Pavel Exner, Vladimir Lotoreichik, Axel Perez-Obiol
On the bound states of magnetic Laplacians on wedges
(779K, PDF)
ABSTRACT. This note is mainly inspired by the conjecture [Conj. 8.10]{R} about the existence of bound states for magnetic Neumann Laplacians
on planar wedges of any aperture $\phi\in (0,\pi)$.
So far, a proof was only obtained for apertures
$\phi\lesssim 0.511\pi$.
The conviction in the validity of this conjecture for apertures
$\phi\gtrsim 0.511\pi$ mainly relied on numerical computations.
In this note we succeed to prove the existence of bound states
for any aperture $\phi \lesssim 0.583
\pi$ using a variational argument
with suitably chosen test functions. Employing some more involved
test functions and combining a variational argument with numerical optimization, we extend this interval up to any aperture
$\phi \lesssim 0.595\pi$. Moreover, we analyze the same question
for closely related problems concerning
magnetic Robin Laplacians on wedges and for magnetic Schr\"odinger operators in the plane with $\delta$-interactions supported on broken lines.