100 years of Weyl’s law
We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl princip...
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| Published in | Bulletin of mathematical sciences Vol. 6; no. 3; pp. 379 - 452 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.10.2016
World Scientific Publishing Co. Pte., Ltd World Scientific Publishing |
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| Abstract | We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schrödinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field. |
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| AbstractList | We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schrödinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field. Abstract We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schrödinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field. |
| Author | Ivrii, Victor |
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| Cites_doi | 10.1070/RM1971v026n06ABEH001274 10.1080/03605300701741099 10.1007/BF01456804 10.1007/BF02391913 10.1090/S0002-9904-1950-09369-0 10.1007/BF01086550 10.2307/2374196 10.1007/BF02097241 10.1002/cpa.3160470406 10.1007/BF03015971 10.1142/S0129055X02001491 10.1016/0001-8708(78)90013-0 10.1080/03605309808821395 10.1007/BF01199396 10.5802/aif.1731 10.1007/BF01081624 10.1002/9783527628025.ch1 10.1007/BF02099414 10.1007/BF01405172 10.1081/PDE-120020493 10.1007/978-3-662-02725-7_34 10.1007/978-3-662-12496-3 10.1090/trans2/150/02 10.1142/S0129055X94000328 10.1112/plms/pdt036 10.1515/crll.1913.143.177 10.1007/BF02055756 10.1090/mmono/155 10.1007/BF01473886 10.1007/978-3-662-06719-2_1 |
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| Keywords | Eigenvalue Counting Function Asymptotic Pointwise Thomas-Fermi Theory Microhyperbolicity Remainder Estimate |
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| References | Lorentz (CR20) 1910; 11 Weyl (CR38) 1913; 143 Ivrii (CR12) 1982; 16 Raikov (CR22) 1998; 23 Seeley (CR30) 1978; 102 Lieb (CR17) 1991 Carleman (CR5) 1936; 88 Avakumovič (CR2) 1956; 65 CR15 CR14 CR13 Seeley (CR31) 1980; 102 CR35 Bach (CR3) 1992; 147 Sommerfeld (CR34) 1910; 11 CR33 CR10 CR32 Raikov (CR23) 1999; 49 Weyl (CR39) 1915; 39 Rozenblum, Tashchiyan (CR27) 2008; 33 Duistermaat, Guillemin (CR7) 1975; 29 Levitan (CR16) 1971; 26 CR4 Weyl (CR37) 1912; 141 CR8 Hörmander (CR9) 1968; 121 CR29 CR28 Raikov, Warzel (CR24) 2002; 14 Courant (CR6) 1920; 7 CR26 CR25 Weyl (CR41) 1950; 56 Lieb, Solovej, Yngvarsson (CR19) 1994; 161 CR40 Melgaard, Rozenblum (CR21) 2003; 28 Arendt, Nittka, Peter, Steiner, Arendt, Schleich (CR1) 2009 Weyl (CR36) 1912; 71 Lieb, Solovej, Yngvarsson (CR18) 1994; 47 Ivrii (CR11) 1980; 14 89_CR28 89_CR26 89_CR25 VG Avakumovič (89_CR2) 1956; 65 M Melgaard (89_CR21) 2003; 28 89_CR29 H Weyl (89_CR38) 1913; 143 G Rozenblum (89_CR27) 2008; 33 89_CR40 JJ Duistermaat (89_CR7) 1975; 29 V Ivrii (89_CR11) 1980; 14 V Ivrii (89_CR12) 1982; 16 HA Lorentz (89_CR20) 1910; 11 W Arendt (89_CR1) 2009 EH Lieb (89_CR19) 1994; 161 EH Lieb (89_CR18) 1994; 47 GD Raikov (89_CR22) 1998; 23 H Weyl (89_CR39) 1915; 39 H Weyl (89_CR36) 1912; 71 L Hörmander (89_CR9) 1968; 121 89_CR15 EH Lieb (89_CR17) 1991 T Carleman (89_CR5) 1936; 88 89_CR14 R Seeley (89_CR30) 1978; 102 R Courant (89_CR6) 1920; 7 R Seeley (89_CR31) 1980; 102 A Sommerfeld (89_CR34) 1910; 11 89_CR13 89_CR35 89_CR8 GD Raikov (89_CR23) 1999; 49 H Weyl (89_CR41) 1950; 56 89_CR33 H Weyl (89_CR37) 1912; 141 89_CR10 89_CR32 89_CR4 GD Raikov (89_CR24) 2002; 14 V Bach (89_CR3) 1992; 147 BM Levitan (89_CR16) 1971; 26 |
