DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN

We show that any symmetric operator H has a dense maximal b-stability domain (i.e. , b∈R1) if and only if H is unbounded from above. This abstract result allows an application to singular perturbed Schrodinger operators which are not semi-bounded from below, i.e., to the so-called "fall to the center problem". It turns out that in this case the regularization problem is always ill-posed which implies that there is no unique "right Hamiltonian" for corresponding perturbed system. We give an example of singular perturbed Schrodinger operator for which stability domains are described explicitly.