A bound for the eigenvalue counting function for Krein-von Neumann and Friedrichs extensions.
File(s)
Accepted version
Author(s)
Ashbaugh, MS
Gesztesy, F
Laptev, A
Mitrea, M
Sukhtaiev, S
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).
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).
Date Issued
2016-09-22
Date Acceptance
2016-09-08
Citation
Advances in Mathematics, 2016, 304, pp.1108-1155
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/
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000398757500025&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Subjects
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
Publication Status
Published