Quantum cosmological Friedman models with an initial singularity

We consider the Wheeler-DeWitt equation H psi = 0 in a suitable Hilbert space. It turns out that this equation has countably many solutions psi(i) which can be considered as eigenfunctions of a Hamilton operator implicitly defined by H. We consider two models, a bounded one, 0 < r < r(0), and an unbounded, 0 < r < infinity, which represent different eigenvalue problems. In the bounded model we look for eigenvalues Lambda(i), where the Lambda(i) are the values of the cosmological constant which we used in the Einstein-Hilbert functional, and in the unbounded model the eigenvalues are given by (- Lambda(i))(-)n-1/n, where Lambda(i) < 0. Note that r is the symbol for the scale factor, usually denoted by a, or a power of it. The psi(i) form a basis of the underlying Hilbert space. We prove furthermore that the implicitly defined Hamilton operator is selfadjoint and that the solutions of the corresponding Schrodinger equation satisfy the Wheeler-DeWitt equation, if the initial values are superpositions of eigenstates. All solutions have an initial singularity in r = 0. Under certain circumstances a smooth transition from big crunch to big bang is possible.