Estimates for the sums of eigenvalues of the fractional laplacian on a bounded domain

Communications in Contemporary Mathematics

Volumn 15, Issue 3, 2013, Pages

  Yildirim Yolcu, Selma ^a   Yolcu, Türkay ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

Berezin Li Yau inequality; eigenvalue; Fractional Laplacian; sums of eigenvalues

Indexed keywords

EID: 84874385940     PISSN: 02191997     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219199712500484     Document Type: Article

References (18)

1. R. Bañuelos and T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 211(2) (2004) 355-423.
2. R. Bañuelos and T. Kulczycki, Eigenvalue gaps for the Cauchy process and a Poincaŕe inequality, J. Funct. Anal. 234 (2006) 199-225.
3. R. Bañuelos, T. Kulczycki and B. Siudeja, On the trace of symmetric stable processes on Lipschitz domains, J. Funct. Anal. 257(10) (2009) 3329-3352.
4. F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 1134-1167.
5. R. Blumenthal and R. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math. 9 (1959) 399-408.
6. E. M. Harrell II and S. Yldrm Yolcu, Eigenvalue inequalities for Klein-Gordon operators, J. Funct. Anal. 256(12) (2009) 3977-3995.
7. A. A. Ilyin, On the spectrum of the Stokes operator, Funktsional. Anal. i Prilozhen. 43(4) (2009) 14-25 (in Russian),
8. Funct. Anal. Appl. 43(4) (2009) 254-263.
9. A. A. Ilyin, Lower bounds for the spectrum of the Laplace and Stokes operators, Discrete Contin. Dyn. Syst. 28(1) (2010) 131-146.
10. H. Kovaŕk, S. Vugalter and T. Weidl, Two dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys. 287(3) (2009) 959-981.
11. N. S. Landkof, Foundations of Modern Potential Theory (Springer-Verlag, New York, 1972).
12. A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997) 531-545.
13. A. Laptev and T. Weidl, Recent results on Lieb-Thirring inequalities, in Jourńees " ́ Equations aux D́eriv́ees Partielles", La Chapelle sur Erdre, 2000, Exp. No. 20 (Univ. Nantes, Nantes, 2000), p. 14.
14. P. Li and S.-T. Yau, On the Schr̈odinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983) 309-318.
15. A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131(2) (2002) 631-636.
16. E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. 49 (2009) 33-44.
17. T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term, in Spectral Theory of Differential Operators, American Mathematical Society Translations, Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 253-263.
18. S. Yldrm Yolcu, An improvement to a Berezin-Li-Yau type inequality, Proc. Amer. Math. Soc. 138(11) (2010) 4059-4066.