIAM Review, 1987, 29:185-197.
[8] Osgood B, Phillips R and Sarnak P. Compact isospectral sets of plane domains[J]. Proc. Nat. Acad. Sci. USA, 1988a, 855359-5361.
[9] Osgood B, Phillips R and Sarnak P. Compact isospectral sets of surfaces[J]. J. Funct. Anal., 1988b, 80212-234.
[10] Sunada T. Riemannian coverings and isospectral manifolds[J]. Ann. of Math., 1985, 121(1):169-186.
[11] Buser P. Isospectral Riemann surfaces[J]. Ann. Inst. Fourier, 1986, 36:167-192.
[12] Buser P. Cayley graphs and planar isospectral domains. Geometry and Analysis on Manifolds, Lecture Notes in Math. 1988, 1339.
[13] Gordon C, Webb D L and Wolpert S. Isospectral plane domains and surfaces via Riemannian orbifolds[J]. Invent. Math., 1992a, 110:1-22.
[14] Gordon C, Webb D L. Isospectral convex domains in Euclidean space[J]. Mathematical Research Letters, 1994, 1:539-545.
[15] Gordon C, Webb D L. Isospectral convex domains in the hyperbolic plane[J]. Proc. Nat. Acad. Sci. USA, 1994, 120(3).
[16] Bérard P. Transplantation et isopectralite[J]. Math. Ann., 1992, 292:547-559.
[17] Buser P, Conway J and Doyle P. Some planar isospectral domains. Int. Math. Res. Notices, 1994, 9:391-400.
[18] Okada Y and Shudo A. Equivalence between isospectrality and iso-length spectrality for a certain class of planar billiard domains[J]. J. Phys. A:Math. Gen., 2001, 34:5911-5922.
[19] Thain A. Classical motion in isospectral billiards[J]. Eur. J. Phys., 2004, 25:633-643.
[20] Okada Y, Shudo A, Tasaki S and Harayama S. ‘Can one hear the shape of a drum?’:revisited[J]. J. Phys. A:Math. Gen., 2005, 38:L163-170.
[21] Dhar A, Rao D M, UdayaShankar N and Sridhar S. Isospectrality in chaotic billiards[J]. Phys. Rev., 2003, 68, 026208.
[22] Gottlied H P W. Isospectral circular membrane[J]. Inverse problems, 2004, 20(1):155.
[23] Knowles I W and McCarthy M L. Isospectral membranes:a connection between shape and density, J. Phys. A:Math. Gen., 2004, 37:8103-8109.
[24] Reuter M, Wolter F E and Peinecke N. Laplace-Beltrami spectra as ‘Shape-DNA’ of surfaces and solids[J]. Computer-Aided Design, 2006, 38:342-366.
[25] Chapman S J. Drums that sound the same[J]. Amer. Math. Monthly. 1995, 102(2):124-138.
[26] Sleeman B D and Chen H. On nonisometric isospectral connected fractal domains[J]. Rev. Mat. Iberoam., 2000, 16:351-361.
[27] Giraud O. Finite geometries and diffractive orbits in isospectral billiards[J]. J. Phys. A:Math. Gen, 2005, 38:L477.
[28] Giraud O and Thas K. Hearing shapes of drums-mathematical and physical aspects of isospectrality[J]. Reviews of modern physics, 2010, 82(3):2213-2255.
[29] Wu H, Sprung D W L and Martorell J. Numerical investigation of iso-spectral cavities built from triangles[J]? Phys. Rev. E., 1995, 51(1):703-708.
[30] Scridhar S and Kudrolli A. Experiments on Not Hearing the shape of Drum[J]. Phys. Rev. Lett., 1995, 72(14):2175-2178.
[31] Driscoll T A. Eigenmodes of isospectral drums[J]. SIAM Rev., 1997, 39(1):1-17.
[32] Descloux J and Tolley M. An accurate algorithm for computing the eigenvalues of a polygonal membrane[J]. Comput. Methods Appl. Mech. Engrg., 1983, 39:37-53.
[33] Betcke T and Trefethen L N. Reviving the Method Of Particular Solutions[J]. SIAM Rev., 2005, 47(3):469-491.
[34] Fox L., Henrici P and Moler C. Approximations and bounds for eigenvalues of elliptic operators[J]. SIAM J. Numer. Anal., 1967, 4(1):89-102.
[35] Even C and Pieranski P. On "hearing the shape of drums":An experimental study using vibrating smectic film. Europhysics Lett., 1999, 47(5):531-537.
[36] Moon C R, Mattos L S, Foster B K, Zeltzer G, Ko W and Manoharan H C. Quantum phase extraction in isospectral electronic nanostructures[J]. Science, 2008, 319:782-787.
[37] Amore P. One cannot hear the density of a drum (and further aspects of isospectrality)[J], Phys. Rev. E., 2013, 88(4):042915.
[38] Courant R. and Hilbert D. Methods of mathematical physics[M]. Vol. I. Interscience Publishers, Inc., New York, N.Y., 1953.
[39] Liu X H, Sun J C and Cao J W. We can't hear the shape of drum:revisited in 3D case. Preprint.
[40] 孙家昶. 非规则区域~Fourier变换与正交多项式[M]. 北京:中国科学技术大学出版社, 2009.
[41] 陈化. 你能听出一面鼓的几何形状吗——谈谈等谱问题[J]. 数学通报, 2014, 53(5).
[1] 孙家昶, 张娅. 二维等谱问题研究的计算数学框架[J]. 计算数学, 2017, 39(3): 229-286.

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