... and its spectrum

All eigenvalues $\lambda < \frac{K \pi}{\ell}^2$ of the equilateral graph $\Gamma$ can be computed with eigvals_quantum. By default, K=3 is applied.

eigvals_quantum(Γ)

First 12 eigenvalues:
12-element Vector{Any}:
 0
 0.1111111111111111
 0.1111111111111111
 0.11111111111111133
 0.4444444444444444
 0.9999999999999993
 1.0
 1.0
 1.7777777777777777
 2.777777777777778
 2.777777777777778
 2.7777777777777795

An eigenfunction basis with all eigenfunctions $\phi_{\lambda}$, $\lambda < \frac{K \pi}{\ell}^2$ can be constructed via eigen_quantum

σ = eigen_quantum(Γ)

First 12 eigenvalues and eigenfunctions:
values:
12-element Vector{Float64}:
 0.0
 0.1111111111111111
 0.1111111111111111
 0.11111111111111133
 0.4444444444444444
 0.9999999999999993
 1.0
 1.0
 1.7777777777777777
 2.777777777777778
 2.777777777777778
 2.7777777777777795
 Coefficients of eigenfunctions ϕ_e = A_e cos(√λ x) + B_e sin (√λ x) for q = 1, …, Q:
A_e:
4×12 SparseMatrixCSC{Float64, Int64} with 24 stored entries:
 0.230329   ⋅    ⋅   2.16983e-16  …  -0.325735   ⋅    ⋅   2.16983e-16
 0.230329   ⋅    ⋅   2.16983e-16     -0.325735   ⋅    ⋅   2.16983e-16
 0.230329   ⋅    ⋅   2.16983e-16     -0.325735   ⋅    ⋅   2.16983e-16
 0.230329   ⋅    ⋅   2.16983e-16     -0.325735   ⋅    ⋅   2.16983e-16
 B_e:
4×12 SparseMatrixCSC{Float64, Int64} with 37 stored entries:
  ⋅   -0.265962  -0.460659     …  -0.265962  -0.460659     -0.188063
  ⋅    0.531923  -5.32401e-17      0.531923  -5.32401e-17  -0.188063
  ⋅   -0.265962   0.460659        -0.265962   0.460659     -0.188063
  ⋅     ⋅          ⋅                ⋅          ⋅            0.56419

Eigenvalues and eigenfunctions are always returned in ascending order. The function allows to explicitly construct a specific eigenfunction:

ϕ_q = eigenfunction(Γ, σ, 5)

4-element Vector{Function}:
 #6 (generic function with 1 method)
 #6 (generic function with 1 method)
 #6 (generic function with 1 method)
 #6 (generic function with 1 method)

It can be vizualized using plot_function_3d

plot_function_3d(Γ, ϕ_q)