| References_xml | – volume: 26 start-page: 165 issue: 6 year: 1971 end-page: 232 ident: CR16 article-title: Asymptotic behaviour of the spectral function of elliptic operator publication-title: Russ. Math. Surv. doi: 10.1070/RM1971v026n06ABEH001274 – volume: 33 start-page: 1048 issue: 4–6 year: 2008 end-page: 1081 ident: CR27 article-title: On the spectral properties of the perturbed Landau Hamiltonian publication-title: Commun. Partial Differ. Equ. doi: 10.1080/03605300701741099 – volume: 11 start-page: 1057 year: 1910 end-page: 1066 ident: CR34 article-title: Die Greensche Funktion der Schwingungsgleichung für ein beliebiges Gebiet publication-title: Physikal. Zeitschr. – volume: 141 start-page: 1 year: 1912 end-page: 11 ident: CR37 article-title: Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung publication-title: J. Für die Angew. Math. – ident: CR4 – ident: CR14 – volume: 71 start-page: 441 year: 1912 end-page: 479 ident: CR36 article-title: Das asymptotische Verteilungsgesetz linearen partiellen Differentialgleichungen publication-title: Math. Ann. doi: 10.1007/BF01456804 – ident: CR10 – volume: 121 start-page: 193 year: 1968 end-page: 218 ident: CR9 article-title: The spectral function of an elliptic operator publication-title: Acta Math. doi: 10.1007/BF02391913 – ident: CR33 – volume: 88 start-page: 119 year: 1936 end-page: 132 ident: CR5 article-title: Über die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen publication-title: Ber. Sachs. Acad. Wiss. Leipzig – volume: 56 start-page: 115 issue: 2 year: 1950 end-page: 139 ident: CR41 article-title: Ramifications, old and new, of the eigenvalue problem publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1950-09369-0 – volume: 14 start-page: 98 issue: 2 year: 1980 end-page: 106 ident: CR11 article-title: Second term of the spectral asymptotic expansion for the Laplace–Beltrami operator on manifold with boundary publication-title: Funct. Anal. Appl. doi: 10.1007/BF01086550 – ident: CR35 – ident: CR29 – volume: 102 start-page: 869 issue: 3 year: 1980 end-page: 902 ident: CR31 article-title: An estimate near the boundary for the spectral function of the Laplace operator publication-title: Am. J. Math. doi: 10.2307/2374196 – volume: 147 start-page: 527 year: 1992 end-page: 548 ident: CR3 article-title: Error bound for the Hartree–Fock energy of atoms and molecules publication-title: Commun. Math. Phys. doi: 10.1007/BF02097241 – ident: CR8 – volume: 47 start-page: 513 year: 1994 end-page: 591 ident: CR18 article-title: Asymptotics of heavy atoms in high magnetic fields: I. Lowest Landau band regions publication-title: Commun. Pure Appl. Math. doi: 10.1002/cpa.3160470406 – ident: CR40 – ident: CR25 – volume: 39 start-page: 1 year: 1915 end-page: 49 ident: CR39 article-title: Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers publication-title: Rend. Circ. Mat. Palermo. doi: 10.1007/BF03015971 – volume: 14 start-page: 1051 year: 2002 end-page: 1072 ident: CR24 article-title: Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials publication-title: Rev. Math. Phys. doi: 10.1142/S0129055X02001491 – volume: 102 start-page: 244 issue: 3 year: 1978 