Conditions of discreteness of the spectrum for Schr\"odinger operator and some optimization problems for capacity and measures

Abstract

For the the Schr\"odinger operator $H=-\Delta+ V(x)\cdot$, acting in the space L_2(\R^d)\,(d\ge 3), with V(x)\ge 0 and V(\cdot)\in L_{1,loc}(\R^d), we obtain some constructive conditions for discreteness of its spectrum. Basing on the Mazya-Shubin criterion for discreteness of the spectrum of $H$ and using the isocapacity inequality and the concept of base polyhedron for the harmonic capacity, we have estimated from below the cost functional of an optimization problem, involved in this criterion, replacing a submodular constrain (in terms of the harmonic capacity) by a weaker but additive constrain (in terms of a measure). By this way we obtain an optimization problem, which can be considered as an infinite-dimensional analogue of the optimal covering problem. We have solved this problem for the case of a non-atomic measure. This approach enables us to obtain for the operator H some sufficient conditions for discreteness of its spectrum in terms of non-increasing rearrangements, with respect to measures from the base polyhedron, for some functions connected with the potential V(x). We construct some counterexamples, which permit to compare our results between themselves and with results of other authors.