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In this work it is studied the Schr\"odinger equation for a non-relativisticparticle restricted to move on a surface $S$ in a three-dimensional Minkowskianmedium $\mathbb{R}_1^3$, i.e., the space $\mathbb{R}^3$ equipped with themetric $\text{diag}(-1,1,1)$. After establishing the consistency of theinterpretative postulates for the new Schr\"odinger equation, namely theconservation of probability and the hermiticity of the new Hamiltonian builtout of the Laplacian in $\mathbb{R}_1^3$, we investigate the confiningpotential formalism in the new effective geometry. Like in the well-knownEuclidean case, it is found a geometry-induced potential acting on the dynamics$V_S = - \frac{\hbar^{2}}{2m} \left(\varepsilon H^2-K\right)$ which, besidesthe usual dependence on the mean ($H$) and Gaussian ($K$) curvatures of thesurface, has the remarkable feature of a dependence on the signature of theinduced metric of the surface: $\varepsilon= +1$ if the signature is $(-,+)$,and $\varepsilon=1$ if the signature is $(+,+)$. Applications to surfaces ofrevolution in $\mathbb{R}^3_1$ are examined, and we provide examples where theSchr\"odinger equation is exactly solvable. It is hoped that our formalism willprove useful in the modeling of novel materials such as hyperbolicmetamaterials, which are characterized by a hyperbolic dispersion relation, incontrast to the usual spherical (elliptic) dispersion typically found inconventional materials.

Schr\"odinger formalism for a particle constrained to a surface in $\mathbb{R}_1^3$