On the bound states of magnetic Laplacians on wedges

TL;DR

This paper proves the existence of bound states for magnetic Laplacians on wedges with apertures up to approximately 0.595π, extending previous results and combining variational methods with computer assistance.

Contribution

It advances the understanding of magnetic Laplacians on wedges by establishing bound states for larger apertures using novel variational and computational techniques.

Findings

Bound states exist for apertures up to approximately 0.595π.

Extended the known range of apertures with bound states beyond previous proofs.

Used variational methods combined with computer-assisted analysis.

Abstract

This paper is mainly inspired by the conjecture 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 paper 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 computer-assistance, we extend this interval up to any aperture $\phi \lesssim 0.595\pi$. Moreover, we analyse 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.