On integral transformations associated with a certain Lagrangian - as a prototype of quantization (Q1069523)

For a finite-dimensional manifold M a function L(\(\gamma\),\({\dot \gamma}\)) on the tangent bundle TM is considered. Under appropriate conditions, the authors construct a \(C^ 0\)-semigroup of bounded linear operators associated with L(\(\gamma\),\({\dot \gamma}\)) and their infinitesimal generators on an intrinsic Hilbert space. M is required to be a smooth, simply-connected and connected d-dimensional manifold. L is assumed in the form L(\(\gamma\),\({\dot \gamma}\))\(=L^ 0(\gamma,{\dot \gamma})-V(\gamma)\), \(L^ 0(\gamma,{\dot \gamma})=1/2\sum_{i,j}g_{ij}(\gamma){\dot \gamma}^ i{\dot \gamma}^ j\) such that \(g=(g_{ij})\) defines a complete Riemannian metric on M. Bounds are required for the sectional curvature and for components of the curvature tensor. Starting from an integral operator, the existence of a semigroup of operators on a Hilbert space \(L^ 2(M,d\mu)\) is shown such that their generators contain the Laplace- Beltrami operator and the real-valued function V in a form similar to standard quantum mechanics.