Preprint: Maximal monotonicity and contraction semigroup for the quantum drift-diffusion (Derrida–Lebowitz–Speer–Spohn) equation

17 August 2026

Together with Daniel Matthes and Giuseppe Savaré, we study the quantum drift-diffusion, or Derrida–Lebowitz–Speer–Spohn (DLSS), equation for a nonnegative density $\varrho$ on a bounded convex domain with Neumann boundary conditions, in the square-root variable $u=\sqrt{\varrho}$ . We show that the DLSS operator, defined and monotone on smooth strictly positive functions, admits a unique maximal monotone extension in $L^{2}(\Omega)$ , explicitly given by the minimal (defect-free) operator plus the normal cone of the positivity constraint.

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