Energy-Dependent Transmission

As a function of the incoming energy, the absolute value of the transmission coefficient is of special interest. When the widths of the left and right arms are small compared with the diameter of the center disk , a resonance is expected if the energy of the incoming wave is near an eigenvalue of the center disk with Dirichlet boundary conditions. Here we will consider the energy range . We start by calculating the lowest eigenstates of the center disk with Dirichlet boundary conditions. The exact eigenvalues for a disk with homogeneous Dirichlet boundary conditions are , where is the th zero of the Bessel function .

The function norm calculates the norm of these states.

We use the function overlap to calculate the overlap integral between the eigenstates of the center disk and the scattering states within it. We form the overlap integral with the expansion functions symbolically. Due to the orthogonality of the angular parts of the eigenfunctions and the scattering state expansion functions, we have only terms to consider.

Now, we calculate the transmission data. To quickly get a rough idea of its energy dependence, we use a smaller number of basis states.

Here are the resulting transmission data and overlap integrals. The left graphic shows the transmission with the scaled absolute values of the overlap integrals. The right graphic shows the transmission with the argument of the transmission coefficient and the arguments of the overlap integrals. We clearly see a strong correlation between the transmission coefficient of the incoming wave in resonance with the eigenstates of the center disk with Dirichlet boundary conditions.