{"id":748,"date":"2020-09-17T03:09:20","date_gmt":"2020-09-17T03:09:20","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=748"},"modified":"2020-09-17T03:09:20","modified_gmt":"2020-09-17T03:09:20","slug":"pujihejianjie","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=748","title":{"rendered":"\u8c31\u51e0\u4f55\u7b80\u4ecb"},"content":{"rendered":"\n

\u8c31\u51e0\u4f55\u662f\u7814\u7a76\u6d41\u5f62\u7684\u51e0\u4f55\u7ed3\u6784\u548c\u6d41\u5f62\u4e0a\u5178\u5219\u7684\u5fae\u5206\u7b97\u5b50(\u4e3b\u8981\u662fLaplace–Beltrami\u7b97\u5b50)\u7684\u8c31\u4e4b\u95f4\u7684\u5173\u7cfb. \u5b83\u4e3b\u8981\u5206\u6210\u4e24\u4e2a\u7814\u7a76\u90e8\u5206:\u76f4\u63a5\u95ee\u9898\u4ee5\u53ca\u53cd\u95ee\u9898.<\/p>\n\n\n\n

\u53cd\u95ee\u9898\u7814\u7a76\u7684\u662f\u80fd\u5426\u4eceLaplace\u7b97\u5b50\u7684\u7279\u5f81\u503c(\u5373\u8c31)\u6765\u786e\u5b9a\u6d41\u5f62\u7684\u51e0\u4f55\u7279\u5f81. \u8fd9\u65b9\u9762\u6700\u65e9\u7684\u7ed3\u679c\u662fWeyl\u7684\u6e10\u8fd1\u516c\u5f0f, \u5b83\u8bf4\u660e\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u6709\u754c\u533a\u57df\u7684\u4f53\u79ef\u53ef\u4ee5\u88ab\u8be5\u533a\u57df\u4e0aLaplace\u65b9\u7a0b\u7684Dirichlet\u8fb9\u503c\u95ee\u9898\u7684\u8c31\u7684\u6e10\u8fd1\u884c\u4e3a\u786e\u5b9a. \u901a\u5e38\u8fd9\u4e00\u95ee\u9898\u88ab\u79f0\u4e3a\u201c\u542c\u9f13\u8fa8\u5f62\u201d[6<\/a>](PDF<\/a>). \u5173\u4e8eWeyl\u6e10\u8fd1\u516c\u5f0f\u7684\u4e00\u4e2a\u6539\u8fdb\u662f\u7531Minakshisundaram\u4ee5\u53caPleijel(Wang zuoqin<\/a>\u7ed9\u51fa\u4e86\u4e00\u4e2a\u70ed\u6838\u8bc1\u660e, \u53c2\u8003Grieser\u7684Notes on heat kernel asymptotics<\/a>)\u5f97\u5230\u7684. \u7531\u8be5\u516c\u5f0f\u53ef\u4ee5\u6784\u9020\u5173\u4e8e\u66f2\u7387\u5f20\u91cf\u53ca\u5176\u9ad8\u9636\u5bfc\u6570\u7684\u5c40\u90e8\u8c31\u4e0d\u53d8\u91cf, \u5b83\u4eec\u53ef\u4ee5\u7528\u6765\u5efa\u7acb\u4e00\u7c7b\u7279\u6b8a\u6d41\u5f62\u7684\u8c31\u521a\u6027\u7ed3\u679c. \u5c3d\u7ba1\u5982\u6b64, Milnor[7<\/a>]\u7684\u7814\u7a76\u8868\u660e, \u5b58\u5728\u7b49\u8c31\u5374\u4e0d\u7b49\u8ddd\u7684\u6d41\u5f62. \u56e0\u6b64\u8c31\u4e0d\u80fd\u5728\u7b49\u8ddd\u610f\u4e49\u4e0b\u5b8c\u5168\u786e\u5b9a\u6d41\u5f62. \u5173\u4e8eMilnor\u7684\u7b49\u8c31\u6d41\u5f62\u7684\u7814\u7a76, Sunada[8<\/a>]\u7ed9\u51fa\u4e86\u4e00\u4e2a\u5177\u4f53\u7684\u529e\u6cd5\u6765\u6784\u9020\u7b49\u8c31\u6d41\u5f62.<\/p>\n\n\n\n

