# Jamming exponent identity — H2

Let \(0<c<1\), \(b=(1+c)/2\), and \(H_-(t)=\mathbf1_{t<0}\) for \(t\ne0\).
Let \(J\in C^1(\mathbb R)\) be \(C^4\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy
\[
\frac c2J''=(-b+cH_-)tJ'+J-H_-+\frac{c^2}{4}(J')^2\qquad(t\ne0).
\]
For some \(C_0,q,\eta>0\), assume
\[
|J(t)-H_-(t)|+\sum_{j=1}^4|J^{(j)}(t)|
\le C_0(1+|t|)^qe^{-\eta t^2}
\]
for all sufficiently large \(|t|\). For \(t\ne0\), put \(f(t)=H_-(t)+cJ''(t)/2\).

Let \(a\in\mathbb R\). Suppose \(p\in C^1(\mathbb R)\) is strictly positive, is \(C^2\) on each open half-line, and satisfies
\[
\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\frac{c^2}{2}(p'J'+pJ'')\qquad(t\ne0).
\]
Assume \(p,p'\) grow at most polynomially.

The following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:
\[
\int_{\mathbb R}p\left\{c\left[(f')^2-2f^2(1-f)\right]-(b-a-c)f(1-f)\right\}\,dt=0.
\]