Topological recursion/quantum curve correspondence

Let $\mathcal{S}$ be an admissible spectral curve with $x,y$ satisfying

$$P(x,y)=0.$$

A quantisation of $P$ is an operator $\hat{P}(\hat{x},\hat{y};\hbar)$ of the form

$$\hat{P}(\hat{x},\hat{y};\hbar)=P(\hat{x},\hat{y})+\sum_{i=1}^{\infty}\hbar^i\hat{P}_i(\hat{x},\hat{y}),$$

where $P(\hat{x},\hat{y})$ is normally ordered and each $\hat{P}_i$ is a normally ordered polynomial of degree at most $\deg P-1$. For a regular point $b\in\Sigma$, let $\psi(z;b)$ be the wavefunction defined by topological recursion, and set $\hat{x}=x\cdot$ and $\hat{y}=\hbar\,d/dx$. Topological recursion/quantum curve correspondence. There exists a regular point $b\in\Sigma$ and a quantisation $\hat{P}(\hat{x},\hat{y};\hbar)$ of $P(x,y)$ such that

$$\hat{P}(\hat{x},\hat{y};\hbar)\psi(z;b)=0.$$

The operator $\hat{P}$ is called the quantum curve or quantum spectral curve. This correspondence gives a precise form to the expectation that topological recursion produces a differential or difference operator annihilating its wavefunction, although the statement is presented here as a conjectural correspondence in general.