Geometrical Versions of improved Berezin-Li-Yau Inequalities

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit. Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.

[1]  E. Lieb,et al.  Segregation in the Falicov--Kimball Model , 2001, math-ph/0107003.

[2]  Reinhard Racke,et al.  Elastic and electro-magnetic waves in infinite waveguides , 2008 .

[3]  Sharp Lieb-Thirring inequalities in high dimensions , 1999, math-ph/9903007.

[4]  Marcel Griesemer,et al.  On the Atomic Photoeffect in Non-relativistic QED , 2009, 0910.1809.

[5]  Thomas Merkle.  On Convergence of Local Averaging Regression Function Estimates for the Regularization of Inverse Problems , 2010 .

[6]  Wolfgang Kimmerle,et al.  On Torsion Subgroups in Integral Group Rings of Finite Groups , 2011 .

[7]  M. Berg.  A uniform bound on trace (etΔ) for convex regions inRn with smooth boundaries , 1984 .

[8]  Antoine Henrot,et al.  Extremum Problems for Eigenvalues of Elliptic Operators , 2006 .

[9]  On Lieb-Thirring Inequalities for Schrödinger Operators with Virtual Level , 2006 .

[10]  T. Merkle,et al.  Conformally closed Poincaré-Einstein metrics with intersecting scale singularities , 2008 .

[11]  Wolfgang Kimmerle,et al.  Finite groups of units and their composition factors in the integral group rings of the groups PSL(2, q) , 2008, 0810.0186.

[12]  Ulrich Brehm,et al.  Lattice triangulations of E 3 and of the 3-torus , 2009 .

[13]  Leander Geisinger,et al.  Institute for Mathematical Physics Universal Bounds for Traces of the Dirichlet Laplace Operator Universal Bounds for Traces of the Dirichlet Laplace Operator , 2022 .

[14]  Markus Stroppel,et al.  Stabilizers of Subspaces under Similitudes of the Klein Quadric, and Automorphisms of Heisenberg Algebras , 2010, 1012.0502.

[15]  H. Weyl.  Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung) , 1912 .

[16]  Hynek Kovařík,et al.  Two-Dimensional Berezin-Li-Yau Inequalities with a Correction Term , 2008, 0802.2792.

[17]  LOWER BOUNDS FOR THE SPECTRUM OF THE LAPLACE AND STOKES OPERATORS , 2009, 0909.2818.

[18]  L. M. Milne-Thomson.  Methoden der mathematischen Physik , 1944, Nature.

[19]  A. Laptev,et al.  A Geometrical Version of Hardy's Inequality , 2002 .

[20]  Robert Osserman,et al.  A note on Hayman's theorem on the bass note of a drum , 1977 .

[21]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.

[22]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[23]  E. Lieb,et al.  On semi-classical bounds for eigenvalues of Schrödinger operators , 1978 .

[24]  Anton E. Mayer.  Theorie der konvexen Körper , 1936 .

[25]  A. Laptev,et al.  Spectral inequalities for Schrödinger operators with surface potentials , 2007, 0711.3473.

[26]  L. Hörmander,et al.  THE ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF PARTIAL DIFFERENTIAL OPERATORS (Translations of Mathematical Monographs 155) , 1998 .

[27]  L. Hörmander,et al.  The spectral function of an elliptic operator , 1968 .

[28]  Yu Safarov,et al.  The Asymptotic Distribution of Eigenvalues of Partial Differential Operators , 1996 .

[29]  Markus Stroppel,et al.  Polarities of Schellhammer Planes , 2011 .

[30]  T. Weidl,et al.  Improved Berezin-Li-Yau inequalities with a remainder term , 2007, 0711.4925.

[31]  A. Melas.  A lower bound for sums of eigenvalues of the Laplacian , 2002 .

[32]  Barbara Kaltenbacher,et al.  On convergence of local averaging regression function estimates for the regularization of inverse problems , 2011 .

[33]  E. Davies.  A review of Hardy inequalities , 1998, math/9809159.

[34]  R. Frank.  A Simple Proof of Hardy-Lieb-Thirring Inequalities , 2008, 0809.3797.

[35]  W. Harro,et al.  Upper bounds for Bermudan options on Markovian data using nonparametric regression and a reduced number of nested Monte Carlo steps , 2009 .

[36]  Rupert L. Frank,et al.  Eigenvalue estimates for Schrödinger operators on metric trees , 2007, 0710.5500.

[37]  F. Berezin.  COVARIANT AND CONTRAVARIANT SYMBOLS OF OPERATORS , 1972 .

[38]  Shing-Tung Yau,et al.  On the Schrödinger equation and the eigenvalue problem , 1983 .

[39]  P. Exner,et al.  Lieb-Thirring Inequalities for Geometrically Induced Bound States , 2004, math-ph/0405046.

[40]  Ari Laptev,et al.  Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces , 1997 .

[41]  Michel L. Lapidus,et al.  Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture , 1991 .

[42]  V. Ivrii.  Microlocal Analysis and Precise Spectral Asymptotics , 1998 .

[43]  Elliott H. Lieb,et al.  Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators , 2006 .

[44]  J. Fleckinger,et al.  Heat Equation on the Triadic Von Koch Snowflake: Asymptotic and Numerical Analysis , 1995 .

[45]  Dmitri Vassiliev,et al.  An example of a two-term asymptotics for the “counting function” of a fractal drum , 1993 .

[46]  Richard B. Melrose,et al.  Weyl''s conjecture for manifolds with concave boundary , 1980 .