= Fluctuation operator of a phi-four kink
{title2=$H_K=-\partial_x^2+4c^2-6c^2\operatorname{sech}^2(cx)$}
Linearizing about the <phi-four kink> gives the displayed one-dimensional <Schrodinger operator> in unit kinetic normalization. Its translational <zero mode in field theory> is proportional to $\operatorname{sech}^2(cx)$. The localized shape eigenfunction $\operatorname{sech}(cx)\tanh(cx)$ has squared frequency $3c^2$, while the continuum starts at $4c^2$. Substitution verifies both bound-state eigenfunctions. The <Pöschl-Teller potential> permits a short completeness argument for the bound modes. In $y=cx$ let $A_\ell=\partial_y+\ell\tanh y$. Then $H_K/c^2=A_2^\dagger A_2$, its partner is $A_2A_2^\dagger=A_1^\dagger A_1+3$, and $A_1A_1^\dagger=-\partial_y^2+1$. The last operator has no bound states. The kernels of $A_1$ and $A_2$ give the two stated modes, while the partner spectra exclude any further normalizable bound modes.
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