[Paper Review] Higher conservation laws for the quantum non-linear Schroedinger equation

Brian Davies, V. E. Korepin|arXiv (Cornell University)|Sep 29, 2011

Cold Atom Physics and Bose-Einstein CondensatesPhysics and Astronomy22 citations

TL;DR

This paper resolves long-standing concerns about the consistency of higher conservation laws in the quantum non-linear Schrödinger equation (QNLS) by constructing explicit forms for $H_3$ and $H_4$ using the Quantum Inverse Scattering Method (QISM). It demonstrates that these operators are not obtainable by normal ordering the classical counterparts, and shows that the quasi-classical limit is correct, confirming the validity of the QISM for QNLS despite prior objections.

ABSTRACT

Quantum non-linear SCHROEDINGER equation is equivalent to Lieb-Liniger model. It has non-trivial conservation laws. Recently these conservation laws were used for evaluation of the three-body recombination rate for interacting gas of quantum bosons. These conservations laws were known already in 1989. Submitted text is retyping of the preprint of Centre for Mathematical Analysis of Australian National University CMA-R33-89. It was discussed in Leningrad Branch of the V.A. Steklov Mathematical Institute at that time. A copy of the original preprint can be found in the section Quantum Inverse Scattering Method of the web-page http://insti.physics.sunysb.edu/~korepin/

Motivation & Objective

• To resolve objections raised by Gutkin (1985, 1988) concerning the consistency of higher conservation laws in the quantum non-linear Schrödinger equation (QNLS).
• To construct explicit, well-defined forms of the higher Hamiltonians $H_3$ and $H_4$ in second-quantized form, avoiding issues from naive normal ordering.
• To verify that the QISM yields the correct higher-order integrals of motion and that their quasi-classical limits match classical expectations.
• To clarify the failure of recovering the asymptotic expansion of the fundamental integral of motion $A(\lambda)$ via normal ordering of the classical expansion.
• To establish that quantum corrections arise in the expansion of $A(\lambda)$, and that the lattice regularisation provides a consistent framework for handling singularities in the continuous limit.

Proposed method

• Construct $H_3$ and $H_4$ explicitly in second-quantized form using the QISM, with $H_3 = \int dx\,\bigl\{\Psi^\dagger(x)\Psi_{xxx}(x) - \frac{3c}{2}\Psi^{\dagger}(x)^2(\Psi(x)^2)_x\bigr\}$.
• Demonstrate that $H_4$ cannot be obtained by normal ordering the classical conserved quantity $H_4^{\text{cl}}$, due to singular operator products.
• Use the Bethe ansatz eigenfunctions $\chi_N$ to verify that $H_3$ and $H_4$ have eigenvalues $i^3\sum \lambda_j^3$ and $\sum \lambda_j^4$, respectively.
• Analyze the asymptotic expansion of the fundamental integral of motion $A(\lambda)$ in inverse powers of $\lambda$, showing it fails to recover via normal ordering.
• Apply the intertwining property of $A(\lambda)$ with free and interacting Hamiltonians, using boundary conditions at $x_j = x_k$ to define the domain of validity.
• Use lattice regularisation as a control mechanism to justify the continuous limit and resolve ordering ambiguities in higher conservation laws.

Experimental results

Research questions

• RQ1 Can explicit, well-defined forms of the higher Hamiltonians $H_3$ and $H_4$ be constructed in the second-quantized formulation of the QNLS?
• RQ2 Is the quasi-classical limit of the higher conservation laws from the QISM consistent with the classical counterparts?
• RQ3 Why does normal ordering of the classical $H_4^{\text{cl}}$ fail to reproduce the correct quantum $H_4$?
• RQ4 Can the asymptotic expansion of the fundamental integral $A(\lambda)$ in inverse powers of $\lambda$ be recovered from normal ordering of the classical expansion?
• RQ5 Do quantum corrections appear in the expansion of $A(\lambda)$, and if so, how are they related to the structure of the QISM?

Key findings

• The second-quantized form of $H_3$ is explicitly given as $H_3 = \int dx\,\bigl\{\Psi^\dagger(x)\Psi_{xxx}(x) - \frac{3c}{2}\Psi^{\dagger}(x)^2(\Psi(x)^2)_x\bigr\}$, which is not derivable by normal ordering the classical $H_3^{\text{cl}}$.
• The operator $H_4$ cannot be written in a form that matches the naive normal ordering of the classical $H_4^{\text{cl}}$, and such expressions lead to ill-defined products like $\delta^2(x_1 - x_2)$.
• The correct $H_4$ is obtained via the QISM and intertwining relations, not by formal quantization of the classical expression.
• The asymptotic expansion of $A(\lambda)$ in inverse powers of $\lambda$ fails to reproduce the correct differential operators when derived via normal ordering, due to singular behavior at particle collision points.
• Quantum corrections arise in the expansion of $A(\lambda)$, starting with $A_3$, and are not captured by classical or naive quantization procedures.
• The QISM produces conservation laws consistent with the Bethe ansatz eigenfunctions and their eigenvalues, confirming the validity of the method despite prior objections.