A bound for the eigenvalue counting function for Krein-von Neumann and Friedrichs extensions

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Title: A bound for the eigenvalue counting function for Krein-von Neumann and Friedrichs extensions
Author(s): Ashbaugh, MS
Gesztesy, F
Laptev, A
Mitrea, M
Sukhtaiev, S
Item Type: Journal Article
Abstract: For an arbitrary open, nonempty, bounded set , , and sufficiently smooth coefficients , we consider the closed, strictly positive, higher-order differential operator in defined on , associated with the differential expression (equations missing) and its Krein–von Neumann extension in . Denoting by , , the eigenvalue counting function corresponding to the strictly positive eigenvalues of , we derive the bound (equations missing)where (with ) is connected to the eigenfunction expansion of the self-adjoint operator in defined on , corresponding to . Here denotes the (Euclidean) volume of the unit ball in (equations missing). Our method of proof relies on variational considerations exploiting the fundamental link between the Krein–von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of in (equations missing) We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension in of (equations missing).
Publication Date: 22-Sep-2016
Date of Acceptance: 8-Sep-2016
URI: http://hdl.handle.net/10044/1/52741
DOI: https://dx.doi.org/10.1016/j.aim.2016.09.11
ISSN: 0001-8708
Publisher: Elsevier
Start Page: 1108
End Page: 1155
Journal / Book Title: Advances in Mathematics
Volume: 304
Copyright Statement: © 2016 Elsevier Inc. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Keywords: Science & Technology
Physical Sciences
Mathematics
Krein and Friedrichs extensions of powers of second-order uniformly elliptic partial differential operators
Bounds on eigenvalue counting functions Spectral analysis
Buckling problem
SCHRODINGER-OPERATORS
EIGENFUNCTION-EXPANSIONS
SINGULAR POTENTIALS
MAGNETIC POTENTIALS
SCATTERING-THEORY
RIESZ MEANS
RESOLVENT
LAPLACIAN
DOMAINS
INEQUALITIES
Science & Technology
Physical Sciences
Mathematics
Krein and Friedrichs extensions of powers of second-order uniformly elliptic partial differential operators
Bounds on eigenvalue counting functions Spectral analysis
Buckling problem
SCHRODINGER-OPERATORS
EIGENFUNCTION-EXPANSIONS
SINGULAR POTENTIALS
MAGNETIC POTENTIALS
SCATTERING-THEORY
RIESZ MEANS
RESOLVENT
LAPLACIAN
DOMAINS
INEQUALITIES
0101 Pure Mathematics
General Mathematics
Publication Status: Published
Appears in Collections:Pure Mathematics
Mathematics
Faculty of Natural Sciences



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