Boundedness of solutions to the Schrödinger equation under Neumann boundary conditions

We deal with Neumann problems for Schrödinger type equations, with non-necessarily bounded potentials, in possibly irregular domains of the n-dimensional Euclidean space. Sharp balance conditions between the regularity of the domain and the integrability of the potential for any solution to be bounded are established. The regularity of the domain is described either through its isoperimetric function or its isocapacitary function. The integrability of the sole negative part of the potential plays a role, and is prescribed via its distribution function. The relevant conditions amount to the membership of the negative part of the potential to a Lorentz type space defined either in terms of the isoperimetric function, or of the isocapacitary function of the domain. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM.