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Register a new userA Magnetic Contribution to the Hardy Inequality
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We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given by Hardy is improved by a non-negative potential term depending on properties of the magnetic field.
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We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensional Euclidean space, we derive a non-trivial magnetic Hardy inequality for a magnetic field that vanishes at infinity and diverges along a plane.
We give a short proof of a recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.
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.
The principal aim of this paper is to employ Bessel-type operators in proving the inequality begin{align*} int_0^pi dx , |f(x)|^2 geq dfrac{1}{4}int_0^pi dx , dfrac{|f(x)|^2}{sin^2 (x)}+dfrac{1}{4}int_0^pi dx , |f(x)|^2,quad fin H_0^1 ((0,pi)), end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schr{o}dinger operator associated with the differential expression begin{align*} tau_s=-dfrac{d^2}{dx^2}+dfrac{s^2-(1/4)}{sin^2 (x)}, quad s in [0,infty), ; x in (0,pi). end{align*} The new inequality represents a refinement of Hardys classical inequality begin{align*} int_0^pi dx , |f(x)|^2 geq dfrac{1}{4}int_0^pi dx , dfrac{|f(x)|^2}{x^2}, quad fin H_0^1 ((0,pi)), end{align*} it also improves upon one of its well-known extensions in the form begin{align*} int_0^pi dx , |f(x)|^2 geq dfrac{1}{4}int_0^pi dx , dfrac{|f(x)|^2}{d_{(0,pi)}(x)^2}, quad fin H_0^1 ((0,pi)), end{align*} where $d_{(0,pi)}(x)$ represents the distance from $x in (0,pi)$ to the boundary ${0,pi}$ of $(0,pi)$.
We consider the Schrodinger operator with constant magnetic field defined on the half-plane with a Dirichlet boundary condition, $H_0$, and a decaying electric perturbation $V$. We analyze the spectral density near the Landau levels, which are thresholds in the spectrum of $H_0,$ by studying the Spectral Shift Function (SSF) associated to the pair $(H_0+V,{H_0})$. For perturbations of a fixed sign, we estimate the SSF in terms of the eigenvalue counting function for certain compact operators. If the decay of $V$ is power-like, then using pseudodifferential analysis, we deduce that there are singularities at the thresholds and we obtain the corresponding asymptotic behavior of the SSF. Our technique gives also results for the Neumann boundary condition.
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