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We consider the Laplace operator in a tubular neighbourhood of a conicalsurface of revolution, subject to an Aharonov-Bohm magnetic field supported onthe axis of symmetry and Dirichlet boundary conditions on the boundary of thedomain. We show that there exists a critical total magnetic flux depending onthe aperture of the conical surface for which the system undergoes an abruptspectral transition from infinitely many eigenvalues below the essentialspectrum to an empty discrete spectrum. For the critical flux we establish aHardy-type inequality. In the regime with infinite discrete spectrum we obtainsharp spectral asymptotics with refined estimate of the remainder andinvestigate the dependence of the eigenvalues on the aperture of the surfaceand the flux of the magnetic field.

Spectral transitions for Aharonov-Bohm Laplacians on conical layers