= Sine-Gordon kink fluctuation operator
{c}
{title2=$\Delta_x=-\partial_x^2+1-2\operatorname{sech}^2x$}
The <second variation> of the <Sine-Gordon theory> action about its static <kink> gives $\Delta_x=-\partial_x^2+1-2\operatorname{sech}^2x$. Let $A=\partial_x+\tanh x$. Then $\Delta_x=A^\dagger A$ and $AA^\dagger=-\partial_x^2+1$. Thus $\Delta_x$ is nonnegative, with its <translational zero mode of a sine-Gordon kink> and a continuum at $\omega^2=k^2+1$. The construction is the <supersymmetric factorization of the one-soliton potential>.
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