Reduction of quantum systems on Riemannian manifolds with symmetry and application to molecular mechanics (Q2737905)

In the present paper the authors give a general formulation to describe quantum mechanics of a reduced system on a Riemannian manifold \(M\) with symmetry Lie group \(G\). More precisely, the reduced quantum system by symmetry is defined on the quotient space \(Q = M/G\), which is in general not smooth, but stratifies into a collection of smooth manifolds of various dimension. If the action of \(G\) is free, reduced quantum systems are set up as quantum systems on the associated vector bundles over \(Q\). Applying the Peter-Weyl theorem to the space of wave functions on \(M\), the authors obtain a decomposition of the space of wave functions into spaces of equivariant functions on \(M\). In this way, the reduced Laplacian is well defined as a self-adjoint operator with the boundary conditions on singular sets of lower dimension. Moreover, applications of the general formalism to molecular mechanics are discussed. A general setting for \(N\)-atomic molecules is established and triatomic molecules are studied in detail.