Branched Quantization
  We propose a method for quantization of Lagrangians for which the Hamiltonian, as a function of momentum, is a branched function with cusps. Appropriate boundary conditions, which we identify, insure unitary time evolution. In special cases a dual (canonical) transformation maps the problem into a problem of quantum mechanics on singular spatial manifolds, which we also develop. Several possible applications are indicated. 
Alfred Shapere and Frank Wilczek
quant-ph cond-mat.other hep-th