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Supersymmetric factorization of the one-soliton potential

Let

A = \frac{d}{d x} + χ tanh (χ x), A^{†} = - \frac{d}{d x} + χ tanh (χ x) .

Then

A^{†} A = - \frac{d ^{2}}{d x ^{2}} + χ^{2} - 2 χ^{2} sech^{2} (χ x), A A^{†} = - \frac{d ^{2}}{d x ^{2}} + χ^{2} .

Thus the one-soliton Schrodinger operator is paired with the free Schrodinger operator. Its normalizable state annihilated by

A

is proportional to

sech (χ x)

.

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