Sine-Gordon kink fluctuation operator (${\Delta }_x=-{\partial }_x^2+1-2{\mathrm{sech}}^{2}x$)

The second variation of the Sine-Gordon theory action about its static kink gives ${\Delta }_x=-{\partial }_x^2+1-2{\mathrm{sech}}^{2}x$. Let $A={\partial }_x+\tanh x$. Then ${\Delta }_x=A^{†}A$ and $A{A}^{†}=-{\partial }_x^2+1$. 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.

The normalized eigenfunction ${\psi }_0(x)=\mathrm{sech}x/\sqrt{2}$ has zero eigenvalue under the Sine-Gordon kink fluctuation operator. It is proportional to ${\partial }_x{\varphi }_K$ and comes from shifting the collective coordinate of the kink. Its frequency is zero, so it contributes no oscillator zero-point energy; it must be handled separately from a Gaussian functional determinant.