Non-Standard Spectral Inequalities and Applications

Abstract

In this thesis, we investigate non-classical spectral inequalities and some applications for second-order differential operators. Obtaining spectral estimates mostly depends on what is known as the semi-classical Weyl’s asymptotic formula which is based on calculating the phase volume (or "numerical range") of the symbol of such operators. However, for the operators with infinite phase volumes, the semi-classical spectral estimates cannot hold and thus calculating the spectral estimates becomes more complicated. These types of operators which this thesis is concerned with. In the first part, we consider a family of Schrödinger operators with increasing electric potential and we obtain non-classical uniform estimates for the Riesz means of the eigenvalues. In the proofs, we consider two main cases depending on the relation between the powers of the potential. In the second part, we consider a family of Baouendi-Grushin type operators that are degenerate. This problem can be classified into three cases: strong degeneracy, weak degeneracy, and intermediate case. For each case, we consider a different relation between the power of the potential and the dimension d. In the strong degeneracy and intermediate cases, we obtain non-classical bounds for the Riesz means of the eigenvalues, however, in the weak degeneracy case we prove that semi-classical estimates can be obtained. At the end of this part, we give an interesting example of a spectral problem for Baouendi-Grushin Type operators whose eigenvalues can be computed explicitly. In the last part of the thesis, we consider a high dimensional class of Baouendi-Grushin operators and apply the so-called coherent state transform to study the asymptotic behavior of the eigenvalues. To this end, we classify the problem into three degenerate cases similar to what has been done in the second part. Consequently, we obtain non-standard Weyl asymptotics for the counting function of the eigenvalues. The techniques of the proofs in all parts lie in reducing the operators into simpler operators with distinct spectral properties that can be identified.