The Asymptotic Distribution of Eigenvalues and Eigenfunctions for Elliptic Boundary Value Problems
1967; Society for Industrial and Applied Mathematics; Volume: 9; Issue: 4 Linguagem: Inglês
10.1137/1009105
ISSN1095-7200
Autores Tópico(s)Differential Equations and Boundary Problems
ResumoNext article The Asymptotic Distribution of Eigenvalues and Eigenfunctions for Elliptic Boundary Value ProblemsColin ClarkColin Clarkhttps://doi.org/10.1137/1009105PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Shmuel Agmon, The angular distribution of eigenvalues of non-selfadjoint elliptic boundary value problems of higher order, Partial differential equations and continuum mechanics, Univ. of Wisconsin Press, Madison, Wis., 1961, 9–18 MR0124617 0111.09503 Google Scholar[2] Shmuel Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15 (1962), 119–147 MR0147774 0109.32701 CrossrefISIGoogle Scholar[3] Shmuel Agmon, On the asymptotic distribution of eigenvalues of differential problems, Seminari 1962/63 Anal. Alg. Geom. e Topol., Vol. 2, Ist. Naz. Alta Mat., Ediz. Cremonese, Rome, 1965, 556–573 MR0188604 Google Scholar[4] Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965v+291 MR0178246 0142.37401 Google Scholar[5] Shmuel Agmon, On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems, Comm. Pure Appl. Math., 18 (1965), 627–663 MR0198287 0151.20203 CrossrefISIGoogle Scholar[6A] S. Agmon, , A. Douglis and , L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623–727 MR0125307 0093.10401 CrossrefISIGoogle Scholar[6B] S. Agmon, , A. Douglis and , L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), 35–92 MR0162050 0123.28706 CrossrefISIGoogle Scholar[7] Shmuel Agmon and , Yakar Kannai, On the asymptotic behavoir of spectral functions and resolvant kernels of elliptic operators, Israel J. Math., 5 (1967), 1–30 MR0218730 0148.13003 CrossrefISIGoogle Scholar[8] A. A. Arsenev, Asymptotic properties of the trace of the spectral function of a self-adjoint elliptic differential operator of second order, Dokl. Akad. Nauk SSSR, 157 (1964), 761–763 MR0177182 0147.09601 Google Scholar[9] A. A. Arsenev, Asymptotic behavior of the spectral function of the Schrödinger operator in an unbounded region, Uspehi Mat. Nauk, 21 (1966), 254–255 MR0206531 Google Scholar[10] Vojislav G. Avakumović, Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z., 65 (1956), 327–344 MR0080862 0070.32601 CrossrefGoogle Scholar[11] Donald Babbitt, The Wiener integral and the Schrödinger operator, Trans. Amer. Math. Soc. 116 (1965), 66-78; correction, ibid., 121 (1966), 549–552 MR0186926 0196.44901 Google Scholar[12] P. B. Bailey and , F. H. Brownell, Removal of the log factor in the asymptotic estimates of polygonal membrane eigenvalues, J. Math. Anal. Appl., 4 (1962), 212–239 10.1016/0022-247X(62)90052-5 MR0145213 0163.13602 CrossrefGoogle Scholar[13] Richard Beals, On eigenvalue distributions for elliptic operators without smooth coefficients, Bull. Amer. Math. Soc., 72 (1966), 701–705 MR0196257 0161.08902 CrossrefISIGoogle Scholar[14] Richard Beals, Classes of compact operators and eigenvalue distributions for elliptic operators, Amer. J. Math., 89 (1967), 1056–1072 MR0225196 0162.46003 CrossrefISIGoogle Scholar[15] Richard Beals, Asymptotic behavior of the Green's function and spectral function of an elliptic operator, J. Functional Analysis, 5 (1970), 484–503 10.1016/0022-1236(70)90021-2 MR0274983 0195.11401 CrossrefGoogle Scholar[16] Yu. M. Berezanskii, Eigenfunction Expansions of Self-Adjoint Operators, Akad. Nauk Ukrainskoi SSR, Naukova Dumka, Kiev, 1965 Google Scholar[17] Gunnar Bergendal, Convergence and summability of eigenfunction expansions connected with elliptic differential operators, Medd. Lunds Univ. Mat. Sem., 15 (1959), 63– MR0105548 0093.06901 Google Scholar[18] A. Boigelot, Comportement