On quantum integrability of the LandauLifshitz model
Abstract
We investigate the quantum integrability of the LandauLifshitz (LL) model and solve the longstanding problem of finding the local quantum Hamiltonian for the arbitrary nparticle sector. The particular difficulty of the LL model quantization, which arises due to the illdefined operator product, is dealt with by simultaneously regularizing the operator product and constructing the selfadjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantummechanical Hamiltonian, are also resolved in our method for the arbitrary nparticle 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 twoparticle sector case. Moreover, we demonstrate the Smatrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the selfadjoint extensions.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 October 2009
 DOI:
 10.1063/1.3231789
 arXiv:
 arXiv:0812.0188
 Bibcode:
 2009JMP....50j3518M
 Keywords:

 integration;
 mathematical operators;
 matrix decomposition;
 quantum theory;
 Smatrix theory;
 03.65.Fd;
 02.30.Cj;
 03.65.Nk;
 02.30.Tb;
 02.10.Yn;
 Algebraic methods;
 Measure and integration;
 Scattering theory;
 Operator theory;
 Matrix theory;
 High Energy Physics  Theory;
 Condensed Matter  Strongly Correlated Electrons;
 Mathematical Physics
 EPrint:
 17 pages