Kernel estimates for Schrödinger type operators with unbounded coefficients and critical exponents

Abstract

We consider the Schrodinger type operator $$(1+|x|^\alpha )\Delta +c|x|^{\alpha -2}$$ , for $$\alpha > 2$$ , $$c<0$$ and $$N>2$$ . Heat kernel estimates of the associated semigroup are obtained using the equivalence between weighted Nash inequalities and “weighted” ultracontractivity of a symmetric Markov semigroup. Moreover we give estimates of the eigenfunctions of the operator for large values of |x|.