Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians
In the paper it is considered a non relativistic, spinless quantum particle in \(\mathbb{R}^d\), (\(d=2,3\)) interacting with a magnetic field \(B\) associated with finitely or infinitely many thin solenoids aligned along the \(x_3\)-axis which pass through the points \(\lambda\) of some discrete subset \(\Lambda\) of \((x_1x_2)\)-plane. The dynamics of the mentioned particle moving in any Aharonov-Bohm solenoids in \(\mathbb{R}^d\) is described by the Schrödinger operator \(H_0^{(d)}=-(\nabla+iA)^2\) acting in \(L^2(\mathbb{R}^d)\), where \(\nabla\) is the gradient in \(\mathbb{R}^d\), and \(A\) is the vector potential associated with \(B\). It is proved the so called diamagnetic inequality \(| e^{-tH_0^{(d)}}u| \leq e^{-tL_0}| u| \) for all \(t\geq 0\) and \(u\in L^2(\mathbb{R}^d)\), here \(L_0\) denotes the negative Laplacian in \(L^2(\mathbb{R}^d)\). By means of this inequality the Lieb-Thirring inequality, for the perturbed Schrödinger operator \(H_0^{(d)}-V\), is proved. Moreover using a new Hardy type inequality CLR-type eigenvalue estimates for \(H_0^{(d)}-V\) are established. Large coupling constant eigenvalue asymptotic formulas for the perturbed operators are also proved.
scientific article; zbMATH DE number 2123138