When considering the heat kernel of a Schr\"odinger operator

$$- \Delta + V(x) $$

where $\Delta$ is the standard Laplacian on ${\mathbb R}^n$ and $V$ is a nonnegative potential function that has nice behavior at infinity (proper, grows polynomially), one usually sees the term


and the asymptotic expansion of the Gaussian integral

$$\int_{{\mathbb R}^n} e^{-t V(x)},\ t \to 0.$$

If $V(x)$ is homogeneous, namely, $V(rx) = r^\alpha V(x)$, the above integral is just a power of $t$ (in particular, no log term). However it seems that in more general case when $V(x)$ is not homogeneous, for example, when

$$V(x, y) = V_1(x, y) + V_2(x, y)$$

which is the sum of two homogeneous polynomials of different degrees, the expansion may have $\log t$ terms.

Question: What is the general form of the asymptotic expansion of this integral as $ t\to 0$?


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