A quantum waveguide with Aharonov Bohm magnetic field

In a previous study \cite{n} 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 $a_0>0$, for which we have absence of the discrete spectrum for $\displaystyle 0<\frac{a}{d}<a_0$. We give a sufficient condition for the existence of discrete eigenvalues.