The ODE/IM Correspondence

The Ordinary Differential Equation/Integrable Model (ODE/IM) correspondence links quantum integrable models with spectral problems akin to Sturm-Liouville type equations. The birth of this research domain can be traced back to 1998 when the first connection was found: the point spectrum of the one-dimensional Schrödinger operator with anharmonic-oscillator potential coincides with the ground-state solution of the Bethe Ansatz equations corresponding to the quantum Korteweg-de Vries model. The area has now matured into an ambitious program to establish a comprehensive correspondence, particularly within the framework of integrable quantum field theories. This article covers the first steps, with an emphasis at various points on the particularly simple case of the simple harmonic oscillator as a way to introduce the main concepts.