Quantum corrections to static solutions of the sine-Gordon and Nahm models via a generalized zeta function

We consider one-dimensional Yang–Mills–Nahm and sine-Gordon models in terms of a class of nonlinear Klein–Gordon–Fock equations. We perform a semiclassical quantization of the models using a generalized zeta function and construct a representation of the quantum theory in terms of the diagonal of the Green's function for the heat equation with an elliptic potential via solutions of the Hermite equation. We formulate an alternative approach based on Baker–Akhiezer functions for the KP equation. We evaluate quantum corrections to the action of the Nahm and sine-Gordon models. We study the fields from the class of elliptic functions. We take extra variables of arbitrary dimensions into account for possible applications of quantized sine-Gordon solitons in solid state physics via the Frenkel–Kontorova model or other models. For the Nahm model, whose field is represented via an elliptic (lemniscate) integral by construction, the Yang–Mills field mass coincides with the correction evaluated as a hyperelliptic integral.