Corrigendum to “Spectral Theory for the Neumann Laplacian on Planar Domains with Horn-Like Ends”

Julian Edward

doi:10.4153/cjm-2000-005-x

Corrigendum to “Spectral Theory for the Neumann Laplacian on Planar Domains with Horn-Like Ends”

Canadian Journal of Mathematics ◽

10.4153/cjm-2000-005-x ◽

2000 ◽

Vol 52 (1) ◽

pp. 119-122

Author(s):

Julian Edward

Keyword(s):

Spectral Theory ◽

Planar Domains ◽

Neumann Laplacian

AbstractErrors to a previous paper (Canad. J. Math. (2) 49(1997), 232–262) are corrected. A non-standard regularisation of the auxiliary operator A appearing in Mourre theory is used.

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Spectral Theory for the Neumann Laplacian on Planar Domains With Horn-Like Ends

Canadian Journal of Mathematics ◽

10.4153/cjm-1997-012-8 ◽

1997 ◽

Vol 49 (2) ◽

pp. 232-262 ◽

Cited By ~ 2

Author(s):

Julian Edward

Keyword(s):

Large Class ◽

Continuous Spectrum ◽

Spectral Theory ◽

Point Spectrum ◽

Pure Point ◽

Pure Point Spectrum ◽

Singular Continuous Spectrum ◽

Finite Multiplicity ◽

Planar Domains ◽

Neumann Laplacian

AbstractThe spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at 0 or ∞. The proof uses Mourre theory.

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From Steklov to Neumann and Beyond, via Robin: The Szegő Way

Canadian Journal of Mathematics ◽

10.4153/s0008414x19000154 ◽

2019 ◽

Vol 72 (4) ◽

pp. 1024-1043 ◽

Cited By ~ 1

Author(s):

Pedro Freitas ◽

Richard S. Laugesen

Keyword(s):

First Eigenvalue ◽

The First Eigenvalue ◽

Planar Domains ◽

Steklov Eigenvalue ◽

Robin Laplacian ◽

Fixed Area ◽

Neumann Laplacian ◽

Simply Connected ◽

Second Eigenvalue ◽

Simply Connected Domains

AbstractThe second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

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