Families of Self-Similar Symmetric Laplacians on the Interval and Sierpinski Gasket
Limiting on the Interval
Below are plots of the eigenfunctions and the csv files of the eigenvalues as the parameters for the interval and SG approach extreme values. They are organized in a similar way as were the non-extreme sets of parameters.
Small p limiting case
p m=1 m=2 m=3 m=4 m=5
10-1 pdf pdf pdf pdf pdf
10-2 pdf pdf pdf pdf pdf
10-3 pdf pdf pdf pdf pdf
10-4 pdf pdf pdf pdf pdf
10-5 pdf pdf pdf pdf pdf
10-6 pdf pdf pdf pdf pdf
10-7 pdf pdf pdf pdf pdf
10-8 pdf pdf pdf pdf pdf
Large r limiting case
p m=1 m=2 m=3 m=4 m=5
1-10-1 pdf pdf pdf pdf pdf
1-10-2 pdf pdf pdf pdf pdf
1-10-3 pdf pdf pdf pdf pdf
1-10-4 pdf pdf pdf pdf pdf
1-10-5 pdf pdf pdf pdf pdf
1-10-6 pdf pdf pdf pdf pdf
1-10-7 pdf pdf pdf pdf pdf
1-10-8 pdf pdf pdf pdf pdf
Eigenvalues for limiting values of p
p=x p=1-x
x=10-1 csv csv
x=10-2 csv csv
x=10-3 csv csv
x=10-4 csv csv
x=10-5 csv csv
x=10-6 csv csv
x=10-7 csv csv
x=10-8 csv csv
Limiting on the Sierpinski Gasket
Small r limiting case
r m=1 m=2 m=3
10-1 zip zip zip
10-2 zip zip zip
10-3 zip zip zip
10-4 zip zip zip
10-5 zip zip zip
10-6 zip zip zip
10-7 zip zip zip
10-8 zip zip zip
Large r limiting case
r m=1 m=2 m=3
101 zip zip zip
102 zip zip zip
103 zip zip zip
104 zip zip zip
105 zip zip zip
106 zip zip zip
107 zip zip zip
108 zip zip zip
Eigenvalues for limiting values of r
r=10-x r=10x
x=1 csv csv
x=2 csv csv
x=3 csv csv
x=4 csv csv
x=5 csv csv
x=6 csv csv
x=7 csv csv
x=8 csv csv
Limiting Ratios
Finally, we can describe the relationship between eigenvalues using the set of ratios between eigenvalues. In particular we can generate plots of thees ratios that resemble barcodes, and examine the limiting behavior as we vary the parameters, for both the interval and the Sierpinski Gasket. Here are such plots for p=0.5,p=0.0001,p=0.9999 on the interval, and r=1,r=0.0001,r=10000 on the sierpinski gasket.