Weyl’s Laws and Connes’ Integration Formulas for Matrix-Valued $$L\!\log \!L$$ -Orlicz Potentials
Tóm tắt
Thanks to the Birman-Schwinger principle, Weyl’s laws for Birman-Schwinger operators yields semiclassical Weyl’s laws for the corresponding Schrödinger operators. In a recent preprint Rozenblum established quite general Weyl’s laws for Birman-Schwinger operators associated with pseudodifferential operators of critical order and potentials that are product of $$L\!\log \!L$$ -Orlicz functions and Alfhors-regular measures supported on a submanifold. In this paper, we show that, for matrix-valued $$L\!\log \!L$$ -Orlicz potentials supported on the whole manifold, Rozenblum’s results are direct consequences of the Cwikel-type estimates on tori recently established by Sukochev–Zanin. As applications we obtain CLR-type inequalities and semiclassical Weyl’s laws for critical Schrödinger operators associated with matrix-valued $$L\!\log \!L$$ -Orlicz potentials. Finally, we explain how the Weyl’s laws of this paper imply a strong version of Connes’ integration formula for matrix-valued $$L\!\log \!L$$ -Orlicz potentials.
Tài liệu tham khảo
citation_journal_title=Funct. Anal. Appl.; citation_title=Estimates for
-numbers, and spectral asymptotics for integral operators of potential type on nonsmooth surfaces; citation_author=MS Agranovich, BA Amosov; citation_volume=30; citation_issue=2; citation_publication_date=1996; citation_pages=75-89; citation_doi=10.1007/BF02509448; citation_id=CR1
citation_journal_title=Funct. Anal. Appl.; citation_title=An estimate of the remainder in the spectral asymptotics of pseudodijferential operators of negative order; citation_author=AS Andreev; citation_volume=20; citation_publication_date=1986; citation_pages=124-125; citation_doi=10.1007/BF01077266; citation_id=CR2
citation_journal_title=Math. USSR Sb.; citation_title=Asymptotics of the spectrum of a compact pseudodifferential operators in a Euclidean domain; citation_author=AS Andreev; citation_volume=65; citation_publication_date=1990; citation_pages=205-228; citation_doi=10.1070/SM1990v065n01ABEH001309; citation_id=CR3
citation_title=Heat Kernels and Dirac Operators; citation_publication_date=1992; citation_id=CR4; citation_author=N Berline; citation_author=E Getzler; citation_author=M Vergne; citation_publisher=Springer
citation_journal_title=Commun. Pure Appl. Math.; citation_title=The negative discrete spectrum of a two-dimensional Schrödinger operator; citation_author=MS Birman, A Laptev; citation_volume=49; citation_publication_date=1996; citation_pages=967-997; citation_doi=10.1002/(SICI)1097-0312(199609)49:9<967::AID-CPA3>3.0.CO;2-5; citation_id=CR5
citation_journal_title=Funct. Anal. Appl.; citation_title=The leading term of the spectral asymptotics for “non-smooth” elliptic problems; citation_author=MS Birman, MZ Solomyak; citation_volume=4; citation_publication_date=1970; citation_pages=265-275; citation_doi=10.1007/BF01075968; citation_id=CR6
citation_journal_title=Trudy Moskov. Mat. Obshch.; citation_title=Spectral asymptotics of nonsmooth elliptic operators. I; citation_author=MSh Birman, MZ Solomyak; citation_volume=27; citation_publication_date=1972; citation_pages=3-52; citation_id=CR7
citation_journal_title=Vestnik Leningrad. Univ.; citation_title=Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols; citation_author=MS Birman, MZ Solomyak; citation_volume=13; citation_issue=3; citation_publication_date=1977; citation_pages=13-21; citation_id=CR8
citation_journal_title=Vestnik Leningrad. Univ. Mat. Mekh. Astronom.; citation_title=Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. II; citation_author=MS Birman, MZ Solomyak; citation_volume=13; citation_publication_date=1979; citation_pages=5-10; citation_id=CR9