end-page: 264 ident: CR30 article-title: A sharp asymptotic estimate for the eigenvalues of the Laplacian in a domain of publication-title: Adv. Math. doi: 10.1016/0001-8708(78)90013-0 – volume: 23 start-page: 1583 year: 1998 end-page: 1620 ident: CR22 article-title: Eigenvalue asymptotics for the Schrödinger operator in strong constant magnetic fields publication-title: Commun. PDE doi: 10.1080/03605309808821395 – volume: 65 start-page: 324 year: 1956 end-page: 344 ident: CR2 article-title: Über die eigenfunktionen auf geschlossen riemannschen mannigfaltigkeiten publication-title: Math. Z. – volume: 7 start-page: 1 year: 1920 end-page: 57 ident: CR6 article-title: Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik publication-title: Mat. Z. doi: 10.1007/BF01199396 – volume: 49 start-page: 1603 year: 1999 end-page: 1636 ident: CR23 article-title: Eigenvalue asymptotics for the Pauli operator in strong non-constant magnetic fields publication-title: Ann. Inst. Fourier doi: 10.5802/aif.1731 – volume: 11 start-page: 1234 year: 1910 end-page: 1257 ident: CR20 article-title: Alte und neue Fragen der Physik publication-title: Physikal. Zeitschr. – ident: CR15 – volume: 16 start-page: 101 issue: 2 year: 1982 end-page: 108 ident: CR12 article-title: Accurate spectral asymptotics for elliptic operators that act in vector bundles publication-title: Funct. Anal. Appl. doi: 10.1007/BF01081624 – start-page: 1 year: 2009 end-page: 71 ident: CR1 article-title: Weyls law: spectral properties of the Laplacian in mathematics and physics publication-title: Mathematical Analysis of Evolution, Information, and Complexity doi: 10.1002/9783527628025.ch1 – ident: CR13 – volume: 161 start-page: 77 year: 1994 end-page: 124 ident: CR19 article-title: Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions publication-title: Commun. Math. Phys. doi: 10.1007/BF02099414 – volume: 143 start-page: 177 year: 1913 end-page: 202 ident: CR38 article-title: Über die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgeometrie publication-title: J. Reine Angew. Math. – ident: CR32 – volume: 29 start-page: 37 issue: 1 year: 1975 end-page: 79 ident: CR7 article-title: The spectrum of positive elliptic operators and periodic bicharacteristics publication-title: Invent. Math. doi: 10.1007/BF01405172 – ident: CR28 – volume: 28 start-page: 697 issue: 3–4 year: 2003 end-page: 736 ident: CR21 article-title: Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rank publication-title: Commun. Partial Differ. Equ. doi: 10.1081/PDE-120020493 – ident: CR26 – year: 1991 ident: CR17 publication-title: The Stability of Matter: From Atoms to Stars (Selecta) doi: 10.1007/978-3-662-02725-7_34 – volume: 88 start-page: 119 year: 1936 ident: 89_CR5 publication-title: Ber. Sachs. Acad. Wiss. Leipzig – volume: 49 start-page: 1603 year: 1999 ident: 89_CR23 publication-title: Ann. Inst. Fourier doi: 10.5802/aif.1731 – volume: 102 start-page: 869 issue: 3 year: 1980 ident: 89_CR31 publication-title: Am. J. Math. doi: 10.2307/2374196 – start-page: 1 volume-title: Mathematical Analysis of Evolution, Information, and Complexity year: 2009 ident: 89_CR1 doi: 10.1002/9783527628025.ch1 – volume: 28 start-page: 697 issue: 3–4 year: 2003 ident: 89_CR21 publication-title: Commun. Partial Differ. Equ. doi: 10.1081/PDE-120020493 – ident: 89_CR13 doi: 10.1007/978-3-662-12496-3 – volume: 26 