Scalar field stochastic dynamics in de Sitter spacetime from exact solutions of quantum deficient oscillators

The stochastic dynamics of a scalar field in de Sitter spacetime can be regarded as a non-perturbative diffusion process, to which exact distribution and correlation functions are constructed by utilising the correspondence between diffusion and Schrödinger equations. The Krein–Adler transformation of the quantum harmonic oscillator deletes several pairs of the energy levels to define anharmonic oscillators that we dub quantum deficient oscillators, based on which this article constructs a new class of exact solutions in stochastic inflation. In addition to the simplest single-well model, an exactly solvable double-well model is also presented. The results are further extended to exactly solvable models with multiple wells, allowing analytical studies on various cosmological phenomenologies.