Refined bounds for the eigenvalues of the Klein-Gordon operator

Yolcu, Türkay

doi:10.1090/S0002-9939-2013-11806-X

Refined bounds for the eigenvalues of the Klein-Gordon operator
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Proc. Amer. Math. Soc. 141 (2013), 4305-4315 Request permission

Abstract:

The aim of this article is twofold. First we establish sharper lower bounds for the sums of eigenvalues of $(-\Delta )^{\frac {1}{2}}|_{D},$ the Klein-Gordon operator restricted to a bounded domain $D\subset {\mathbb R}^d,$ than the bounds obtained in works by E. Harrell; S. Yıldırım Yolcu; and G. Wei, H. Sun, and L. Zeng. Then we study upper bounds for the sums of negative powers of the eigenvalues of $(-\Delta )^{\frac {1}{2}}|_{D}.$

References

F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134–1167 (Russian). MR 0350504
R. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math. 9 (1959), 399–408. MR 107298, DOI 10.2140/pjm.1959.9.399
Evans M. Harrell II and Selma Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon operators, J. Funct. Anal. 256 (2009), no. 12, 3977–3995. MR 2521917, DOI 10.1016/j.jfa.2008.12.008
Peter Li and Shing Tung Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), no. 3, 309–318. MR 701919, DOI 10.1007/BF01213210
Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
Antonios D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003), no. 2, 631–636. MR 1933356, DOI 10.1090/S0002-9939-02-06834-X
Selma Yildirim Yolcu, An improvement to a Berezin-Li-Yau type inequality, Proc. Amer. Math. Soc. 138 (2010), no. 11, 4059–4066. MR 2679626, DOI 10.1090/S0002-9939-2010-10419-7
S. Yıldırım Yolcu and T. Yolcu, Estimates on the eigenvalues of the clamped plate problem on domains in Euclidean spaces, J. Math. Phys. 54 (2013), no. 4, 043515, DOI 10.1063/1.4801446.
Selma Yildirim Yolcu and Türkay Yolcu, Bounds for the eigenvalues of the fractional Laplacian, Rev. Math. Phys. 24 (2012), no. 3, 1250003, 18. MR 2902846, DOI 10.1142/S0129055X12500031
S. Yıldırım Yolcu and T. Yolcu, Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Comm. Contemp. Math. 15 (2013), no. 3, 1250048, DOI 10.1142/S0219199712500484.
Selma Yıldırım Yolcu and Türkay Yolcu, Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators, J. Math. Phys. 53 (2012), no. 4, 043508, 17. MR 2953152, DOI 10.1063/1.3701978
G. Wei, H. J. Sun, L. Zeng, Lower bounds for Laplacian and fractional Laplacian eigenvalues, preprint, 2011, http://arxiv.org/abs/1112.4571.
V. Vougalter, Sharp semiclassical bounds for the moments of eigenvalues for some Schroedinger type operators with unbounded potentials, preprint, (2011), http://www.ma.utexas.edu/mp_arc/c/11/11-155.pdf.