asymptotique du noyau spectral de $-\Delta$ sous des conditions aux limites de Dirichlet-Neumann dans un ouvert donné, Bull. Soc. Roy. Sci. Liège, 28 (1959), 175–179 MR0114036 0092.31503 Google Scholar[19] Félix Browder, Le problème des vibrations pour un opérateur aux dérivées partielles self-adjoint et du type elliptique, à coefficients variables, C. R. Acad. Sci. Paris, 236 (1953), 2140–2142 MR0058089 0051.32802 Google Scholar[20] Felix E. Browder, The asymptotic distribution of eigenfunctions and eigenvalues for semi-elliptic differential operators, Proc. Nat. Acad. Sci. U.S.A., 43 (1957), 270–273 MR0090741 0077.30002 CrossrefISIGoogle Scholar[21] Felix E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann., 142 (1960/1961), 22–130 10.1007/BF01343363 MR0209909 0104.07502 CrossrefISIGoogle Scholar[22] Felix E. Browder, Asymptotic distribution of eigenvalues and eigenfunctions for non-local elliptic boundary value problems. I, Amer. J. Math., 87 (1965), 175–195 MR0174858 0143.14403 CrossrefISIGoogle Scholar[23] F. H. Brownell, An extension of Weyl's asymptotic law for eigenvalues, Pacific J. Math., 5 (1955), 483–499 MR0080252 0066.08602 CrossrefGoogle Scholar[24] F. H. Brownell, Extended asymptotic eigenvalue distributions for bounded domains in n-space, J. Math. Mech., 6 (1957), 119–166 MR0084686 0084.30901 ISIGoogle Scholar[25] F. H. Brownell and , C. W. Clark, Asymptotic distribution of the eigenvalues of the lower part of the Schrödinger operator spectrum, J. Math. Mech., 10 (1961), 525–527 MR0125316 0148.09502 Google Scholar[26] S. Bochner, Zeta functions and Green's functions for linear partial differential operators of elliptic type with constant coefficients, Ann. of Math. (2), 57 (1953), 32–56 MR0054830 0050.08001 CrossrefISIGoogle Scholar[27A] F. J. Bureau, Asymptotic representation of the spectral function of self-adjoint elliptic operators of the second order with variable coefficients, J. Math. Anal. Appl., 1 (1960), 423–483 10.1016/0022-247X(60)90014-7 MR0157103 0099.30402 CrossrefGoogle Scholar[27B] F. J. Bureau, Asymptotic representation of the spectral function of self-adjoint elliptic operators of the second order with variable coefficients. II, J. Math. Anal. Appl., 4 (1962), 181–192 10.1016/0022-247X(62)90048-3 MR0157104 0103.06801 CrossrefGoogle Scholar[28] F. J. Bureau and , E. J. Pellicciaro, Asymptotic behavior of the spectral matrix of the operator of elasticity, J. Soc. Indust. Appl. Math., 10 (1962), 1–18 10.1137/0110001 MR0160034 0109.08502 LinkISIGoogle Scholar[29] T. Carleman, Sur les équations intégrales singulières à noyau réel et symétrique, Uppsala University Årsskrift, 1923 Google Scholar[30] T. Carleman, Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes, 8th Scand. Mat. Congress (1934), Collected Works, Malmö, 1960, 471–483 Google Scholar[31] T. Carleman, Über die asymptotische Verteilung des Eigenwerte partieller Differentialgleichungen, Ber. des Sächs. Akad. d. Wiss. Leipzig, 88 (1936), 119–132, Collected Works, Malmö, 1960, pp. 483–497 0017.11402 Google Scholar[32] T. W. Chaundy and , J. B. McLeod, Eigenvalues in problems with spherical symmetry and an associated integral, Quart. J. Math. Oxford Ser. (2), 14 (1963), 205–212 MR0153968 0125.05801 CrossrefISIGoogle Scholar[33] Colin Clark, An asymptotic formula for the eigenvalues of the Laplacian operator in an unbounded domain, Bull. Amer. Math. Soc., 72 (1966), 709–712 MR0196258 0145.14802 CrossrefISIGoogle Scholar[34] Colin Clark and , Denton Hewgill, One can hear whether a drum has finite area, Proc. Amer. Math. Soc., 18 (1967), 236–237 MR0213757 0146.34904 CrossrefISIGoogle Scholar[35] Colin Clark, The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems, SIAM Rev., 9 (1967), 627–646 10.1137/1009105 MR0510064 0159.19704 LinkISIGoogle Scholar[36] R. Courant, Über