Analyze a Sturm–Liouville Operator with an Asymmetric Potential
Find the five smallest periodic eigenvalues and eigenfunctions of a Sturm–Liouville operator.
Specify a Sturm–Liouville operator.
In[1]:=
![Click for copyable input](assets.en/analyze-a-sturm-liouville-operator-with-an-asymmet/In_18.png)
V[x_] := Cos[x] + x;
\[ScriptCapitalL] = -u''[x] - (V''[x] - V'[x]^2) u[x];
Specify a periodic boundary condition.
In[2]:=
![Click for copyable input](assets.en/analyze-a-sturm-liouville-operator-with-an-asymmet/In_19.png)
\[ScriptCapitalB] = u[0] == u[2 \[Pi]];
Find the five smallest eigenvalues and eigenfunctions.
In[3]:=
![Click for copyable input](assets.en/analyze-a-sturm-liouville-operator-with-an-asymmet/In_20.png)
{vals, funs} =
NDEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x], {x, 0, 2 \[Pi]}, 5];
Inspect the eigenvalues.
In[4]:=
![Click for copyable input](assets.en/analyze-a-sturm-liouville-operator-with-an-asymmet/In_21.png)
vals
Out[4]=
![](assets.en/analyze-a-sturm-liouville-operator-with-an-asymmet/O_10.png)
Visualize the eigenfunctions.
In[5]:=
![Click for copyable input](assets.en/analyze-a-sturm-liouville-operator-with-an-asymmet/In_22.png)
Plot[funs, {x, 0, 2 \[Pi]}]
Out[5]=
![](assets.en/analyze-a-sturm-liouville-operator-with-an-asymmet/O_11.png)