citation_journal_title=Sib. Math. J.; citation_title=Asymptotics of the spectrum of variational problems on solutions of elliptic equations; citation_author=MS Birman, MZ Solomyak; citation_volume=20; citation_publication_date=1979; citation_pages=1-15; citation_doi=10.1007/BF00976125; citation_id=CR10
Birman, M.Sh., Solomyak, M.Z.: Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory. Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974 (Russian). American Mathematical Society Translations, Series 2, 114. American Mathematical Society, Providence, R.I., 1980
citation_title=Spectral Theory of Self-adjoint Operators in Hilbert Space; citation_publication_date=1987; citation_id=CR12; citation_author=MS Birman; citation_author=MZ Solomyak; citation_publisher=D. Reidel
Birman M.Sh., Solomyak M.Z.: Schrödinger operator. Estimates for number of bound states as function-theoretical problem. Spectral theory of operators (Novgorod, 1989), 1–54. Amer. Math. Soc. Transl. Ser. 2, 150, Amer. Math. Soc., Providence, RI, 1992
citation_journal_title=J. Sov. Math.; citation_title=Asymptotic behavior of the spectrum of the scattering matrix; citation_author=MS Birman, DR Yafaev; citation_volume=25; citation_publication_date=1984; citation_pages=793-814; citation_doi=10.1007/BF01788912; citation_id=CR14
citation_journal_title=Ark. Mat.; citation_title=The negative discrete spectrum of the operator
in
for
even and
; citation_author=MS Birman, A Laptev, M Solomyak; citation_volume=35; citation_issue=1; citation_publication_date=1997; citation_pages=87-126; citation_doi=10.1007/BF02559594; citation_id=CR15
citation_journal_title=Commun. Math. Phys.; citation_title=Estimates and extremals for zeta function determinants on four-manifolds; citation_author=T Branson, S-YA Chang, P Yang; citation_volume=149; citation_publication_date=1992; citation_pages=241-262; citation_doi=10.1007/BF02097624; citation_id=CR16
citation_journal_title=J. Math. Phys.; citation_title=Bound states in one and two spatial dimensions; citation_author=K Chadan, NN Khuri, A Martin, TT Wu; citation_volume=44; citation_publication_date=2003; citation_pages=406-422; citation_doi=10.1063/1.1532538; citation_id=CR17
citation_journal_title=Rocky Mt. J. Math.; citation_title=A new proof of the Cwikel–Lieb–Rosenbljum bound; citation_author=JG Conlon; citation_volume=15; citation_publication_date=1985; citation_pages=117-122; citation_doi=10.1216/RMJ-1985-15-1-117; citation_id=CR18
citation_journal_title=Commun. Math. Phys.; citation_title=The action functional in non-commutative geometry; citation_author=A Connes; citation_volume=117; citation_publication_date=1988; citation_pages=673-683; citation_doi=10.1007/BF01218391; citation_id=CR19
citation_title=Noncommutative Geometry; citation_publication_date=1994; citation_id=CR20; citation_author=A Connes; citation_publisher=Academic Press Inc
citation_journal_title=Ann. Math. (2); citation_title=Weak type estimates for singular values and the number of bound states of Schrödinger operators; citation_author=M Cwikel; citation_volume=106; citation_issue=1; citation_publication_date=1977; citation_pages=93-100; citation_doi=10.2307/1971160; citation_id=CR21
citation_journal_title=Commun. Math. Phys.; citation_title=An uncertainty principle for fermions with generalized kinetic energy; citation_author=I Daubechies; citation_volume=90; citation_publication_date=1983; citation_pages=511-520; citation_doi=10.1007/BF01216182; citation_id=CR22
citation_title=Weyl’s Formula for a Class of Pseudodifferential Operators with Negative Order on . Pseudodifferential Operators (Oberwolfach: Lecture Notes in Math); citation_publication_date=1986; citation_id=CR23; citation_author=M Dauge; citation_author=D Robert; citation_publisher=Springer
citation_journal_title=C. R. Acad. Sci. Paris Sér. A–B; citation_title=Existence de traces non normales; citation_author=J Dixmier; citation_volume=262; citation_publication_date=1966; citation_pages=A1107-A1108; citation_id=CR24
