Parameter Integration

In its simplest form, the method of parameter integration yields a fundamental solution E of a product P 1(∂)P 2(∂) of differential operators as a simple integral with respect to $$\lambda$$ over fundamental solutions $$E_{\lambda }$$ of the squared convex sums $${\bigl (\lambda P_{1}(\partial ) + (1-\lambda )P_{2}(\partial )\bigr )}^{2}$$ . Heuristically, this relies on the representations of E and of $$E_{\lambda }$$ as inverse Fourier transforms, i.e., $$\displaystyle{\mathcal{F}E = \frac{1} {P_{1}(\text{i}\xi )P_{2}(\text{i}\xi )}\mathop{ =}\limits^{ (\text{F})}\int _{0}^{1} \frac{\text{d}\lambda } {{\bigl (\lambda P_{1}(\xi ) + (1-\lambda )P_{2}(\xi )\bigr )}^{2}} =\int _{ 0}^{1}\mathcal{F}E_{\lambda }\,\text{d}\lambda }$$ where the equation (F) is Feynman’s first formula, see (3.1.1) below (for the name cf. Schwartz [245], Ex. I-8, p. 72). Note that Eq. (3.1.1) boils down to the formula $$a^{-1} - b^{-1} =\int _{ a}^{b}x^{-2}\text{d}x,$$ 0