My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether it is self-adjoint or not (we are all physicists). If a mathematician can give a definite answer to it for even simple surfaces such as cylindrical and spherical, he has then a nice paper.

The standard representation of the curved smooth surface $M$ embedded\ in $ R^{3}$ is,

$\mathbf{r}(\xi ,\zeta )\mathbf{=}\left( x(\xi ,\zeta ),y(\xi ,\zeta ),z(\xi ,\zeta )\right)$.

The covariant derivatives of $\mathbf{r}$ are $\mathbf{r}_{\mu }=\partial \mathbf{r}/ \partial x^{\mu }$ .

The contravariant derivatives

$\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}_{\upsilon }$

is the generalized inverse of the covariant ones $\mathbf{r}_{\mu }$.

The unit normal vector at point $(\xi ,\zeta )$ is $\mathbf{n=r}^{\xi } \times \mathbf{r}^{\zeta }/ \sqrt{g}$.

The Hermitian Cartesian momentum $\mathbf{p}$ takes a compact form,

$\mathbf{p=}-i\hbar (\mathbf{r}^{\mu }\partial _{\mu }+H\mathbf{n),}$

where $H$ is the mean curvature of the surface. When the motion is constraint-free or in a flat plane, i.e., when $H=0$, the constraint induced terms $H\mathbf{n}$ vanish. Then the Cartesian momentum operator reproduces its usual form as, $\mathbf{p=}-i\hbar \nabla $.

For a particle moves on the surface of a sphere of radius $r$, $ x=r\sin \theta \cos \varphi ,\text{ }y=r\sin \theta \sin \varphi ,\text{ }z=r\cos \theta$,

the hermitian operators for Cartesian momenta $p_{i}$ are respectively,

$p_{x} =-\frac{i\hbar }{r}(\cos \theta \cos \varphi \frac{\partial }{\partial \theta }-\frac{\sin \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }-\sin \theta \cos \varphi ), $

$p_{y} =-\frac{i\hbar }{r}(\cos \theta \sin \varphi \frac{\partial }{\partial \theta }+\frac{\cos \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }-\sin \theta \sin \varphi ), $

$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$

On the spherical surface, the complete set of the spherical harmonics defines the Hilbert space.

Refs.

2003, Liu Q H and Liu T G, Int. Quantum Hamiltonian for the Rigid Rotator, J. Theoret. Phys. 42(2003)2877.

2004, Liu Q H, Hou J X, Xiao Y P and Li L X, Quantum Motion on 2D Surface of Nonspherical Topology, Int. J. Theoret. Phys. 43(2004)1011.

2005, Xiao Y P, Lai M M, Hou J X, Chen X W and Liu Q H, A Secondary Operator Ordering Problem for a Charged Rigid Planar Rotator in Uniform Magnetic Field, Comm. Theoret. Phys. 44(2005)49.

2006a, Lai M M, Wang X, Xiao Y P and Liu Q H, Gauge Transformation and Constraint Induced Operator Ordering for Charged Rigid Planar Rotator in Uniform Magnetic Field, Comm. Theoret. Phys. 46(2006) 843.

2006b, Wang X, Xiao Y P, Liu T G, Lai M M and Rao, Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.

2006c, Liu Q H, Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, Int. J. Theoret. Phys. 45(2006)2167.

2007, Liu Q H., Tong C L., Lai M M., Constraint-induced mean curvature dependence of Cartesian momentum operators J. Phys. A 40(2007)4161.

2010, Zhu X M, Xu M and Liu Q H, Wave packets on spherical surface viewed from expectation values of Cartesian variables, Int. J. Geom. Meth. Mod. Phys., 7(2010)411-423.