Schrödinger formalism for a particle constrained to a surface in R13

In this work it is studied the Schrödinger equation for a non-relativistic particle restricted to move on a surface S in a three-dimensional Minkowskian medium R_1^3, i.e., the space R^3 equipped with the metric diag(−1,1,1). After establishing the consistency of the interpretative postulates for the new Schrödinger equation, namely the conservation of probability and the hermiticity of the new Hamiltonian built out of the Laplacian in R_1^3, we investigate the confining potential formalism in the new effective geometry. Like in the well-known Euclidean case, it is found a geometry-induced potential acting on the dynamics V_S = −ℏ^2/2m (ε H^2 − K) which, besides the usual dependence on the mean (H) and Gaussian (K) curvatures of the surface, has the remarkable feature of a dependence on the signature of the induced metric of the surface: ε=+1 if the signature is (−,+), and ε=1 if the signature is (+,+). Applications to surfaces of revolution in R_1^3 are examined, and we provide examples where the Schrödinger equation is exactly solvable. It is hoped that our formalism will prove useful in the modeling of novel materials such as hyperbolic metamaterials, which are characterized by a hyperbolic dispersion relation, in contrast to the usual spherical (elliptic) dispersion typically found in conventional materials.