Fluctuation operator of a phi-four kink ($H_K=-{\partial }_x^2+4c^2-6c^2{\mathrm{sech}}^2(cx)$)

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 ${\mathrm{sech}}^2(cx)$. The localized shape eigenfunction $\mathrm{sech}(cx)\tanh (cx)$ 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 $H_K/c^2=A_2^{†}{A}_2$, its partner is $A_2{A}_2^{†}=A_1^{†}{A}_1+3$, and $A_1{A}_1^{†}=-{\partial }_y^2+1$. 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.