The Laplace Transform and Quantum Curves

A Laplace transform that maps the topological recursion (TR) wavefunction to its $x$-$y$ swap dual is defined. This transform is then applied to the construction of quantum curves. General results are obtained, including a formula for the quantisation of many spectral curves of the form $e^x{P}_2(e^y)-P_1(e^y)=0$ where $P_1$ and $P_2$ are coprime polynomials; an important class that contains interesting spectral curves related to mirror symmetry and knot theory that have, heretofore, evaded the general TR-based methods previously used to derive quantum curves. Quantum curves known in the literature are reproduced, and new quantum curves are derived. In particular, the quantum curve for the $T$-equivariant Gromov-Witten theory of $\mathbb{P}(a,b)$ is obtained.