Properties of the Discrete and Continuous Spectrum of Differential Operators

Abstract: This thesis contains three scientific papers devoted to the study of different spectral theoretical aspects of differential operators in Hilbert spaces.The first paper concerns the magnetic Schrödinger operator (i∇ + A)2 in L2(ℝn). It is proved that given certain conditions on the decay of A, the set [0,∞) is an essential support of the absolutely continuous part of the spectral measure corresponding to the operator.The second paper considers a regular d-dimensional metric tree Γ and defines Schrödinger operators - Δ - V on it.  Here, V is a symmetric, non-negative potential on Γ. It is assumed that V decays like lxl-Γ at infinity, where 1 < γ ≤ d ≤2, γ ≠ 2. A weak coupling constant α is introduced in front of V, and the asymptotics of the bottom of the spectrum as α → 0+ is described.The third, and last, paper revolves around fourth-order differential operators in the space L2(ℝn), where n = 1 or n = 3.  In particular, the operator (-Δ)2 - C|x|-4 - V(x) is studied, where C is the sharp constant in the Hardy-Rellich inequality. A Lieb-Thirring inequality for this operator is proved, and as a consequence a Sobolev-type inequality is obtained.