Are these operators defined on 2D surfaces self-adjoint? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:55:07Z http://mathoverflow.net/feeds/question/37704 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37704/are-these-operators-defined-on-2d-surfaces-self-adjoint Are these operators defined on 2D surfaces self-adjoint? QHLIU 2010-09-04T09:11:54Z 2010-09-11T14:03:35Z <p>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.</p> <p>The standard representation of the curved smooth surface $M$ embedded\ in $R^{3}$ is,</p> <p>$\mathbf{r}(\xi ,\zeta )\mathbf{=}\left( x(\xi ,\zeta ),y(\xi ,\zeta ),z(\xi ,\zeta )\right)$.</p> <p>The covariant derivatives of $\mathbf{r}$ are $\mathbf{r}_{\mu }=\partial \mathbf{r}/ \partial x^{\mu }$ .</p> <p>The contravariant derivatives </p> <p>$\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}_{\upsilon }$ </p> <p>is the generalized inverse of the covariant ones $\mathbf{r}_{\mu }$.</p> <p>The unit normal vector at point $(\xi ,\zeta )$ is $\mathbf{n=r}^{\xi } \times \mathbf{r}^{\zeta }/ \sqrt{g}$. </p> <p>The Hermitian Cartesian momentum $\mathbf{p}$ takes a compact form,</p> <p>$\mathbf{p=}-i\hbar (\mathbf{r}^{\mu }\partial _{\mu }+H\mathbf{n),}$</p> <p>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, $\mathbf{p=}-i\hbar \nabla$.</p> <p>For a particle moves on the surface of a sphere of radius $r$, $x=r\sin \theta \cos \varphi ,\text{ }y=r\sin \theta \sin \varphi ,\text{ }z=r\cos \theta$,</p> <p>the hermitian operators for Cartesian momenta $p_{i}$ are respectively, </p> <p>$p_{x} =-\frac{i\hbar }{r}(\cos \theta \cos \varphi \frac{\partial }{\partial \theta }-\frac{\sin \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }-\sin \theta \cos \varphi ),$</p> <p>$p_{y} =-\frac{i\hbar }{r}(\cos \theta \sin \varphi \frac{\partial }{\partial \theta }+\frac{\cos \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }-\sin \theta \sin \varphi ),$</p> <p>$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$</p> <p>On the spherical surface, the complete set of the spherical harmonics defines the Hilbert space.</p> <hr> <p>Refs.</p> <p>2003, Liu Q H and Liu T G, Int. Quantum Hamiltonian for the Rigid Rotator, J. Theoret. Phys. 42(2003)2877.</p> <p>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.</p> <p>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.</p> <p>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. </p> <p>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. </p> <p>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.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/37704/are-these-operators-defined-on-2d-surfaces-self-adjoint/38408#38408 Answer by Martin Gisser for Are these operators defined on 2D surfaces self-adjoint? Martin Gisser 2010-09-11T14:03:35Z 2010-09-11T14:03:35Z <p>If H is bounded and the surface is geodesically complete, then I bet the operator is self-adjoint. The proofs of such theorems rest on an approximation of 1 by smooth compactly supported functions with bounded derivatives, constructed from the Riemannian distance function. Should be quite simple.</p> <p>I have no library access currently. Do you have a link to 2007?</p> <p>The trick should be found in one of the following (but I couldn't check).</p> <p>M.P. Gaffney, A Special Stokes's Theorem for Complete Riemannian Manifolds, Ann. of Math. (2) 60 (1954), 140 145.</p> <p>P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973), 401 414.</p> <p>R.S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48 79.</p>