dg.differential geometry - Are these operators defined on 2D surfaces self-adjoint?

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,

$mathbf{p=}-ihbar (mathbf{r}^{mu }partial _{mu }+Hmathbf{n),}$

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 $Hmathbf{n}$ vanish. Then the Cartesian momentum operator reproduces its usual form as, $mathbf{p=}-ihbar nabla $.

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

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

$p_{x} =-frac{ihbar }{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{ihbar }{r}(cos theta sin varphi frac{partial }{partial theta }+frac{cos varphi }{sin theta }frac{partial }{partial varphi }-sin theta sin varphi ), $

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

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

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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.