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This paper deals with a general method for the reduction of quantum systemswith symmetry. For a Riemannian manifold M admitting a compact Lie group G asan isometry group, the quotient space Q = M/G is not a smooth manifold ingeneral but stratified into a collection of smooth manifolds of variousdimensions. If the action of the compact group G is free, M is made into aprincipal fiber bundle with structure group G. In this case, reduced quantumsystems are set up as quantum systems on the associated vector bundles over Q =M/G. This idea of reduction fails, if the action of G on M is not free.However, the Peter-Weyl theorem works well for reducing quantum systems on M.When applied to the space of wave functions on M, the Peter-Weyl theoremprovides the decomposition of the space of wave functions into spaces ofequivariant functions on M, which are interpreted as Hilbert spaces for reducedquantum systems on Q. The concept of connection on a principal fiber bundle isgeneralized to be defined well on the stratified manifold M. Then the reducedLaplacian is well defined as a self-adjoint operator with the boundaryconditions on singular sets of lower dimensions. Application to quantummolecular mechanics is also discussed in detail. In fact, the reduction ofquantum systems studied in this paper stems from molecular mechanics. If onewishes to consider the molecule which is allowed to lie in a line when it is inmotion, the reduction method presented in this paper works well.Comment: 33 pages, no figure

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