Numerical computation for pyramid quantum dot

We present a simple and efficient numerical method for the simulation of the three-dimensional pyramid quantum dot which is placed in a cuboid box. The corresponding Schrödinger equation is discretized using the finite volume method with uniform mesh in Cartesian coordinates and the interface conditions are incorporated into the discretization scheme without explicitly enforcing them. The resulting matrix eigenvalue problem is then solved using a Jacobi-Davidson based method. Both linear and quintic polynomial eigenvalue problems are simulated. The scheme is 2nd order accurate and converges extremely fast. The superior performance is a combined effect of the uniform spacing of the grids and the nice structure of the resulting matrices. We have successfully simulated a variety of test problems, including a quintic polynomial eigenvalue problem with more than 32 million variables.