This paper establishes a rigorous well-posedness theory for constrained nonlinear Schrödinger equations using m-accretive operator theory. The approach unifies mass-preserving and energy-preserving constraints within a single operator-theoretic framework, circumventing limitations of Strichartz-based methods. We prove local and global existence, uniqueness, and continuous dependence for subcritical nonlinearities in Sobolev spaces. The framework characterizes constraint manifolds as nonlinear projections compatible with the m-accretive structure and provides convergence rates for implicit Euler discretization.
Key findings
The constrained Laplacian operator restricted to mass-preserving or energy-preserving subspaces admits an m-accretive extension, providing a semigroup-theoretic foundation for constrained dynamics.
Local well-posedness is established for subcritical power nonlinearities in H^s(R^N) with s > N/2, ensuring classical solutions without relying on Strichartz estimates.
The Lagrange multiplier λ(u) is characterized as a continuous functional of the solution, ensuring the constraint is dynamically maintained for all time.
The constrained NLS generates a strongly continuous nonlinear semigroup on appropriate Sobolev spaces with explicit convergence rates for the implicit Euler approximation.
Results extend to quasilinear variants and apply to physical systems including Bose-Einstein condensates with fixed particle number and nonlinear optical systems with power constraints.
Limitations & open questions
Theory is restricted to subcritical nonlinearities and sufficiently high Sobolev regularity (s > N/2).
Does not cover critical or supercritical cases where traditional Strichartz estimates may be essential.
Constraint manifolds must be compatible with the m-accretive operator structure, limiting applicability to certain geometric constraints.