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We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. In particular, we use a careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator.
@article{10_4007_annals_2018_187_2_1,
author = {Luc Hillairet and Chris Judge},
title = {Hyperbolic triangles without embedded eigenvalues},
journal = {Annals of mathematics},
pages = {301--377},
publisher = {mathdoc},
volume = {187},
number = {2},
year = {2018},
doi = {10.4007/annals.2018.187.2.1},
mrnumber = {3744854},
zbl = {06841542},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.2.1/}
}
TY - JOUR AU - Luc Hillairet AU - Chris Judge TI - Hyperbolic triangles without embedded eigenvalues JO - Annals of mathematics PY - 2018 SP - 301 EP - 377 VL - 187 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.2.1/ DO - 10.4007/annals.2018.187.2.1 LA - en ID - 10_4007_annals_2018_187_2_1 ER -
%0 Journal Article %A Luc Hillairet %A Chris Judge %T Hyperbolic triangles without embedded eigenvalues %J Annals of mathematics %D 2018 %P 301-377 %V 187 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.2.1/ %R 10.4007/annals.2018.187.2.1 %G en %F 10_4007_annals_2018_187_2_1
Luc Hillairet; Chris Judge. Hyperbolic triangles without embedded eigenvalues. Annals of mathematics, Tome 187 (2018) no. 2, pp. 301-377. doi: 10.4007/annals.2018.187.2.1
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