Scattering problems on noncompact graphs
This paper is devoted to the spectral theory and the theory of scattering for the generalized one dimensional Schrödinger operators associated with connected graphs \(\Gamma\) with a finite number of lines, some of them possibly semi infinite, in the following way: with each line \(\ell_ i\), identified with an interval \([o_ i,s_ i]\subset {\mathbb{R}}\) is associated an operator \({\mathcal L}_ i=-d^ 2/dx^ 2_ i+u_ i(x_ i)\) with \(u_ i\in L^{\infty}_{loc}((o_ i,s_ i))\); with the graph \(\Gamma\) is then associated the (self adjoint) operator \(\tilde {\mathcal L}_{\Gamma}\) in \(\oplus_{i}L^ 2((o_ i,s_ i))\equiv L^ 2(\Gamma)\) defined by the differential expression \(\oplus_{i}{\mathcal L}_ i\) and the boundary conditions expressing at each vertex that the wave functions associated with the lines ending at that vertex take the same value, and that their derivatives add up to zero. That model is relevant to describe the vibration of electrons in a complex molecule. For compact \(\Gamma\), \(\tilde {\mathcal L}_{\Gamma}\) has compact resolvent. For non compact \(\Gamma\), the authors develop the spectral theory and the theory of scattering for \(\tilde {\mathcal L}_{\Gamma}\), generalizing the corresponding theory for Schrödinger operators in \({\mathbb{R}}\). They define Jost solutions, which are outgoing on n-1 of the n semi infinite lines, and are the sum of a normalized ingoing and a reflected part of the n-th one; they define transmission and reflection coefficients and Jost functions; they give various expressions for the resolvent kernel and the S matrix in terms of such quantities as well as some properties of their poles. For clarity, the theory is presented on a number of representative examples rather than in full generality.
scattering for the generalized one dimensional Schrödinger operators associated with connected graphs with a finite number of lines
vibration of electrons in a complex molecule
scientific article; zbMATH DE number 4078053