Bounding the Bogoliubov coefficients

While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little work seems to have been put into the complementary problem of establishing rigourous bounds on the exact solutions. We have in mind either bounds on parametric amplification and the related quantum phenomenon of particle production (as encoded in the Bogoliubov coefficients), or bounds on transmission and reflection coefficients. Modifying and streamlining an approach developed by one of the present authors [M. Visser, Phys. Rev. A 59 (1999) 427–438, arXiv:quant-ph/9901030], we investigate this question by developing a formal but exact   solution for the appropriate second-order linear ODE in terms of a time-ordered exponential of 2×22×2 matrices, then relating the Bogoliubov coefficients to certain invariants of this matrix. By bounding the matrix in an appropriate manner, we can thereby bound the Bogoliubov coefficients.