Don't normalize eigenfunctions in examples (#882)

* don't re-normalize eigenfunctions in examples

* use real part of eigenvalues

Author: jishnub

Project.toml

@@ -22,7 +22,7 @@ StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 
 [compat]
 AbstractFFTs = "1.0"
-ApproxFunBase = "0.8.20"
+ApproxFunBase = "0.8.24"
 ApproxFunBaseTest = "0.1"
 ApproxFunFourier = "0.3"
 ApproxFunOrthogonalPolynomials = "0.5, 0.6"

examples/Eigenvalue.jl

@@ -9,7 +9,7 @@ import Plots
 using LinearAlgebra: norm
 p = Plots.plot(V, legend=false, ylim=(-Inf, λ[22]))
 for k=1:20
-    Plots.plot!(real(v[k]/norm(v[k]) + λ[k]))
+    Plots.plot!(real(v[k]) + real(λ[k]))
 end
 p
 
@@ -28,21 +28,21 @@ Plots.plot(λ, title = "Eigenvalues", legend=false)
 include("Eigenvalue_well_barrier.jl")
 
 # We plot the first few eigenfunctions offset by their eigenvalues.
-# The eigenfunctions appear in odd-even pairs as expected.
+# The eigenfunctions appear in odd-even pairs by construction.
 
 import Plots
 using LinearAlgebra: norm
 p1 = Plots.plot(Vfull, legend=false, ylim=(-Inf, λ[14]), title="Eigenfunctions",
     seriestype=:path, linestyle=:dash, linewidth=2)
 Plots.vline!([-Lx/2, Lx/2], color=:black)
 for k=1:12
-    Plots.plot!(real(v[k]/norm(v[k]) + λ[k]))
+    Plots.plot!(real(v[k]) + λ[k])
 end
 
 # Zoom into the ground state:
 p2 = Plots.plot(Vfull, legend=false, ylim=(-Inf, λ[3]), title="Ground state",
     seriestype=:path, linestyle=:dash, linewidth=2)
 Plots.vline!([-Lx/2, Lx/2], color=:black)
-Plots.plot!(real(v[1]/norm(v[1]) + λ[1]), linewidth=2)
+Plots.plot!(real(v[1]) + λ[1], linewidth=2)
 
 Plots.plot(p1, p2, layout = 2)

examples/Eigenvalue_well_barrier.jl

@@ -35,13 +35,14 @@ Scomplement = Legendre(-Lx/2..0);
 # and those of even orders are even functions
 # Using this, for the odd solutions, we negate the even-order coefficients to construct the odd image in `-Lx/2..0`
 function oddimage(f, Scomplement)
-	coeffs = [(-1)^isodd(m) * c for (m,c) in enumerate(coefficients(f))]
-	Fun(Scomplement, coeffs)
-end;
+    coeffs = [(-1)^isodd(m) * c for (m,c) in enumerate(coefficients(f))]
+    Fun(Scomplement, coeffs)
+end
 voddimage = oddimage.(vodd, Scomplement);
 
 # Construct the functions over the entire domain `-Lx/2..Lx/2` as piecewise sums over the two half domains `-Lx/2..0` and `0..Lx/2`
-voddfull = voddimage .+ vodd;
+# The eigenfunctions `vodd` are normalized on the half-domain, so we normalize the sum by dividing it by `√2`
+voddfull = (voddimage .+ vodd)./√2;
 
 # Even solutions, with a Neumann condition at `0` representing the symmetry of the function
 B = [lneumann(S); rdirichlet(S)];
@@ -51,16 +52,16 @@ Seig = ApproxFun.SymmetricEigensystem(H, B);
 
 # For the even solutions, we negate the odd-order coefficients to construct the even image in `-Lx/2..0`
 function evenimage(f, Scomplement)
-	coeffs = [(-1)^iseven(m) * c for (m,c) in enumerate(coefficients(f))]
-	Fun(Scomplement, coeffs)
-end;
+    coeffs = [(-1)^iseven(m) * c for (m,c) in enumerate(coefficients(f))]
+    Fun(Scomplement, coeffs)
+end
 vevenimage = evenimage.(veven, Scomplement);
-vevenfull = vevenimage .+ veven;
+vevenfull = (vevenimage .+ veven)./√2;
 
 # We interlace the eigenvalues and eigenvectors to obtain the entire spectrum
 function interlace(a::AbstractVector, b::AbstractVector)
-	vec(permutedims([a b]))
-end;
+    vec(permutedims([a b]))
+end
 λ = interlace(λeven, λodd);
 v = interlace(vevenfull, voddfull);