The trace formula for the nonlinear Klein-Gordon equation. (English) Zbl 0807.35132

Buslaev, V. S.; Rybakina, E. A.

Birman, M. Sh. (ed.), Wave propagation. Scattering theory. Transl. from the Russian by Peter Zhevandrov. Transl. ed. by Simeon Ivanov. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 157, 191-213 (1993).

Summary: A trace formula for the nonlinear Klein-Gordon equation is obtained. According to this formula, the regularized determinant of the Jacobi operator (i.e., the operator in variations) on a fixed trajectory of the nonlinear Klein-Gordon equation \(\square u + m^ 2u + V'(u) = 0\) is equal to the determinant of the following mapping: initial momentum (for a fixed coordinate) \(\to\) final coordinate.
The regularization of the determinant of the differential operator (apart from the standard procedure of passage to the “determinant of the perturbation”) contains elements having the same technical origin as the procedure of regularization of divergences appearing when the field under consideration is quantized.
For the entire collection see [Zbl 0788.00073].

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
70H03	Lagrange’s equations

Keywords:

Lagrangian mechanics; Hamiltonian mechanics; regularized determinant of the Jacobi operator