Hamiltonian of the quantized non-linear Schroedinger equation
As a classical equation, the non-linear Schroedinger equation has been studied successfully by means of inverse scattering. A complete family of conserved quantities was exhibited using this technique; however, in the absence of a quantum Inverse Scattering transform, the quantized field equations must be handled differently. The Lagrangian formalism permits the construction of a Hamiltonian associated to the NLSE, which has been explicitly spectrally decomposed by Gaudin in the repulsive case c > 0. Both the attractive, c < 0, and repulsive cases have been studied when the field is restricted to a finite interval with periodic boundary conditions, and some properties of the thermodynamic limit have been exhibited. Our purpose is to treat the attractive case in the quantized system strictly in the thermodynamic limit, explicitly exhibiting all bound states in the course of finding an explicit unitary transformation of the physical Hilbert space to a certain momentum space, which reduces the Hamiltonian to a multiplication operator. Additionally, a so-called multi-channel scattering matrix is calculated, which shows that bound states remain bound through the scattering experiment, and which exhibits the transformation of incoming scattering states to outgoing scattering states as multiplication by a function of a very simple form, in the K-space. Last we perform some calculations to verify whether the conserved functions of phi and phi* found above for the classical problem (or any other functionals) can be quantized to yield unbounded operators which transform into multiplication operators in the K-domain (momentum space).
- Research Organization:
- California Univ., Los Angeles (USA)
- OSTI ID:
- 5260428
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
GENERAL PHYSICS
SCHROEDINGER EQUATION
HAMILTONIANS
BOUND STATE
QUANTUM MECHANICS
SCATTERING
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL OPERATORS
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS
WAVE EQUATIONS
657002* - Theoretical & Mathematical Physics- Classical & Quantum Mechanics