We study spectral asymptotics for a large class of differential operators on an open subset of $\mathbb {R}^d$ with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operators with non-homogeneous symbols. Based on a sharp estimate for the sum of the eigenvalues we establish the first term of the semiclassical asymptotics. This generalizes Weyl's law for the Laplace operator.
REFERENCES
1.
F. A.
Berezin
, “Convex functions of operators
,” Mat. Sb. (N.S.)
88
(130
), 268
–276
(1972
).2.
M. Š.
Birman
and M. Z.
Solomjak
, Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory
, American Mathematical Society Translations, Series 2
Vol. 114
(American Mathematical Society
, Providence, RI
, 1980
).3.
R. M.
Blumenthal
and R. K.
Getoor
, “The asymptotic distribution of the eigenvalues for a class of Markov operators
,” Pacific J. Math.
9
, 399
–408
(1959
).4.
E.
Di Nezza
, G.
Palatucci
, and E.
Valdinoci
, “Hitchhiker's guide to the fractional Sobolev spaces
,” Bull. Sci. Math.
136
(5
), 521
–573
(2012
).5.
R. L.
Frank
and L.
Geisinger
, “Refined semiclassical asymptotics for fractional powers of the Laplace operator
,” J. Reine Angew. Math. (to be published); preprint arXiv:1105:5181.6.
R. L.
Frank
and L.
Geisinger
, Two-Term Spectral Asymptotics for the Dirichlet Laplacian on a Bounded Domain
, Mathematical Results in Quantum Physics
(World Scientific Publishing
, Hackensack, NJ
, 2011
), pp. 138
–147
.7.
R. L.
Frank
, “Remarks on eigenvalue estimates and semigroup domination
,” Spectral and Scattering Theory for Quantum Magnetic Systems
, Contemporary Mathematics
Vol. 500
(American Mathematical Society
, Providence, RI
, 2009
), pp. 63
–86
.8.
L.
Geisinger
, A.
Laptev
, and T.
Weidl
, “Geometrical versions of improved Berezin-Li-Yau inequalities
,” J. Spectr. Theory
1
(1
), 87
–109
(2011
).9.
A. N.
Hatzinikitas
, “Spectral properties of the Dirichlet operator
,” $\sum _{i=1}^d (-\partial _i^2)^s$
on domains in d-dimensional Euclidean spaceJ. Math Phys.
54
, 103501
(2013
).10.
L.
Hörmander
, The Analysis of Linear Partial Differential Operators
(Springer-Verlag
, Berlin
, 1985
), Vol. 4
.11.
V. Ja.
Ivrii
, Microlocal Analysis and Precise Spectral Asymptotics
, Springer Monographs in Mathematics
(Springer-Verlag
, Berlin
, 1998
).12.
J.
Korevaar
, “Tauberian theory
,” Grundlehren der Mathematischen Wissenschaften
, Fundamental Principles of Mathematical Sciences
Vol. 329
(Springer-Verlag
, Berlin
, 2004
).13.
H.
Kovařík
, S.
Vugalter
, and T.
Weidl
, “Two dimensional Berezin-Li-Yau inequalities with a correction term
,” Commun. Math. Phys.
287
(3
), 959
–981
(2009
).14.
A.
Laptev
, “Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces
,” J. Funct. Anal.
151
(2
), 531
–545
(1997
).15.
P.
Li
and S. T.
Yau
, “On the Schrödinger equation and the eigenvalue problem
,” Commun. Math. Phys.
88
(3
), 309
–318
(1983
).16.
E. H.
Lieb
, “The classical limit of quantum spin systems
,” Commun. Math. Phys.
31
, 327
–340
(1973
).17.
E. H.
Lieb
and M.
Loss
, Analysis
, 2nd ed., Graduate Studies in Mathematics
Vol. 14
(American Mathematical Society
, Providence, RI
, 2001
).18.
A. D.
Melas
, “A lower bound for sums of eigenvalues of the Laplacian
,” Proc. Am. Math. Soc.
131
(2
), 631
–636
(2003
) (electronic).19.
M.
Reed
and B.
Simon
, Methods of Modern Mathematical Physics IV. Analysis of Operators
(Academic Press
, 1978
).20.
Y.
Safarov
and D.
Vassiliev
, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators
, Translations of Mathematical Monographs
Vol. 155
(American Mathematical Society
, Providence, RI
, 1997
).21.
T.
Weidl
, “Improved Berezin-Li-Yau inequalities with a remainder term
,” Am. Math. Soc. Transl.
225
(2
), 253
–263
(2008
).22.
H.
Weyl
, “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)
,” Math. Ann.
71
(4
), 441
–479
(1912
).23.
S.
Yildirim Yolcu
and T.
Yolcu
, “Bounds for the eigenvalues of the fractional Laplacian
,” Rev. Math. Phys.
24
(3
), 1250003
(2012
).24.
S.
Yildirim Yolcu
and T.
Yolcu
, “Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain
,” Commun. Contemp. Math.
15
(3
), 1250048
(2013
).25.
T.
Yolcu
, “Refined bounds for the eigenvalues of the Klein-Gordon operator
,” Proc. Am. Math. Soc.
141
(12
), 4305
–4315
(2013
).© 2014 AIP Publishing LLC.
2014
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