On the spectrum of the Schrodinger Operator with Aharonov-Bohm Magnetic Field in quantum waveguide with Neumann window

In a previous study [5] we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disc window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip. We proved that such system exhibits discrete eigenvalues below the essential spectrum for any a > 0. In the present work we study the effect of a magnetic filed of Aharonov-Bohm type when the magnetic field is turned on this system. Precisely we prove that in the presence of such magnetic filed there is some critical values of a0 > 0, for which we have absence of the discrete spectrum for 0 < a d < a0. We give a sufficient condition for the existence of discrete eigenvalues. AMS Classification: 81Q10 (47B80, 81Q15)