Calculate Fourier Coefficients

To determine the coefficients , we must construct equations encoding the continuity of the solution and its first derivative. As the joining curves, we will use the parts of the boundary of the center disk that intersects both arms. Along these arcs, we demand that the first Fourier coefficients of the difference between the solution ansatz of the arms and of the disk vanish.

We denote by the angle from the positive -axis to the intersection of the center disk with the upper boundary of the right arm.

We use these global parameters: the energy , the number of states in the three regions , , and , the radius and width and , and .

To abbreviate the following inputs, we define these solution ingredients: a left running mode, a right running mode, and a mode for the center disk. Though we restrict ourselves to solutions symmetric with respect to the -axis, the antisymmetric states can be obtained by replacing cos with sin in the fundamental modes.

Here are the basis functions for the expansion in the three regions.

On the left arc, we multiply with the first of the functions L, R, that is of the form (they form an overcomplete system but cover the arc in a homogeneous manner). Later, we will use some of the Fourier coefficients (the cos-modes with odd that are complete for the symmetric states we are looking for) and demand that they vanish.

Here are these Fourier modes. Because the solution of the problem has to fulfill homogeneous Dirichlet boundary conditions along the boundaries of the center disk, we will later use the cos-modes that vanish at the endpoints of these arcs.

The function cond[leftRight, , , {cs, n}] represents the continuity of the th mode on the left or right arm arc of the th derivative ( or in the following) arising from the th function of the Fourier series with trigonometric function cs. When the Fourier coefficients can be calculated in closed form symbolically, we will do so. Some Fourier coefficients have to be calculated through numerical integration. The function condI represents the inhomogeneous part of the linear equation arising from the incoming wave.

We proceed in a completely analogous manner for the right arc (where there is no inhomogeneous part).

The homogeneous Dirichlet boundary conditions along the horizontal boundaries of the left and right arms are fulfilled automatically through the form of the ansatz. To fulfill the Dirichlet boundary conditions along the upper and lower arcs bounding the center disk, we again multiply with functions from an overcomplete set of Fourier-like functions. Later, we will use the sin-modes for all , which form a complete system along the upper and lower arcs of the center disk.