On spectral asymptotic for the second-derivative operators

Abstract

In this work we focus on spectral asymptotic for the second derivative operators. Here we study Schrödinger operator with zero-range potentials, because this operator has great importance for understanding the solvable problems in quantum mechanics and atomic physics. It appears in different models such as the mathematical physics, applied mathematics and theoretical physics. We have two objectives in this work. We first demonstrated that this operator has a continuous spectrum contains an infinite number of bands separated by gaps. We then explained that the bands to gaps ratio tends to zero under certain conditions.