citation_journal_title=Adv. Math.; citation_title=Commutator structure of operator ideals; citation_author=K Dykema, T Figiel, G Weiss, M Wodzicki; citation_volume=185; citation_publication_date=2004; citation_pages=1-79; citation_doi=10.1016/S0001-8708(03)00141-5; citation_id=CR25
citation_journal_title=Proc. Nat. Acad. Sci. USA; citation_title=Maximum properties and inequalities for the eigenvalues of completely continuous operators; citation_author=K Fan; citation_volume=37; citation_publication_date=1951; citation_pages=760-766; citation_doi=10.1073/pnas.37.11.760; citation_id=CR26
citation_journal_title=Bull. Am. Math. Soc.; citation_title=The uncertainty principle; citation_author=CL Fefferman; citation_volume=9; citation_publication_date=1983; citation_pages=129-206; citation_doi=10.1090/S0273-0979-1983-15154-6; citation_id=CR27
citation_journal_title=Comment. Math. Helv.; citation_title=Sharp borderline Sobolev inequalities on compact Riemannian manifolds; citation_author=L Fontana; citation_volume=68; citation_publication_date=1993; citation_pages=415-454; citation_doi=10.1007/BF02565828; citation_id=CR28
citation_journal_title=J. Spectr. Theory; citation_title=Cwikel’s theorem and the CLR inequality; citation_author=R Frank; citation_volume=4; citation_publication_date=2014; citation_pages=1-21; citation_doi=10.4171/JST/59; citation_id=CR29
citation_title=Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs; citation_publication_date=1969; citation_id=CR30; citation_author=IC Gohberg; citation_author=MG Krein; citation_publisher=American Mathematical Society
citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=Negative eigenvalues of two-dimensional Schrödinger operators; citation_author=A Grigor’yan, N Nadirashvili; citation_volume=217; citation_publication_date=2015; citation_pages=975-1028; citation_doi=10.1007/s00205-015-0848-z; citation_id=CR31
citation_journal_title=Commun. Partial Differ. Equ.; citation_title=Spectral asymptotics for nonsmooth singular Green operators; citation_author=G Grubb; citation_volume=39; citation_issue=3; citation_publication_date=2014; citation_pages=530-573; citation_doi=10.1080/03605302.2013.864207; citation_id=CR32
citation_journal_title=Adv. Math.; citation_title=A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues; citation_author=V Guillemin; citation_volume=55; citation_issue=2; citation_publication_date=1985; citation_pages=131-160; citation_doi=10.1016/0001-8708(85)90018-0; citation_id=CR33
Huang, J., Sukochev, F., Zanin D.: Operator
$$\theta $$
-Hölder functions with respect to
$$\Vert \cdot \Vert _p$$
,
$$0 \le p \le \infty $$
.
arXiv:2110.09708
, 31p
Hundertmark, D., Kunstmann, P., Ried, T., Vulgater, S.: Cwikel’s bound reloaded.
arXiv:1809.05069
, 30 pp
citation_title=Microlocal Analysis, Sharp Spectral Asymptotics and Applications II. Functional Methods and Eigenvalue Asymptotics; citation_publication_date=2019; citation_id=CR36; citation_author=V Ivrii; citation_publisher=Springer
citation_journal_title=Adv. Math.; citation_title=Traces of compact operators and the noncommutative residue; citation_author=N Kalton, S Lord, D Potapov, F Sukochev; citation_volume=235; citation_publication_date=2013; citation_pages=1-55; citation_doi=10.1016/j.aim.2012.11.007; citation_id=CR37
citation_journal_title=J. Math. Appl.; citation_title=On estimates for the number of negative eigenvalues of two-dimensional Schrödinger operators with potentials supported by Lipschitz curves; citation_author=M Karuhanga; citation_volume=456; citation_publication_date=2017; citation_pages=1365-1379; citation_id=CR38
citation_journal_title=J. Math. Phys.; citation_title=On negative eigenvalues of two-dimensional Schrödinger operators with singular potentials; citation_author=M Karuhanga, E Shargorodsky; citation_volume=61; citation_publication_date=2020; citation_pages=26; citation_doi=10.1063/5.0004481; citation_id=CR39