\u76f4\u63a5\u95ee\u9898\u7814\u7a76\u7684\u662f\u4ece\u6d41\u5f62\u7684\u51e0\u4f55\u7ed3\u6784\u5f97\u5230\u6d41\u5f62\u7684\u8c31\u7684\u884c\u4e3a. \u8fd9\u65b9\u9762\u7ecf\u5178\u7684\u7ed3\u679c\u662fCheeger\u5f97\u5230\u7684Cheeger\u4e0d\u7b49\u5f0f[3<\/a>], \u5b83\u7ed9\u51fa\u4e86\u6d41\u5f62\u7684\u7b2c\u4e00\u7279\u5f81\u503c\u4e8e\u7b49\u5468\u5e38\u6570\u4e4b\u95f4\u7684\u5173\u7cfb. \u81ea\u6b64\u4ee5\u540e, \u8fd9\u65b9\u9762\u6709\u5f88\u591a\u7814\u7a76, \u4f8b\u5982Brooks[1<\/a>]\u548cBuser[2<\/a>]\u7684\u7ed3\u679c.<\/p>\n\n\n\n

\u5047\u8bbe$D\\subset \\mathbb{R}^2$\u662f\u4e00\u4e2a\u5e73\u9762\u6709\u754c\u533a\u57df, \u5176\u8fb9\u754c\u662f\u7531\u9010\u6bb5\u5149\u6ed1\u66f2\u7ebf\u6784\u6210\u7684, \u5047\u8bbe$D$\u5173\u4e8eDirichlet\u6216\u8005Neumann\u8fb9\u754c\u95ee\u9898(Laplace\u7b97\u5b50)\u7684\u7279\u5f81\u6839\u8bb0\u4e3a$\\mu_1^2\\leq\\mu_2^2\\leq\\cdots$, \u4e3a\u4e86\u7814\u7a76$\\mu_n$\u5f53$n\\to\\infty$\u65f6\u7684\u6e10\u8fd1\u884c\u4e3a, \u8003\u8651\u7279\u5f81\u503c\u8ba1\u6570\u51fd\u6570$\\mathcal{N}_D(\\mu)={}^\\#\\left\\{ n : \\mu_n<\\mu \\right\\}$. \u5219\u5e73\u9762Weyl\u6e10\u8fd1\u516c\u5f0f[9<\/a>, 5<\/a>]\u53ef\u8868\u793a\u4e3a $$ \\mathcal{N}_D(\\mu)=\\frac{\\mathrm{Area}(D)}{4\\pi}\\mu^2\\pm\\frac{\\mathrm{Length}(\\partial D)}{4\\pi}\\mu+o(\\mu). $$ \u5176\u4e2d, Dirichlet\u8fb9\u754c\u53d6$-$\u53f7, \u800cNeumann\u8fb9\u754c\u53d6$+$\u53f7. \u6700\u8fd1wang zuoqin[4<\/a>]\u7b49\u6539\u8fdb\u4e86\u4f59\u9879\u7684\u4f30\u8ba1.<\/p>\n

\n\t
\u53c2\u8003\u6587\u732e<\/div>\n\t
  1. R. Brooks<\/span>, The spectral geometry of a tower of coverings<\/a>, J. Differential Geom.<\/span> 23<\/span>(1986<\/span>), no. 1<\/span>, 97---107<\/span>. MR840402<\/a><\/li>
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  5. V. Ivrii<\/span>, 100 years of Weyl s law<\/a>, Bull. Math. Sci.<\/span> 6<\/span>(2016<\/span>), no. 3<\/span>, 379---452<\/span>. MR3556544<\/a><\/li>
  6. M. Kac<\/span>, Can one hear the shape of a drum?<\/a>, Amer. Math. Monthly<\/span> 73<\/span>(1966<\/span>), no. 4, part II<\/span>, 1---23<\/span>. MR201237<\/a><\/li>
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  8. T. Sunada<\/span>, Riemannian coverings and isospectral manifolds<\/a>, Ann. of Math. (2)<\/span> 121<\/span>(1985<\/span>), no. 1<\/span>, 169---186<\/span>. MR782558<\/a><\/li>
  9. H. Weyl<\/span>, Uber die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetze<\/a>, J. Reine Angew. Math.<\/span> 143<\/span>(1913<\/span>), 177---202<\/span>. MR1580880<\/a><\/li><\/ol><\/div>","protected":false},"excerpt":{"rendered":"

    \u8c31\u51e0\u4f55\u662f\u7814\u7a76\u6d41\u5f62\u7684\u51e0\u4f55\u7ed3\u6784\u548c\u6d41\u5f62\u4e0a\u5178\u5219\u7684\u5fae\u5206\u7b97\u5b50(\u4e3b\u8981\u662fLaplace–Beltrami\u7b97\u5b50)\u7684\u8c31… \u7ee7\u7eed\u9605\u8bfb\u8c31\u51e0\u4f55\u7b80\u4ecb<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[142,141],"class_list":["post-748","post","type-post","status-publish","format-standard","hentry","category-math","tag-weyljianjingongshi","tag-pujihe","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/748","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=748"}],"version-history":[{"count":5,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/748\/revisions"}],"predecessor-version":[{"id":753,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/748\/revisions\/753"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=748"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=748"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=748"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}