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Register a new userEigenvalue estimates for Schroedinger operators on metric trees
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We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit.
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We study Schroedinger operators with Robin boundary conditions on exterior domains in $R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.
Consider a regular $d$-dimensional metric tree $Gamma$ with root $o$. Define the Schroedinger operator $-Delta - V$, where $V$ is a non-negative, symmetric potential, on $Gamma$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-gamma}$ at infinity, where $1 < gamma leq d leq 2, gamma eq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-Delta - V$. In other words, we will describe the asymptotical behavior of $inf sigma(-Delta - alpha V)$ as $alpha to 0+$
We find sharp conditions on the growth of a rooted regular metric tree such that the Neumann Laplacian on the tree satisfies a Hardy inequality. In particular, we consider homogeneous metric trees. Moreover, we show that a non-trivial Aharonov-Bohm magnetic field leads to a Hardy inequality on a loop graph.
We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furthermore prove a lower bound for the first magnetic Neumann eigenvalue in the case of constant field.
We show that for a Jacobi operator with coefficients whose (j+1)th moments are summable the jth derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known dispersive estimates with integrable time decay for the time dependent Jacobi equation in the resonant case.
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