die Abhängigkeit der Schwingungszahlen einer Membran von ihrer Begrenzung und über asymptotische Eigenwertverteilung, Göttinger Nachr., (1919), 255–264 Google Scholar[37] R. Courant, Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik, Math. Z., 7 (1920), 1–57 10.1007/BF01199396 MR1544417 CrossrefGoogle Scholar[38] R. Courant, Über die Schwingungen eingespannter Platten, Math. Z., 15 (1922), 195–200 10.1007/BF01494393 MR1544567 CrossrefGoogle Scholar[39] R. Courant and , D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953xv+561 MR0065391 0053.02805 Google Scholar[40] M. S. P. Eastham, The distribution of the eigenvalues in certain eigenvalue problems, Quart. J. Math. Oxford Ser. (2), 15 (1964), 251–257 MR0165234 0173.13601 CrossrefISIGoogle Scholar[41] Gunnar Ehrling, On a type of eigenvalue problems for certain elliptic differential operators, Math. Scand., 2 (1954), 267–285 MR0067311 0056.32304 CrossrefGoogle Scholar[42] B. V. Fedosov, Asymptotic formulae for the eigenvalues of the Laplace operator in the case of a polygonal domain, Dokl. Akad. Nauk SSSR, 151 (1963), 786–789 MR0157095 0134.09303 Google Scholar[43] B. V. Fedosov, Asymptotic formulae for eigenvalues of the Laplace operator for a polyhedron, Dokl. Akad. Nauk SSSR, 157 (1964), 536–538 MR0164129 0133.36101 Google Scholar[44] G. Freud, Restglied eines Tauberschen Satzes. III, Acta Math. Acad. Sci. Hungar., 5 (1954), 275–289 10.1007/BF02020414 MR0072283 CrossrefGoogle Scholar[45] Tord Ganelius, On the remainder in a Tauberian theorem, Kungl. Fysiog. Sällsk. i Lund Förh., 24 (1954), 6 pp. (1955) MR0071562 0057.09204 Google Scholar[46] Lars Gårding, Dirichlet's problem and the vibration problem for linear elliptic partial differential equations with constant coefficients, Proceedings of the Symposium on Spectral Theory and Differential Problems, Oklahoma Agricultural and Mechanical College, Stillwater, Okla., 1951, 291–299 MR0052659 0067.07701 Google Scholar[47] Lars Gårding, On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differential operators, Math. Scand., 1 (1953), 237–255 MR0064980 0053.39102 CrossrefGoogle Scholar[48] Lars Gårding, On the asymptotic properties of the spectral function belonging to a self-adjoint semi-bounded extension of an elliptic differential operator, Kungl. Fysiog. Sällsk. i Lund Förh., 24 (1954), 18 pp. (1955) MR0071626 0058.08802 Google Scholar[49] M. G. Gasymov and , B. M. Levitan, The asymptotic behaviour of the spectral functions of the Schrödinger operator near a planar part of the boundary, Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 527–552 MR0165233 Google Scholar[50] I. M. Gelfand and , A. M. Jaglom, Integration in functional spaces and its applications in quantum physics, J. Mathematical Phys., 1 (1960), 48–69 10.1063/1.1703636 MR0112604 0092.45105 CrossrefISIGoogle Scholar[51] I. M. Glazman, Direct Methods in the Qualitative Spectral Analysis of Singular Differential Operators, Fizmatgiz, Moscow, 1963 Google Scholar[52] I. M. Glazman and , B. Ja. Skaček, On the discrete part of the spectrum of the Laplacian in limit-cylindrical regions, Dokl. Akad. Nauk SSSR, 147 (1962), 760–763 MR0180754 0168.08902 Google Scholar[53] V. N. Gorčakov, Asymptotic properties of the spectral function of hypoelliptic operators, Dokl. Akad. Nauk SSSR, 160 (1965), 746–749 MR0180760 0143.13401 Google Scholar[54] S. I. Grinberg, Certain asymptotic formulae for the spectra of the first and third boundary value problems, Teor. Funkcii˘ Funkcional. Anal. i Priložen. Vyp., 3 (1966), 117–134 MR0204888 Google Scholar[55] Dieter Gromes, Über die asymptotische Verteilung der Eigenwerte des Laplace-Operators für Gebiete auf der Kugeloberfläche, Math. Z., 94 (1966), 110–121 10.1007/BF01118974 MR0199575 0146.35002 CrossrefISIGoogle Scholar[56] G. H. Hardy and , J. E. Littlewood, Notes on the theory of series XI: On Tauberian theorems, Proc. London Math. Soc. (2), 30 (1930), 23–37 CrossrefGoogle Scholar[57] Denton Hewgill, On the eigenvalues of the Laplacian in an unbounded domain, Arch. Rational Mech. Anal., 27 (1967), 153–164 10.1007/BF00281340 MR0217456 0155.17102 CrossrefISIGoogle Scholar[58] Lars Hörmander, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965–1966), Belfer Graduate School of Science, Yeshiva Univ., New York, 1969, 155–202 MR0257589 Google Scholar[59] S. Ikehara, An extension of Landau's theorem in the analytic theory of numbers, J. Math. and Phys., 10 (1931), 1–12 0001.12902 CrossrefGoogle Scholar[60] D. S. Jones, The eigenvalues of $\nabla\sp 2u+\lambda u=0$ when the boundary conditions are given on semi-infinite domains, Proc. Cambridge Philos. Soc., 49 (1953), 668–684 MR0058086 0051.07704 CrossrefISIGoogle Scholar[61] M. Kac, On some connections between probability theory and differential and integral equations, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, 189–215 MR0045333 0045.07002 Google Scholar[62] Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1–23 MR0201237 0139.05603 CrossrefISIGoogle Scholar[63] J. Karamata, Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, welche die Laplacesche und Stieltjessche Transformation betreffen, J. Reine Angew. Math., 164 (1931), 27–39 0001.27302 Google Scholar[64] M. V. Keldysh, On the eigenvalues and eigenfunctions of some classes of nonself-adjoint operators, Dokl. Akad. Nauk SSSR, 77 (1951), 11–14 Google Scholar[65] A. G. Kostjučenko, The asymptotic distribution of eigenvalues of elliptic operators, Dokl. Akad. Nauk SSSR, 158 (1964), 41–44 MR0167713 0135.32007 Google Scholar[66] Takeshi Kotake and , Mudumbai S. Narasimhan, Regularity theorems for fractional powers of a linear elliptic operator, Bull. Soc. Math. France, 90 (1962), 449–471 MR0149329 0104.32503 CrossrefGoogle Scholar[67] N. V. Kuznecov, Asymptotic distribution of eigenfrequencies of a plane membrane in the case of separable variables, Differencialnye Uravnenija, 2 (1966), 1385–1402 MR0206524 0202.25201 Google Scholar[68] B. M. Levitan, On expansion in eigenfunctions of the Laplace operator, Mat. Sb. N.S., 35(77) (1954), 267–316 MR0071624 Google Scholar[69] B. M. Levitan, On the asymptotic behavior of the spectral function and expansion in eigenfunctions of the equation $\Delta u+ \{ \lambda -q(x\sb{1},x\sb{2},x\sb{3})\}u=0$, Amer. Math. Soc. Transl. (2), 20 (1962), 1–53 MR0136856 0122.33703 Google Scholar[70] B. M. Levitan, On expansion in eigenfunctions of the Schrödinger operator in the case of a potential increasing without bound, Dokl. Akad. Nauk SSSR (N.S.), 103 (1955), 191–194; erratum 105 (1955), 620 MR0074678 Google Scholar[71] B. M. Levitan, Certain questions in the spectral theory of self-adjoint differential operators, Uspehi Mat. Nauk (N.S.), 11 (1956), 117–144 MR0104043 Google Scholar[72] B. M. Levitan, On the asymptotic behavior of Green's function and its expansion in eigenfunctions of Schrödinger's equation, Mat. Sb. N. S., 41(83) (1957), 439–458 MR0089353 Google Scholar[73] V. A. Marčenko, Theorems of Tauberian type in spectral analysis of differential operators, Izv. Akad. Nauk SSSR. Ser. Mat., 19 (1955), 381–422 MR0076147 Google Scholar[74] H. P. McKean, Jr. and , I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry, 1 (1967), 43–69 MR0217739 0198.44301 CrossrefGoogle Scholar[75] J. B. McLeod, The distribution of the eigenvalues for the hydrogen atom and similar cases, Proc. London Math. Soc. (3), 11 (1961), 139–158 MR0123054 0094.06103 CrossrefGoogle Scholar[76] S. Minakshisundaram, A generalization of Epstein zeta functions. With a supplementary note by Hermann Weyl, Canadian J. Math., 1 (1949), 320–327 MR0032861 0034.05103 CrossrefISIGoogle