citation_journal_title=Few Body Syst.; citation_title=Bound states in
dimensions (especially
and
); citation_author=NN Khuri, A Martin, TT Wu; citation_volume=31; citation_publication_date=2002; citation_pages=83-89; citation_doi=10.1007/s006010200003; citation_id=CR40
citation_journal_title=J. Spectr. Theory; citation_title=On spectral estimates for two-dimensional Schrödinger operators; citation_author=A Laptev, M Solomyak; citation_volume=3; citation_issue=4; citation_publication_date=2013; citation_pages=505-515; citation_doi=10.4171/JST/53; citation_id=CR41
citation_journal_title=Ann. Global Anal. Geom.; citation_title=On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols; citation_author=M Lesch; citation_volume=17; citation_issue=2; citation_publication_date=1999; citation_pages=151-187; citation_doi=10.1023/A:1006504318696; citation_id=CR42
citation_journal_title=J. Anal. Math.; citation_title=The Rozenblum–Lieb–Cwikel inequality for Markov generators; citation_author=D Levin, M Solomyak; citation_volume=71; citation_publication_date=1997; citation_pages=173-193; citation_doi=10.1007/BF02788029; citation_id=CR43
citation_journal_title=Commun. Math. Phys.; citation_title=On the Schrödinger equation and the eigenvalue problem; citation_author=P Li, ST Yau; citation_volume=88; citation_issue=3; citation_publication_date=1983; citation_pages=309-318; citation_doi=10.1007/BF01213210; citation_id=CR44
citation_journal_title=Bull. Am. Math. Soc.; citation_title=Bounds on the eigenvalues of the Laplace and Schrödinger operators; citation_author=EH Lieb; citation_volume=82; citation_publication_date=1976; citation_pages=751-752; citation_doi=10.1090/S0002-9904-1976-14149-3; citation_id=CR45
Lieb, E.H.: The number of bound states of one-body Schrödinger operators and the Weyl problem. Proceedings of Symposia in Pure Mathematics XXXVI. American Mathematical Society, Providence, RI, 1980. 241–252
Lord, S., Sukochev, F., Zanin, D.: Singular traces: theory and applications. De Gruyter Studies in Mathematics, Vol. 46, Walter de Gruyter, 2012, Theory and applications
McDonald, E., Ponge, R.: Cwikel estimates and negative eigenvalues of Schrödinger operators on noncommutative tori.
arXiv:2102.12021
, 28 pp
McDonald, E., Ponge: Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tor.
arXiv:2103.16869
, 32 pp
McDonald, E., Sukochev, F., Zanin, D.: Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry.
arXiv:2106.02235
, 31 pp
citation_journal_title=J. Math. Sci.; citation_title=Bargmann type estimates of the counting function for general Schrödinger operators; citation_author=S Molchanov, B Vainberg; citation_volume=184; citation_publication_date=2012; citation_pages=457-508; citation_doi=10.1007/s10958-012-0877-1; citation_id=CR51
citation_journal_title=J. Funct. Anal.; citation_title=On critical cases of Sobolev’s inequalities; citation_author=T Ozawa; citation_volume=127; citation_issue=2; citation_publication_date=1995; citation_pages=259-269; citation_doi=10.1006/jfan.1995.1012; citation_id=CR52
Ponge, R.: Connes’ integration and Weyl’s laws.
arXiv:2107.01242
, 29pp
citation_journal_title=J. Anal. Math.; citation_title=Traces on pseudodifferential operators and sums of commutators; citation_author=R Ponge; citation_volume=110; citation_publication_date=2010; citation_pages=1-30; citation_doi=10.1007/s11854-010-0001-8; citation_id=CR54
Ponge, R.: Zeta functions and Weyl’s laws on noncommutative tori
citation_journal_title=Soviet Math. (Iz. VUZ); citation_title=Distribution of the discrete spectrum of singular differential operators; citation_author=GV Rozenblum; citation_volume=20; citation_publication_date=1976; citation_pages=63-71; citation_id=CR56
Rozenblum, G.: Eigenvalues of singular measures and Connes noncommutative integration.
