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Register a new userLimits of Quantum Graph Operators With Shrinking Edges
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We address the question of convergence of Schrodinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graphs edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. We use a combination of functional-analytic bounds on the edges of the graph and Lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence. This condition encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges.
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We consider the Schrodinger operator $H_{eta W} = -Delta + eta W$, self-adjoint in $L^2({mathbb R}^d)$, $d geq 1$. Here $eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{eta W}$ near the origin, and due to the irregular decay of $eta W$, we encounter some non semiclassical phenomena. In particular, $H_{eta W}$ has less eigenvalues than suggested by the semiclassical intuition.
We study the direct and inverse scattering problem for the one-dimensional Schrodinger equation with steplike potentials. We give necessary and sufficient conditions for the scattering data to correspond to a potential with prescribed smoothness and prescribed decay to their asymptotics. These results are important for solving the Korteweg-de Vries equation via the inverse scattering transform.
We consider a $2times 2$ system of 1D semiclassical differential operators with two Schrodinger operators in the diagonal part and small interactions of order $h^ u$ in the off-diagonal part, where $h$ is a semiclassical parameter and $ u$ is a constant larger than $1/2$. We study the absence of resonance near a non-trapping energy for both Schrodinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by $Mhlog(1/h)$ and the coefficient $M$ is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hamiltonians.
We study perturbations of the self-adjoint periodic Sturm--Liouville operator [ A_0 = frac{1}{r_0}left(-frac{mathrm d}{mathrm dx} p_0 frac{mathrm d}{mathrm dx} + q_0right) ] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.
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 $(0,pi)$.
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