Scholar[77] S. Minakshisundaram, Lattice point problems and eigenvalue problems, Proceedings of the Symposium on Spectral Theory and Differential Problems, Oklahoma Agricultural and Mechanical College, Stillwater, Okla., 1951, 325–332 MR0043320 0067.08103 Google Scholar[78] S. Minakshisundaram and , Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian J. Math., 1 (1949), 242–256 MR0031145 0041.42701 CrossrefISIGoogle Scholar[79] Sigeru Mizohata, Sur les propriétés asymptotiques des valeurs propres pour les opérateurs elliptiques, J. Math. Kyoto Univ., 4 (1964/1965), 399–428 MR0185270 0147.09602 CrossrefGoogle Scholar[80] Sigeru Mizohata and , Reiko Arima, Propriétés asymptotiques des valeurs propres des opérateurs elliptiques auto-adjoints, J. Math. Kyoto Univ., 4 (1964), 245–254 MR0179464 0137.30102 CrossrefGoogle Scholar[81] Edward Nelson, Feynman integrals and the Schrödinger equation, J. Mathematical Phys., 5 (1964), 332–343 10.1063/1.1704124 MR0161189 0133.22905 CrossrefISIGoogle Scholar[82] Nils Nilsson, Some estimates for eigenfunction expansions and spectral functions corresponding to elliptic differential operators, Math. Scand., 9 (1961), 107–121 MR0124612 0098.06801 CrossrefGoogle Scholar[83] Jan Odhnoff, Operators generated by differential problems with eigenvalue parameter in equation and boundary condition, Medd. Lunds Univ. Mat. Sem., 14 (1959), 79 pp; corrections, unbound insert MR0109249 0138.36103 Google Scholar[84] J. Peetre, Remark on eigenfunction expansions for elliptic operators with constant coefficients, Math. Scand., 15 (1964), 83–92 MR0178252 0131.09802 CrossrefGoogle Scholar[85] Jaak Peetre, Some estimates for spectral functions, Math. Z., 92 (1966), 146–153 10.1007/BF01312433 MR0192368 0151.20204 CrossrefISIGoogle Scholar[86] Åke Pleijel, Propriétés asymptotiques des fonctions fondamentales du problème des vibrations dans un corps élastique, Ark. Mat., Astr. Fys., 26 (1939), 9– MR0000338 0021.31803 Google Scholar[87] Åke Pleijel, Propriétés asymptotiques des fonctions et valeurs propres de certains problèmes de vibrations, Ark. Mat. Astr. Fys., 27A (1940), 101– MR0003911 0023.12402 Google Scholar[88] Åke Pleijel, Sur la distribution des valeurs propres de problèmes régis par l'équation $\Delta u+\lambda k(x,y) u=0$, Ark. Mat. Astr. Fys., 29B (1943), 8– MR0012359 0028.06601 Google Scholar[89] Åke Pleijel, Asymptotic relations for the eigenfunctions of certain boundary problems of polar type, Amer. J. Math., 70 (1948), 892–907 MR0027410 0036.06802 CrossrefISIGoogle Scholar[90] Åke Pleijel, On the eigenvalues and eigenfunctions of elastic plates, Comm. Pure Appl. Math., 3 (1950), 1–10 MR0037459 0040.05403 CrossrefISIGoogle Scholar[91] Åke Pleijel, Green's functions and asymptotic distribution of eigenvalues and eigenfunctions, Proceedings of the Symposium on Spectral Theory and Differential Problems, Oklahoma Agricultural and Mechanical College, Stillwater, Okla., 1951, 439–454 MR0047892 0068.08503 Google Scholar[92] Åke Pleijel, Sur les valeurs et les fonctions propres des membranes vibrantes, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], 1952 (1952), 173–180 MR0052019 0048.08004 Google Scholar[93] Åke Pleijel, On Green's functions and the eigenvalue-distribution of the three-dimensional membrane equation, Tolfte Skandinaviska Matematikerkongressen, Lund, 1953, Lunds Universitets Matematiska Institution, Lund, 1954, 222–240 MR0074674 0056.09701 Google Scholar[94] Åke Pleijel, A study of certain Green's functions with applications in the theory of vibrating membranes, Ark. Mat., 2 (1954), 553–569 MR0061257 0055.08801 CrossrefGoogle Scholar[95] Å. Pleijel, On the problem of improving Weyl's law for the asymptotic eigenvalue distributionConvegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954, Edizioni Cremonese, Roma, 1955, 69–75 MR0074675 