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arXiv:0812.0188 · Journal of Mathematical Physics

On quantum integrability of the Landau–Lifshitz model

A. Melikyan, Aleksandr Pinzul

Published Oct 1, 2009• 14 citations
arXivHTMLPDF

Abstract

We investigate the quantum integrability of the Landau–Lifshitz (LL) model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin [Lett. Math. Phys. 15, 357 (1988)] and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions.

DOI
10.1063/1.3231789

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We investigate the quantum integrability of the Landau–Lifshitz (LL) model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin [Lett. Math. Phys. 15, 357 (1988)] and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions.
We solve the operator ordering problem for the quantum continuous integrable su(1,1) Landau–Lifshitz model, and give a prescription to obtain the quantum trace identities, and the spectrum for the higher-order local charges. We also show that this method, based on operator regularization and renormalization, which guarantees quantum integrability, as well as the construction of self-adjoint extensions, can be used as an alternative to the discretization procedure, and unlike the latter, is based only on integrable representations.
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We study self-adjoint operators defined by factorizing second-order differential operators in first-order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum-mechanical models such as the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero–Sutherland type.
The properties of exactly integrable two-dimensional quantum systems are reviewed and discussed. The nature of exact integrability as a physical phenomenon and various aspects of the mathematical formalism are explored by discussing several examples, including detailed treatments of the nonlinear Schr\"odinger (delta-function gas) model, the massive Thirring model, and the six-vertex (ice) model. The diagonalization of a Hamiltonian by Bethe's Ansatz is illustrated for the nonlinear Schr\"od\'{\i}nger model, and the integral equation method of Lieb for obtaining the spectrum of the many-body system from periodic boundary conditions is reviewed. Similar methods are applied to the massive Thirring model, where the fermion-antifermion and bound-state spectrum are obtained explicitly by the integral equation method. After a brief review of the classical inverse scattering method, the quantum inverse method for the nonlinear Schr\"odinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique. In the quantum inverse method, an auxiliary linear problem is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length. The particular operators for which the string is infinitely long (free boundary conditions) or forms a closed loop around a cylinder (periodic boundary conditions) correspond to the quantized scattering data and have a special significance. One of them creates the Bethe eigenstates, while the other is the generating function for an infinite number of conservation laws. The analogous operators on a lattice are constructed for the symmetric six-vertex model, where the object which corresponds to a solution of the auxiliary linear problem is a string of vertices contracted over horizontal links (arrows). The relationship between the quantum inverse method and the transfer matrix formalism is exhibited. The inverse Gel'fand-Levitan transform which expresses the local field operator as a functional of the quantized scattering data is formulated for the nonlinear Schr\"odinger equation, and some interesting properties of this transformation are noted, including its reduction to a Jordan-Wigner transformation in the limit of infinitely repulsive coupling.
In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square integrable and then can be associated with a physical state. However, an analysis based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.
We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional non-relativistic motion of a particle in the potential field , For g 2 >0 and g 1 <0, the potential is known as the Kratzer potential V K ( x ) and is usually used to describe molecular energy and structure, interactions between different molecules and interactions between non-bonded atoms. We construct all self-adjoint Schrödinger operators with the potential V ( x ) and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying self-adjoint extensions by (asymptotic) self-adjoint boundary conditions. Solving spectral problems, we follow Krein's method of guiding functionals. This work is a continuation of our previous works devoted to the Coulomb, Calogero and Aharonov–Bohm potentials.