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Register a new userEigenvalue inequalities and absence of threshold resonances for waveguide junctions
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Added by
Konstantin Pankrashkin
Publication date
2016
fields
Physics
and research's language is
English
Authors
Konstantin Pankrashkin
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Let $Lambdasubset mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $Lambda$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-Delta u= u u$ in $Lambda$ and vanishing at the boundary, where $ u$ is the bottom of the essential spectrum of the Dirichlet Laplacian in $Lambda$. We derive a sufficient condition for the absence of threshold resonances in terms of the Laplacian eigenvalues on the center. The proof is elementary and is based on the min-max principle. Some two- and three-dimensional examples and applications to the study of Laplacians on thin networks are discussed.
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Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrodinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $n geq 3$, the absence of resonances at $pm m$ for massive Dirac operators (with mass $m > 0$) in dimensions $n geq 5$, and recall the well-known case of absence of zero-energy resonances for Schrodinger operators in dimension $n geq 5$.
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.
In the presence of the homogeneous electric field ${bf E}$ and the homogeneous perpendicular magnetic field ${bf B}$, the classical trajectory of a quantum particle on ${mathbb R}^2$ moves with drift velocity $alpha$ which is perpendicular to the electric and magnetic fields. For such Hamiltonians the absence of the embedded eigenvalues of perturbed Hamiltonian has been conjectured. In this paper one proves this conjecture for the perturbations $V(x, y)$ which have sufficiently small support in direction of drift velocity.
We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_circ ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.
We consider the 3D Schrodinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $Omega_{rm in} subset {mathbb R}^3$. We introduce the Krein spectral shift functions $xi(E;H_pm,H_0)$, $E geq 0$, for the operator pairs $(H_pm,H_0)$, and study their singularities at the Landau levels $Lambda_q : = b(2q+1)$, $q in {mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. We show that $xi(E;H_+,H_0)$ remains bounded as $E uparrow Lambda_q$, $q in {mathbb Z}_+$ being fixed, and obtain three asymptotic terms of $xi(E;H_-,H_0)$ as $E uparrow Lambda_q$, and of $xi(E;H_pm,H_0)$ as $E downarrow Lambda_q$. The first two terms are independent of the perturbation while the third one involves the {em logarithmic capacity} of the projection of $Omega_{rm in}$ onto the plane perpendicular to $B$.
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