0068.08501 Google Scholar[96] Åke Pleijel, On the eigenfunctions of the membrane equation in a singular case, Proceedings of the conference on differential equations (dedicated to A. Weinstein), University of Maryland Book Store, College Park, Md., 1956, 197–207 MR0080859 0071.09901 Google Scholar[97] A˙ke Pleijel, Certain indefinite differential eigenvalue problems—the asymptotic distribution of their eigenfunctions, Partial differential equations and continuum mechanics, Univ. of Wisconsin Press, Madison, Wis., 1961, 19–37 MR0123854 0111.09502 Google Scholar[98A] A˙ke Pleijel, A bilateral Tauberian theorem, Ark. Mat., 4 (1963), 561–571 (1963) MR0158200 0107.32001 Google Scholar[98B] A˙ke Pleijel, Remark to my previous paper on a bilateral Tauberian theorem, Ark. Mat., 4 (1963), 573–574 (1963) MR0158201 0107.32002 Google Scholar[99] Daniel Ray, On spectra of second-order differential operators, Trans. Amer. Math. Soc., 77 (1954), 299–321 MR0066539 0058.32901 CrossrefISIGoogle Scholar[100] Franz Rellich, Das Eigenwertproblem von Δu+λu=0 in Halbröhren, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, 329–344 MR0023428 0035.06402 Google Scholar[101] R. T. Seeley, The powers A8 of an elliptic operator A, to appear Google Scholar[102] B. Ja. Skaček, Asymptotic behaviour of the negative part of the spectrum of higher-order singular differential operators, Dopovidi Akad. Nauk Ukraïn. RSR, 1964 (1964), 14–17 MR0165393 Google Scholar[103] I. A. Solomešč, Eigenvalues of some degenerate elliptic equations, Mat. Sb. (N.S.), 54 (96) (1961), 295–310 MR0159111 Google Scholar[104] I. A. Solomešč, The asymptotic behavior of eigenvalues of bilinear forms associated with certain elliptic equations which degenerate on the boundary, Dokl. Akad. Nauk SSSR, 144 (1962), 727–729 MR0140825 0171.08403 Google Scholar[105] I. A. Solomešč, On the asymptotic behaviour of eigenvalues of bilinear forms associated with degenerate elliptic equations, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom, 19 (1964), 163–166 MR0170097 Google Scholar[106] E. C. Titchmarsh, Eigenfunction Expansions, Vol. I and II, Oxford University Press, Oxford, 1953 and 1958 Google Scholar[107A] E. C. Titchmarsh, On the eigenvalues in problems with spherical symmetry, Proc. Roy. Soc. London. Ser. A, 245 (1958), 147–155 MR0096032 0084.09401 CrossrefISIGoogle Scholar[107B] E. C. Titchmarsh, On the eigenvalues in problems with spherical symmetry. II, Proc. Roy. Soc. London Ser. A, 251 (1959), 46–54 MR0104045 0084.30801 CrossrefGoogle Scholar[107C] E. C. Titchmarsh, On the eigenvalues in problems with spherical symmetry. III, Proc. Roy. Soc. London. Ser. A, 252 (1959), 436–444 MR0108634 0092.10503 CrossrefISIGoogle Scholar[108] Bui An Ton, Asymptotic distribution of eigenvalues and eigenfunctions for general linear elliptic boundary value problems, Trans. Amer. Math. Soc., 122 (1966), 516–546 MR0198016 0139.05802 CrossrefISIGoogle Scholar[109] J. S. de Wet and , F. Mandl, On the asymptotic distribution of eigenvalues, Proc. Roy. Soc. London. Ser. A., 200 (1950), 572–580 MR0035369 0040.33702 CrossrefISIGoogle Scholar[110] H. Weyl, Über die asymptotische Verteilung der Eigenwerte, Göttinger Nachr., (1911), 110–117 Google Scholar[111] Hermann Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441–479 10.1007/BF01456804 MR1511670 CrossrefISIGoogle Scholar[112] H. Weyl, Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung, J. Reine Angew. Math., 141 (1912), 1–11 Google Scholar[113] H. Weyl, Über das Spektrum der Hohlraumstrahlung, J. Reine Angew. Math., 141 (1912), 163–181 Google Scholar[114] H. Weyl, Über die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetze, J. Reine Angew. Math., 143 (1913), 177–202 CrossrefGoogle Scholar[115] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers, Rend. Circ. Mat. Palermo, 