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A remark on the asymptotic properties of eigenvalues and the lattice point problem. (English) Zbl 0918.35100

Summary: The asymptotic distribution of eigenvalues of elliptic selfadjoint operators on the flat torus is discussed. A relation between a geometrical property of the operators and the error terms in the distribution formulas is given in the case when the operators have constant coefficients. As a corollary, the error terms can be determined only by the order of the operators and the dimension of the torus. This result also gives an information on the number of lattice points inside convex or nonconvex bodies in \(\mathbb{R}^n\).

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H16 Nonconvex bodies
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[1] J CHEN, The lattice-points in a circle, Sci Sinica 12 (1963), 633-649
[2] J J DUISTERMAAT AND V. W GuiLLEMiN, The spectrum of positive elliptic operators and periodi bicharacteristics, Invent Math. 29 (1975), 39-79 · Zbl 0307.35071 · doi:10.1007/BF01405172
[3] E HLAWKA, Uber Integrale auf konvexen Krper I, Monatsh Math 54 (1950), 1-3 · Zbl 0036.30902 · doi:10.1007/BF01304101
[4] L Hrmander, The spectral function of an elliptic operator, Acta Math 121 (1968), 193-21 · Zbl 0164.13201 · doi:10.1007/BF02391913
[5] H IWANIEC AND C. J MozzocHi, On the divisor andcircle problems, J Number Theory 29(1988), 60-9 · Zbl 0644.10031 · doi:10.1016/0022-314X(88)90093-5
[6] S KOBAYASHI AND K. NoMizu, Foundation of Differential Geometry (Vol II), Interscience, Ne York, 1969 · Zbl 0526.53001
[7] E LANDAU, Vorlesungen iber Zahlentheorie (Vol II), Hirzel, Leipzig, 192
[8] W MULLER AND W. G NowAK, Lattice points in planar domains: Applications of Huxley’s discret Hardy-Littlewood Method, Lecture Notes in Math 1452 (1990), 139-164 · Zbl 0715.11054
[9] B RANDOL, On the asymptotic behavior of the Fourier transform of the indicator function of a conve set, Trans Amer Math Soc 139 (1969), 279-285 JSTOR: · Zbl 0183.26905 · doi:10.2307/1995320
[10] W SIERPINSKI, Prace Mat Fiz 17 (1906), 77-11
[11] E M STEIN, Harmonic Analysis, Princeton Univ Press, Princeton, 199
[12] M SUGIMOTO, A priori estimates for higher order hyperbolic equations, Math Z 215(1994), 519-53 · Zbl 0790.35063 · doi:10.1007/BF02571728
[13] M SUGIMOTO, Estimates for hyperbolic equations with non-convex characteristics, Math. Z 22 (1996), 521-531 · Zbl 0867.35013 · doi:10.1007/PL00004265
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