This is not the answer to my question, rather an explanation into its background.

Google AI tells me that in Dirac bracket notation, the nonlinear Schrödinger equation (NLSE) is expressed as:

iħ ∂|ψ⟩/∂t = (H + V(|ψ⟩))|ψ⟩,

where |ψ⟩ represents the quantum state, H is the Hamiltonian operator, V is the nonlinear potential, and ħ is the reduced Planck constant.

Dirac Notation:

Dirac notation, also known as bra-ket notation, is a mathematical formalism used to represent quantum states (kets: |ψ⟩) and their duals (bras: ⟨ψ|).

Equation Breakdown:

· iħ ∂|ψ⟩/∂t: This term represents the time evolution of the quantum state |ψ⟩, where i is the imaginary unit, ħ is the reduced Planck constant, and ∂/∂t is the partial derivative with respect to time.

· H: This is the Hamiltonian operator, which represents the total energy of the system.

· V(|ψ⟩): This term represents the nonlinear potential, which depends on the quantum state |ψ⟩ itself.

ð V(||ψ⟩) = constant*⟨ψ|ψ⟩ = for NLS.

ð V(||ψ⟩) = constant*log(⟨ψ|ψ⟩ ) for LogSE.

However, that must be wrong. Apart from the linearity constraint, there is an important associativity rule in Dirac bra ket:

https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation#Associativity

which does not hold for nonlinear operators.

Hence my question. I know the LogSE has been treated in Sobolev space but is there a way of extending Dirac bra ket to express the time evolution of the LogSE?