39 (1915), 1–49 CrossrefGoogle Scholar[116] Hermann Weyl, Ramifications, old and new, of the eigenvalue problem, Bull. Amer. Math. Soc., 56 (1950), 115–139 MR0034940 0041.21003 CrossrefISIGoogle Scholar[117] Norbert Wiener, Tauberian theorems, Ann. of Math. (2), 33 (1932), 1–100 MR1503035 0004.05905 CrossrefGoogle Scholar[118] Wen-Lin Yin, The lattice-points in a circle, Sci. Sinica, 11 (1962), 10–15 MR0159792 0158.29803 Google Scholar[A1] Marcel Berger, Sur le spectre d'une variété riemannienne, C. R. Acad. Sci. Paris Sér. A-B, 263 (1966), A13–A16 MR0202090 0141.38203 ISIGoogle Scholar[A2] P. Greiner, On zeta functions connected with elliptic differential operators, to appear Google Scholar Next article FiguresRelatedReferencesCited ByDetails Spectral invariants of the perturbed polyharmonic Steklov problemCalculus of Variations and Partial Differential Equations, Vol. 61, No. 4 | 5 May 2022 Cross Ref Sharp estimates for approximation numbers of non-periodic Sobolev embeddingsJournal of Complexity, Vol. 54 | 1 Oct 2019 Cross Ref Estimates of Wiman–Valiron type for evolution equationsDifferential Equations, Vol. 53, No. 8 | 14 September 2017 Cross Ref On asymptotic properties of biharmonic Steklov eigenvaluesJournal of Differential Equations, Vol. 261, No. 9 | 1 Nov 2016 Cross Ref Nonharmonic Analysis of Boundary Value ProblemsInternational Mathematics Research Notices, Vol. 2016, No. 12 | 2 September 2015 Cross Ref Asymptotic behavior of eigenvalues of hydrogen atom equationBoundary Value Problems, Vol. 2015, No. 1 | 2 June 2015 Cross Ref Asymptotic expansion of the trace of the heat kernel associated to the Dirichlet-to-Neumann operatorJournal of Differential Equations, Vol. 259, No. 7 | 1 Oct 2015 Cross Ref Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck OperatorsFoundations of Computational Mathematics, Vol. 12, No. 5 | 24 May 2012 Cross Ref Regularized trace of the inverse of the dirichlet laplacianCommunications on Pure and Applied Mathematics, Vol. 64, No. 8 | 29 March 2011 Cross Ref Vacuum stress and closed paths in rectangles, pistons and pistolsJournal of Physics A: Mathematical and Theoretical, Vol. 42, No. 15 | 24 March 2009 Cross Ref Estimate of the second term in the spectral asymptotic of Cauchy transformJournal of Functional Analysis, Vol. 249, No. 1 | 1 Aug 2007 Cross Ref Weyl formula: Experimental test of ray splitting and corner correctionsPhysical Review E, Vol. 72, No. 5 | 17 November 2005 Cross Ref The Cesàro behaviour of distributionsProceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 454, No. 1977 | 8 September 1998 Cross Ref Elliptic Boundary ProblemsPartial Differential Equations IX | 1 Jan 1997 Cross Ref The resolvent parametrix of the general elliptic linear differential operator: a closed form for the intrinsic symbolTransactions of the American Mathematical Society, Vol. 310, No. 2 | 1 January 1988 Cross Ref Temperature, periodicity and horizonsPhysics Reports, Vol. 152, No. 3 | 1 Aug 1987 Cross Ref Chaos on the pseudospherePhysics Reports, Vol. 143, No. 3 | 1 Nov 1986 Cross Ref Can one hear the dimension of a fractal?Communications in Mathematical Physics, Vol. 104, No. 1 | 1 Mar 1986 Cross Ref Spectral asymptotics of differential and pseudodifferential operators. IJournal of Soviet Mathematics, Vol. 31, No. 4 | 1 Nov 1985 Cross Ref Schrödinger semigroups for vector fieldsJournal of Mathematical Physics, Vol. 26, No. 3 | 1 Mar 1985 Cross Ref Perturbational approach for the curvature correction to the surface tension via the Hartree-Fock approximation for the plane surfaceAnnals of Physics, Vol. 159, No. 2 | 1 Feb 1985 Cross Ref The Relation Between Quantum Mechanics And Classical Mechanics: A Survey Of Some Mathematical AspectsChaotic Behavior in Quantum Systems | 1 Jan 1985 Cross Ref Eigenvalues of the Laplacian in