# Solving a Eigenvalue Problem

$$\textbf{Eigenvalue problem on a unit square \Omega = [0,1]^2 :}$$

Consider the eigenvalue problem with the Dirichlet boundary condition that is, $$-Lu = \lambda u$$ where $$L = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$$.

The boundary condition is that $$u=0$$ on $$\partial \Omega$$.

I am computing the $$\textbf{eigenvalues}$$ and $$\textbf{eigenfunctions}$$ of Laplacian numerically in Mathematica. Specially, I am interested about the $$\textbf{multiplicity}$$ of a specific eigenvalue. I already know that Tally or Count can certainly count the occurrances of eigenvalues in the list called vals.

{ℒ, ℬ} = {-Laplacian[u[x, y], {x, y}],
DirichletCondition[u[x, y] == 0, True]};

{vals, funs} =
DEigensystem[{ℒ, ℬ},
u[x, y], {x, 0, 1}, {y, 0, 1}, _];

Tally[vals]


In the eigenvalue problem there are infinite numbers of eigenvalues. So, We can not list all of them. I want to find out $$\textbf{multiplicity}$$ of a specific eigenvalue that counts the total number of occurrances. For example, the multiplicity of the eigenvalue $$5 \pi^2$$ is $$2$$. Similarly, I want to find out the multiplicity of the eigenvalue $$50 \pi^2$$ is $$3$$.

• Edit your question to include your Mathematica code rather than pictures of the code. Feb 15 '21 at 17:51
• Convert your cells to InputForm and look at the Markdown help Feb 15 '21 at 18:24
• Feb 15 '21 at 19:30

With the exact solutions

u[{nx_, ny_}, {x_, y_}] = Sin[nx π x] Sin[ny π y];


and $$\{n_x,n_y\}\in\mathbb{N}^2$$, the eigenvalues are

λ[nx_, ny_] = (nx^2 + ny^2) π^2;


Test:

Assuming[Element[nx | ny, PositiveIntegers],
{u[{nx, ny}, {x, 0}], u[{nx, ny}, {x, 1}],
u[{nx, ny}, {0, y}], u[{nx, ny}, {1, y}]} // FullSimplify]
(*    {0, 0, 0, 0}    *)

-D[u[{nx, ny}, {x, y}], {x, 2}] - D[u[{nx, ny}, {x, y}], {y, 2}] ==
λ[nx, ny] * u[{nx, ny}, {x, y}] // FullSimplify
(*    True    *)


The number of pairs $$\{n_x,n_y\}$$ equal to $$\lambda/\pi^2$$ is therefore given by OEIS A063725. You can calculate the multiplicity of a given value of $$i=\lambda/\pi^2$$ directly from the generating function,

g[q_] = (EllipticTheta[3, q] - 1)^2/4;
m[i_Integer?NonNegative] := SeriesCoefficient[g[q], {q, 0, i}]


For example, the multiplicity of $$\lambda=50\pi^2$$ is 3, as you know:

m[50]
(*    3    *)


There is no eigenvalue at $$326\pi^2$$, on the other hand:

m[326]
(*    0    *)


More generally, the $$d$$-dimensional generating function

g[d_, q_] = ((EllipticTheta[3, q] - 1)/2)^d;


allows us to find the multiplicity of the eigenvalue $$\lambda=i\pi^d$$ directly:

m[d_Integer?NonNegative, i_Integer?NonNegative] :=
SeriesCoefficient[g[d, q], {q, 0, i}]


For example, the $$d=17$$-dimensional hypercube has $$1\,416\,786\,753\,216$$ eigenvalues $$\lambda=250\pi^{17}$$:

m[17, 250]
(*    1416786753216    *)


If you want the actual solutions, not just the number of solutions, PowersRepresentations is a good starting point (but it gives solutions containing zeros, which we need to filter out):

s[d_Integer?NonNegative, i_Integer?NonNegative] :=
Select[PowersRepresentations[i, d, 2], Min[#] > 0 &]


For example, the eigenvalue $$\lambda=50\pi^2$$ in $$d=2$$ dimensions can be composed in two ordered ways, as you know:

s[2, 50]
(*    {{1, 7}, {5, 5}}    *)


Enumerating all permutations of these ordered combinations gives the full multiplicity of 3:

t[d_Integer?NonNegative, i_Integer?NonNegative] :=
Join @@ Permutations /@ s[d, i]

t[2, 50]
(*    {{1, 7}, {7, 1}, {5, 5}}    *)


In $$d=17$$ dimensions the eigenvalue $$\lambda=250\pi^{17}$$ has 999 ordered solutions,

s[17, 250]
(*    {{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 14},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 10, 10},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 15},
...
{3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5},
{3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6}}    *)


which then become the aforementioned $$1\,416\,786\,753\,216$$ unordered solutions from t[17, 250].

• We should work with the generating function explicitly because it gives a very efficient recipe, especially in the high-dimensional case. Just look at the given example for the 17-dimensional hypercube: manually summing and counting is going to be very tedious and difficult, whereas the call to m[17, 1000] takes less than half a second. The same is true, to some extent, for your case $d=2$ when you work with large values of $\lambda$. Feb 16 '21 at 8:32
• Sorry, this is not the right place for an introduction to generating functions or Jacobi theta functions. Feb 16 '21 at 9:31