arXiv:2103.02067
, 33 pp. To appear in J. Spectr. Theory
Rozenblum, G., Shargorodsky, E.: Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case. Partial differential equations, spectral theory, and mathematical Physics. The Ari Laptev anniversary volume, EMS Series of Congress Reports, Vol. 18, EMS Publishing, 2021, pp. 331–353
citation_journal_title=St. Petersburg Math. J.; citation_title=The Cwikel–Lieb–Rozenblyum estimator for generators of positive semigroups and semigroups dominated by positive semigroups; citation_author=G Rozenblum, M Solomyak; citation_volume=9; citation_issue=6; citation_publication_date=1998; citation_pages=1195-1211; citation_id=CR59
Rozenblum, G., Tashchiyan., G.: Eigenvalues of the Birman-Schwinger operator for singular measures: the noncritical case.
arXiv:2107.04682
, 40pp
citation_journal_title=Soviet Math. Dokl.; citation_title=Distribution of the discrete spectrum of singular differential operators; citation_author=GV Rozenbum; citation_volume=13; citation_publication_date=1972; citation_pages=245-249; citation_id=CR61
citation_journal_title=Adv. Math.; citation_title=Banach limits and traces on
; citation_author=E Semenov, F Sukochev, A Usachev, D Zanin; citation_volume=285; citation_publication_date=2015; citation_pages=568-628; citation_doi=10.1016/j.aim.2015.08.010; citation_id=CR62
citation_journal_title=Proc. Lond. Math. Soc.; citation_title=On negative eigenvalues of two-dimensional Schrödinger operators; citation_author=E Shargorodsky; citation_volume=108; citation_publication_date=2014; citation_pages=441-483; citation_doi=10.1112/plms/pdt036; citation_id=CR63
citation_title=Pseudodifferential Operators and Spectral Theory; citation_publication_date=2001; citation_id=CR64; citation_author=MA Shubin; citation_publisher=Springer
citation_journal_title=Trans. Am. Math. Soc.; citation_title=Analysis with weak trace ideals and the number of bound states of Schrödinger operators; citation_author=B Simon; citation_volume=224; citation_publication_date=1976; citation_pages=367-380; citation_id=CR65
Simon, B.: Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005
citation_title=Convexity: An Analytic Viewpoint. Cambridge Tracts in Mathematics; citation_publication_date=2011; citation_id=CR67; citation_author=B Simon; citation_publisher=Cambridge University Press
citation_title=Operator Theory: A Comprehensive Course in Analysis, Part 4; citation_publication_date=2015; citation_id=CR68; citation_author=B Simon; citation_publisher=American Mathematical Society
citation_journal_title=Isr. J. Math.; citation_title=Piecewise-polynomial approximation of functions from
,
, and applications to the spectral theory of the Schrödinger operator; citation_author=M Solomyak; citation_volume=86; citation_publication_date=1994; citation_pages=253-275; citation_doi=10.1007/BF02773681; citation_id=CR69
citation_journal_title=Proc. Lond. Math. Soc.; citation_title=Spectral problems related to the critical exponent in the Sobolev embedding theorem; citation_author=M Solomyak; citation_volume=71; citation_issue=1; citation_publication_date=1995; citation_pages=53-75; citation_doi=10.1112/plms/s3-71.1.53; citation_id=CR70
citation_journal_title=Proc. Am. Math. Soc.; citation_title=An estimate for the number of bound states of the Schrödinger operator in two dimensions; citation_author=M Stoiciu; citation_volume=132; citation_publication_date=2004; citation_pages=1143-1151; citation_doi=10.1090/S0002-9939-03-07257-5; citation_id=CR71
Sukochev, F., Zanin, D.: Connes integration formula without singular traces.
arXiv:2103.08817
, 19 pp
Sukochev, F., Zanin, D.: Cwikel–Solomyak estimates on tori and Euclidean spaces.
arXiv:2008.04494
, 31pp
Wodzicki, M.: Spectral asymmetry and noncommutative residue (in Russian). Habilitation Thesis, Steklov Institute, Soviet Academy of Sciences, Moscow (1984)
Wodzicki, M.: Noncommutative residue. I. Fundamentals.
$$K$$
-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., Vol. 1289, Springer, Berlin, 1987, pp. 320–399