= Solution
Use the <Fourier transform> convention $\hat f(s)=\int_{\mathbb R}e^{-isx}f(x)\,dx$. First observe that $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$ satisfies the differential equation, decay, and normalization, using the <Gaussian integral>. The uniqueness allowed in the question identifies it with $f$. In particular it is a <Schwartz function>, so the derivative and multiplication rules below are justified.
Taking the <Fourier transform> of the differential equation gives
$$
-s^2\hat f+\hat f+\hat f''=0.
$$
The <Riemann-Lebesgue lemma> gives decay of $\hat f$. Every decaying solution $h$ of this equation is proportional to $e^{-s^2/2}$: setting $w=h'+sh$ gives $w'=sw$, and integration yields
$$
h(s)=e^{-s^2/2}\left(D+C\int_0^se^{t^2}\,dt\right).
$$
The term with $C\ne0$ grows like $Ce^{s^2/2}/(2s)$ as $s\to+\infty$, so decay forces $C=0$. Thus $\hat f=\lambda f$ for a scalar $\lambda$. This establishes proportionality before using <Fourier inversion> to calculate the scalar.
Back to article page
