= Normalizable zero mode of a polynomial factorized Hamiltonian
{title2=$\psi_0(x)=C\exp[-\hbar^{-1}\int_0^xf(t)\,dt]$}
For real <polynomial> $f(x)=cx^n+\cdots$, $c\ne0$, the zero mode of $H=(p+if)(p-if)/(2m)$ is normalizable on $\mathbb R$ exactly when $n$ is odd and $c>0$. Its squared modulus has leading exponential $\exp[-2cx^{n+1}/((n+1)\hbar)]$. Odd $n$ gives the same tail sign at both ends; positivity of $c$ makes both tails decay. This kernel is one-dimensional because $(p-if)\psi=0$ is a first-order <differential equation>. On the <operator domain> where integration by parts is valid, the energy form is $\|(p-if)\psi\|^2/(2m)\geq0$.
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