Quantum problems with special singular potentials and Hilbert spaces with indefinite scalar products.

Plan of the talk. 1. Classical Kepler problem. The difference between 1/r and 1/r^2 potentials: in the second case the particle may fall to the centre. 2. Spin zero particle in quantum mechanics. Schrodinger equation and Hilbert space of states. Connection between the conservation of mass and self-adjointness of the quantum Hamiltonian. 3. Eigenfunction decomposition for Schrodinger operator with sufficiently good potentials. The spectral measure. 4. Quantum Kepler problem. For strong attractors the Schrodinger operator becomes non-self-adjoint. 5. Special class of singular 1-dimensional Schrodinger operators - spectrally meromorphic operators. The natural indefinite scalar product for this problem.