Sharp Gaussian upper bounds for Schrödinger semigroups with singular potentials and mixed boundary conditions

Key Points

• This research aims to establish sharp Gaussian upper bounds for Schrödinger semigroups with singular potentials. It investigates the conditions under which these bounds hold.
• Established Gaussian bounds for Schrödinger semigroups generated by the operator -Δ_D + V on (0,∞)^d with Dirichlet boundary conditions.
• Analyzed singular complex-valued potentials that satisfy dissipativity and Kato conditions, ensuring integrability.
• Proved bounds on the integral kernel of the perturbed semigroup.
• The integral kernel of the perturbed semigroup satisfies the sharp Gaussian upper bound as specified.
• Conditions for the potential V, including Re V ≤ 0 and the weighted global Kato condition, are met.
• Demonstrated that the Gaussian bounds are essential for understanding the behavior of the semigroups.