We analyze a family of singular Schrdinger operators with local singular interactions supported by a hypersurface n, n 2, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of the interaction is characterized by four real parameters, the earlier studied case of - and -interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.

We present a 3D topological picture-language for quantum information. Our approach combines charged excitations carried by strings, with topological properties that arise from embedding the strings in the interior of a 3D manifold with boundary. A quon is a composite that acts as a particle. Specifically, a quon is a hemisphere containing a neutral pair of open strings with opposite charge. We interpret multiquons and their transformations in a natural way. We obtain a type of relation, a string–genus “joint relation,” involving both a string and the 3D manifold. We use the joint relation to obtain a topological interpretation of the C* Hopf algebra relations, which are widely used in tensor networks. We obtain a 3D representation of the controlled NOT (CNOT) gate that is considerably simpler than earlier work, and a 3D topological protocol for teleportation.

In this article, we classify all standard invariants that can arise from a composed inclusion of an A3with an A4subfactor. More precisely, if N⊂P is an A3subfactor and P⊂M is an A4subfactor, then only four standard invariants can arise from the composed inclusion N⊂M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A4subfactors.