17-45 Pavel Exner, Vladimir Lotoreichik, Axel Perez-Obiol

On the bound states of magnetic Laplacians on wedges (779K, PDF) Apr 10, 17

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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.

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