Definition and Self-Adjointness of the Stochastic Airy Operator

In this note, it is shown that the stochastic Airy operator, which is the "Schr\"odinger operator" on the half-line whose potential term consists of Gaussian white noise plus a linear term tending to $+\infty$, can naturally be defined as a generalized Sturm-Liouville operator and that it is self-adjoint and has purely discrete spectrum with probability one. Thus "stochastic Airy spectrum" of Ram\'irez, Rider and Vir\'ag is the spectrum of an operator in the ordinary sense of the word.