Quantum theory of nonlocal nonlinear Schrodinger equation

TL;DR

This paper quantizes a nonlocal nonlinear Schrödinger model, identifies issues with the canonical inner product, and resolves them to show the model is equivalent to local bosonic systems with delta interactions, solved exactly via Bethe ansatz.

Contribution

It provides an exact quantization of the nonlocal nonlinear Schrödinger equation and resolves inner product issues, revealing its equivalence to local bosonic models.

Findings

Inner product problem is resolved by redefining the inner product.

The nonlocal model is shown to be equivalent to local bosons with delta interaction.

Exact eigenstates are obtained using Bethe ansatz.

Abstract

Nonlocal nonlinear Schrodinger model is quantised and exactly solved using the canonical framework. It is found that the usual canonical quantisation of the model leads to a theory with pathological inner product. This problem is resolved by constructing another inner product over the vector space of the theory. The resultant theory is found to be identical to that of nonrelativistic bosons with delta function interaction potential, devoid of any nonlocality. The exact eigenstates are found using the Bethe ansatz technique.