×

Scattering problems on noncompact graphs. (English. Russian original) Zbl 0659.47006

Theor. Math. Phys. 74, No. 3, 230-240 (1988); translation from Teor. Mat. Fiz. 74, No. 3, 345-359 (1988).
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.
Reviewer: J.Ginibre

MSC:

47A40 Scattering theory of linear operators
47E05 General theory of ordinary differential operators
81U20 \(S\)-matrix theory, etc. in quantum theory
34L99 Ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] B. S. Pavlov and M. D. Faddeev, Teor. Mat. Fiz.,55, 257 (1983).
[2] N. I. Akhiezer and I. M. Glazman, The Theory of Linear Operators in Hilbert Space, Pitman, Boston, Mass. (1981). · Zbl 0467.47001
[3] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 3, Academic Press, New York (1979). · Zbl 0405.47007
[4] L. D. Faddeev, Tr. Mosk. Inst. Akad. Nauk SSSR,73, 314 (1964).
[5] B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory [in Russian], Nauka, Moscow (1970). · Zbl 0225.47019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.