Kuba Kominiarczuk and Leszek Stawikowski, two high school students participated in the Workshop organized in the Institute of Theoretical Physics under patronage of Fundusz na Rzecz Dzieci in March 2003 and publicly presented results of their work in May 2003. Their presentation included the two files that are available through links provided below.
The first file is a collection of slides with formulas explaining their calculation. Particularly interesting is the slide No. 19. This slide shows dependence of the coupling constant g_lambda on the logarithm of the parameter lambda. This parameter and the coupling constant are introduced by applying similarity renormalization group procedure to the s-wave Schroedinger equation for one particle moving on the plane in the presence of a delta-function potential. The discretization is obtained by replacing momentum squared of the particle by b^n where b ~ 2 and n is an integer. Different curves correspond to different sizes of the initial cutoff in the model. The cutoff parameter is defined as the maximal allowed momentum squared. The cutoff equals b^p and the exponent p is varied. The initial couplings are adjusted to always produce the same bound state energy no matter what the initial cutoff p is. In all cases, the similarity RG group produces functions g_lambda that fall on the same curve when lambda becomes much smaller than the initial cutoff. This example illustrates how universality shows up in the similarity RG procedure.
The second file is a video clip showing the evolution of matrix elements of the discretized delta-function Hamiltonian H when lambda varies from infinity to zero. The "horizontal" plane represents the matrix table and the "vertical" coordinate is a measure of the magnitude of matrix elements. The matrix elements are suitably rescaled in order to avoid very large and very small numbers. The height of a bar with horizontal coordinates mn is equal to
H_mn [delta_mn /(1-g) - (1-delta_mn) /g ] / b^((m+n)/2) ,
where g = g_infinity is the coupling constant in the initial Hamiltonian.