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.

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V[x_] := Cos[x] + x; \[ScriptCapitalL] = -u''[x] - (V''[x] - V'[x]^2) u[x];

Specify a periodic boundary condition.

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\[ScriptCapitalB] = u[0] == u[2 \[Pi]];

Find the five smallest eigenvalues and eigenfunctions.

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{vals, funs} = NDEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]}, u[x], {x, 0, 2 \[Pi]}, 5];

Inspect the eigenvalues.

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vals

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Visualize the eigenfunctions.

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Plot[funs, {x, 0, 2 \[Pi]}]

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