Two DimensionsJ. R. Kuttler and V. G. SigillitoSIAM Review, Vol. 26, No. 2 | 10 July 2006AbstractPDF (2896 KB)Asymptotic formulas for the eigenvalues of the Sturm-Liouville operatorSiberian Mathematical Journal, Vol. 24, No. 4 | 1 Jan 1984 Cross Ref Schrödinger spectral kernels: High order asymptotic expansionsJournal of Mathematical Physics, Vol. 24, No. 6 | 1 Jun 1983 Cross Ref Time evolution kernels: uniform asymptotic expansionsJournal of Mathematical Physics, Vol. 24, No. 5 | 1 May 1983 Cross Ref The second term of the asymptotic distribution of eigenvalues of the laplacian in the polygonal domainCommunications in Partial Differential Equations, Vol. 8, No. 15 | 8 May 2007 Cross Ref The Local Geometric Asymptotics of Continuum Eigenfunction Expansions I. OverviewS. A. FullingSIAM Journal on Mathematical Analysis, Vol. 13, No. 6 | 17 February 2012AbstractPDF (2508 KB)Asymptotics of the spectral function of an elliptic differential operator of second orderJournal of Soviet Mathematics, Vol. 19, No. 4 | 1 Jan 1982 Cross Ref Asymptotic formulae with remainder estimates for eigenvalues of schrödinger operatorsCommunications in Partial Differential Equations, Vol. 7, No. 1 | 23 December 2010 Cross Ref The Local Asymptotics of Continuum Eigenfunciton ExpansionsSpectral Theory of Differential Operators, Proceedings of the Conference held at the University of Alabama in Birmingham | 1 Jan 1981 Cross Ref PHYSICAL STATES AND RENORMALIZED OBSERVABLES IN QUANTUM FIELD THEORIES WITH EXTERNAL GRAVITY**Report presented at the Seminar on Quantum Gravity, Moscow, Dec. 5–7, 1978, and at the New Orleans Conference on Quantum Theory and Gravitation, May 23–26, 1979. Research supported in part by National Science Foundation Grant No. PHY 77-01432.Quantum Theory and Gravitation | 1 Jan 1980 Cross Ref On the 𝐿^{∞}-convergence of Galerkin approximations for second-order hyperbolic equationsMathematics of Computation, Vol. 34, No. 150 | 1 January 1980 Cross Ref Asymptotic behavior of the spectrum of differential equationsJournal of Soviet Mathematics, Vol. 12, No. 3 | 1 Sep 1979 Cross Ref Finite temperature and boundary effects in static space-timesJournal of Physics A: Mathematical and General, Vol. 11, No. 5 | 18 January 2001 Cross Ref A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of R3Advances in Mathematics, Vol. 29, No. 2 | 1 Feb 1978 Cross Ref Planck's radiation law for finite cavities and related problemsInfrared Physics, Vol. 16, No. 1-2 | 1 Jan 1976 Cross Ref PLANCK'S RADIATION LAW FOR FINITE CAVITIES AND RELATED PROBLEMSInternational Conference on Infrared Physics (CIRP) | 1 Jan 1976 Cross Ref Abelian and tauberian theorems for the Stefan-Boltzmann formulaPhysica, Vol. 67, No. 2 | 1 Jul 1973 Cross Ref Asymptotic Eigenvalue Distribution for the Wave Equation in a Cylinder of Arbitrary Cross SectionPhysical Review A, Vol. 6, No. 6 | 1 December 1972 Cross Ref Progress in Weyl's problem achieved by computational methodsComputer Physics Communications, Vol. 4, No. 2 | 1 Nov 1972 Cross Ref Edge and corner contributions in Weyl's problemPhysics Letters A, Vol. 38, No. 7 | 1 Mar 1972 Cross Ref An asymptotic expansion for the heat equationArchive for Rational Mechanics and Analysis, Vol. 41, No. 3 | 1 Jan 1971 Cross Ref A comparison theorem for operators with compact resolventProceedings of the American Mathematical Society, Vol. 27, No. 3 | 1 January 1971 Cross Ref Distribution of eigenfrequencies for the wave equation in a finite domainAnnals of Physics, Vol. 60, No. 2 | 1 Oct 1970 Cross Ref Volume 9, Issue 4| 1967SIAM Review627-777 History Submitted:03 April 1967Published online:18 July 2006 InformationCopyright © 1967 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1009105Article page range:pp. 627-646ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics
Referência(s)