Numerical simulations for the Toda lattices Hamiltonian system: Higher order symplectic illustrative perspective.

In this paper we apply some higher order symplectic numerical methods to analyze the dynamics of 3-site Toda lattices (reduced to relative coordinates). We present benchmark numerical simulations that has been generated from the HOMsPY (Higher Order Methods in Python) library. These results provide detailed information of the underlying Hamiltonian system.

Quantum loop expansion to high orders, extended Borel summation, and comparison with exact results.

We compare predictions of the quantum loop expansion to (essentially) infinite orders with (essentially) exact results in a simple quantum mechanical model. We find that there are exponentially small corrections to the loop expansion, which cannot be explained by any obvious "instanton"-type corrections. It is not the mathematical occurrence of exponential corrections but their seeming lack of any physical origin which we find surprising and puzzling.