### A duality between Schrödinger operators on graphs and certain Jacobi matrices

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We consider a class of semilinear elliptic equations of the form$$-{\epsilon}^{2}\Delta u(x,y)+a\left(x\right){W}^{\text{'}}\left(u(x,y)\right)=0,\phantom{\rule{1.0em}{0ex}}(x,y)\in {\mathbb{R}}^{2}$$where $\epsilon \>0$, $a:\mathbb{R}\to \mathbb{R}$ is a periodic, positive function and $W:\mathbb{R}\to \mathbb{R}$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={({s}^{2}-1)}^{2}$. We look for solutions to (1) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty $ uniformly with respect to $y\in \mathbb{R}$. We show via variational methods that if $\epsilon $ is sufficiently small and $a$ is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

We consider a class of semilinear elliptic equations of the form 15.7cm -${\epsilon}^{2}\Delta u(x,y)+a\left(x\right){W}^{\text{'}}\left(u(x,y)\right)=0,\phantom{\rule{1.0em}{0ex}}(x,y)\in {\mathbb{R}}^{2}$ where $\epsilon >0$, $a:\mathbb{R}\to \mathbb{R}$ is a periodic, positive function and $W:\mathbb{R}\to \mathbb{R}$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={({s}^{2}-1)}^{2}$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty $ uniformly with respect to $y\in \mathbb{R}$. We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct...

We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

We aim here at analyzing the fundamental properties of positive semidefinite Schrödinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment...

In 1986 P. Šeba in the classic paper considered one-dimensional pseudo-Hamiltonians containing the first derivative of the Dirac delta function. Although the paper contained some inaccuracy, it was one of the starting points in approximating one-dimension self-adjoint couplings. In the present paper we develop the above results to the case of quantum systems with complex geometry.