start-page: 165 issue: 6 year: 1971 ident: 89_CR16 publication-title: Russ. Math. Surv. doi: 10.1070/RM1971v026n06ABEH001274 – volume: 47 start-page: 513 year: 1994 ident: 89_CR18 publication-title: Commun. Pure Appl. Math. doi: 10.1002/cpa.3160470406 – ident: 89_CR25 – ident: 89_CR4 – ident: 89_CR10 – ident: 89_CR28 doi: 10.1090/trans2/150/02 – volume: 56 start-page: 115 issue: 2 year: 1950 ident: 89_CR41 publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1950-09369-0 – volume: 147 start-page: 527 year: 1992 ident: 89_CR3 publication-title: Commun. Math. Phys. doi: 10.1007/BF02097241 – ident: 89_CR8 doi: 10.1142/S0129055X94000328 – ident: 89_CR32 doi: 10.1112/plms/pdt036 – ident: 89_CR14 – ident: 89_CR35 – ident: 89_CR33 – volume: 121 start-page: 193 year: 1968 ident: 89_CR9 publication-title: Acta Math. doi: 10.1007/BF02391913 – volume: 143 start-page: 177 year: 1913 ident: 89_CR38 publication-title: J. Reine Angew. Math. doi: 10.1515/crll.1913.143.177 – ident: 89_CR40 doi: 10.1007/BF02055756 – volume: 23 start-page: 1583 year: 1998 ident: 89_CR22 publication-title: Commun. PDE doi: 10.1080/03605309808821395 – volume: 71 start-page: 441 year: 1912 ident: 89_CR36 publication-title: Math. Ann. doi: 10.1007/BF01456804 – volume: 14 start-page: 1051 year: 2002 ident: 89_CR24 publication-title: Rev. Math. Phys. doi: 10.1142/S0129055X02001491 – volume: 16 start-page: 101 issue: 2 year: 1982 ident: 89_CR12 publication-title: Funct. Anal. Appl. doi: 10.1007/BF01081624 – volume: 161 start-page: 77 year: 1994 ident: 89_CR19 publication-title: Commun. Math. Phys. doi: 10.1007/BF02099414 – volume: 33 start-page: 1048 issue: 4–6 year: 2008 ident: 89_CR27 publication-title: Commun. Partial Differ. Equ. doi: 10.1080/03605300701741099 – volume: 141 start-page: 1 year: 1912 ident: 89_CR37 publication-title: J. Für die Angew. Math. – volume: 29 start-page: 37 issue: 1 year: 1975 ident: 89_CR7 publication-title: Invent. Math. doi: 10.1007/BF01405172 – volume-title: The Stability of Matter: From Atoms to Stars (Selecta) year: 1991 ident: 89_CR17 – ident: 89_CR29 doi: 10.1090/mmono/155 – volume: 11 start-page: 1057 year: 1910 ident: 89_CR34 publication-title: Physikal. Zeitschr. – ident: 89_CR15 – volume: 65 start-page: 324 year: 1956 ident: 89_CR2 publication-title: Math. Z. doi: 10.1007/BF01473886 – volume: 7 start-page: 1 year: 1920 ident: 89_CR6 publication-title: Mat. Z. doi: 10.1007/BF01199396 – volume: 102 start-page: 244 issue: 3 year: 1978 ident: 89_CR30 publication-title: Adv. Math. doi: 10.1016/0001-8708(78)90013-0 – volume: 14 start-page: 98 issue: 2 year: 1980 ident: 89_CR11 publication-title: Funct. Anal. Appl. doi: 10.1007/BF01086550 – volume: 11 start-page: 1234 year: 1910 ident: 89_CR20 publication-title: Physikal. Zeitschr. – volume: 39 start-page: 1 year: 1915 ident: 89_CR39 publication-title: Rend. Circ. Mat. 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| Snippet | We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the... Abstract We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to... |
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| Title | 100 years of Weyl’s law |
| URI | https://link.springer.com/article/10.1007/s13373-016-0089-y https://www.proquest.com/docview/1827830100 https://doaj.org/article/28ace18888fa4bc7b269c43dff4cd37c |
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