Schrödinger operators on a sphere

Question

if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on the azimuthal coordinate explicitely, then the substitution $\psi(\theta,\phi) = \Theta(\theta)e^{i n \phi}$ leaves you with the differential equation

$$- \Theta''(\theta)- \cot(\theta) \Theta'(\theta) + \frac{n^2}{\sin^2(\theta)} \Theta(\theta)+ V(\theta) \Theta(\theta) = E \Theta(\theta).$$

For $n=0$ this simplifies to $$- \Theta''(\theta)- \cot(\theta) \Theta'(\theta) + V(\theta) \Theta(\theta) = E \Theta(\theta).$$

Has this equation ever been studied? I mean if $V \in C^{\infty}$, then it should have a discrete spectrum, as the initial PDE has a discrete spectrum. But is there a further literature on this kind of problem?

I mean, by just googling Schrödinger operator on sphere, I got this result for example on arxiv, but if you are aware of better overview articles, please do not hesitate to give them to me. But apparently, the special case, where you can separate that easily is not (as far as I see) that well-studied or am I wrong?

Answer 1

I presume that $V$ is real valued. Then the equation on the sphere is $L\psi=E\psi$, an eigenvalue equation for the self-adjoint operator $L$, with compact resolvant. Therefore the eigenvalues are real and the normalized eigen-functions form an orthonormal basis of $L^2(S)$. If $H\subset L^2$ is an invariant subspace, then you can choose the basis in such a way that some of the eigen-functions form a basis of $H$. This applies to the subspace of functions of the form $\Theta(\theta)e^{in\phi}$.

Edit OK, the reduced equation is $\ell^n\Theta=E\Theta$, an eigenvalue problem for an ordinary differential linear second-order operator. Thus $\ell^n$ is Liouville type. It can be written as a self-adjoint operator for some suitable weight. Liouville's theory tells you that at fixed $n$, the eigenvalues $$E^n_0<E^n_1<\cdots$$ are simple, and the eigenfunction $\Theta^n_k$ vanishes precisely at $k$ points. Is it what you wished ?

Answer 2

From mathematical side, spectral problem for zonal Schrödinger operators on n-spheres is studied in http://projecteuclid.org/euclid.cmp/1104200938 (Zonal Schrödinger operators on the n-sphere: inverse spectral problem and rigidity, by David Gurarie). See also http://msp.org/pjm/1997/177-1/p01.xhtml (A Borg–Levinson theorem for Bessel operators, by Robert Carlson).

From the physics side, two methods were developed to study systems constrained on a curved surface. A method due to DeWitt http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29.377 (Dynamical Theory in Curved Spaces. I. A Review of the Classical and Quantum Action Principles) considers the dynamics as truly two-dimensional, while the approach of da Costa http://journals.aps.org/pra/abstract/10.1103/PhysRevA.23.1982 (Quantum mechanics of a constrained particle) assumes starting from the three dimensional problem and then reduces it to a two-dimensional one by a confining procedure. See http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.100.230403 (Schrödinger Equation for a Particle on a Curved Surface in an Electric and Magnetic Field, by Giulio Ferrari and Giampaolo Cuoghi) for further references.

Quantum mechanics on a sphere (or on a more general manifolds) is by no means a trivial generalization of the quantum mechanics in Euclidean configuration space. Such subtleties as inequivalent quantization schemes and induced gauge structures do appear. See, for example, http://arxiv.org/abs/hep-th/9306098 (Gauge Field, Parity and Uncertainty Relation of Quantum Mechanics on $S^1$, by Shogo Tanimura).