Sweep completed on December 14, 2016.

Unit-edged Regular Pentagon, both Dirichlet and Neumann boundary conditions

Bob Jones

Summer 2016

This page simply lists the eigenvalues corresponding to each boundary condition (Dirichlet D or Neumann N) within each symmetry class (A,B,C,S) for the regular pentagon, i.e., the values of $\lambda$ which are solutions to $$ (\Delta+\lambda)\Psi(\mathbf{r}) = 0\qquad\mbox{in $\Omega$, with} \qquad\mbox{D: } \Psi=0\quad\mbox{or}\quad \qquad\mbox{N: } \frac{\partial\Psi}{\partial n}=0 \qquad \mbox{on $\partial\Omega$} $$ where $\Omega$ is the unit-edged regular pentagon, $\partial\Omega$ is its boundary, and $n$ the outward pointing normal.

I also include Weyl counting function plots which provide some numerical evidence that this set of numbers is complete.

Since the edge-length of the regular pentagon is one, the lowest Dirichlet eigenvalue is about 11.00. Also of note, for this tally, I do include the trivial NS $\lambda=0$ eigenvalue.

The total number of distinct eigenvalues with $\lambda<10^6$ is $$ \#(10^6) = \#_\mathrm{D}(10^6) + \#_\mathrm{N}(10^6) = 81,913 + 82,360 = 164,303 $$ where $\#(\lambda)$ counts the number of distinct eigenvalues less than $\lambda$ (here a continuous parameter).

In addition, for each of the eight towers, I include the next one up beyond $\lambda=10^6$, making a total of 164,311 eigenvalues displayed on this page.

For a glimpse of what the four symmetry classes (A,B,C,S) look like, the plots of the eigenfunctions for the lowest 802 Dirichlet eigenvalues here.

I used the GSVD method to sweep over the interval $\lambda=0$ to just over 1,000,000. I used various but related methods to identify the eigenvalues by watching the smallest singular value as I "swept" over the interval from $\lambda=0$ to $10^6$.

Actual number of distinct

Dirichlet eigenvalueswith $\lambda< 1,000,000$: $$ 13531+27302+27304+13776= 81,913$$Actual number of

Dirichlet eigenfunctionswith $\lambda< 1,000,000$: $$ 13531+2(27302+27304)+13776= 136,519$$Weyl's two term counting formula (disregarding constant term) for

Dirichlet eigenvalues: $$ N_w(10^6) = \frac{1}{4\pi}\,\left\{ A \times (10^6) - 5 \times (10^3)\right\} = 136,513.4$$ where $A=10\tan(3\pi/10)/8\approx 1.7204774005$ and 5 are the area and perimeter of the pentagon.

Actual number of distinct

Neumann eigenvalueswith $\lambda< 1,000,000$: $$ 13610+ 27460 + 27464 + 13856 = 82,390 $$Actual number of

Neumann eigenfunctionswith $\lambda< 1,000,000$: $$ 13610+ 2(27460 + 27464) + 13856 = 137,314 $$Weyl's two term counting formula (disregarding constant term) for

Neumann eigenvalues: $$ N_w(10^6) = \frac{1}{4\pi}\,\left\{ A \times (10^6) + 5 \times (10^3)\right\} = 137,309.2$$ where $A=10\tan(3\pi/10)/8\approx 1.7204774005$ and 5 are the area and perimeter of the pentagon.

Total of Dirichlet eigenvalues in these lists $= 81,917$

DA DB DC DS (13,532) (27,303) (27,305) (13,777)

[L.DA.dat]

[L.DB.dat]

[L.DC.dat]

[L.DS.dat]

Total number$^\dagger$ of Neumann eigenvalues in these lists $= 82,394$

NA NB NC NS (13,611) (27,461) (27,465) (13,857)$^\dagger$

[L.NA.dat]