## Final Report Summary - GRAPH-COUPL (Spectral analysis of graph-coupled systems)

The project dealt with the mathematical analysis of thin branched structures. Such structures have become very popular in Mathematical

Physics during recent years. One reason is the increasing technological feasability of manufacturing nano-structures at an atomic level, making a detailed quantum mechanical analysis of the problem necessary. Thin branched structures occur e.g. in micro-electronics or in opto-electronics; and a theoretical analysis is necessary in order to understand the behaviour of such media or to engeneer materials with certain properties. One is for example interested whether a material conducts or transmits light or not, e.g. semi-conductors or photonic crystals.

The research topics have a close relationship to some areas of nano-electronics, which is one of the priority research themes in

Europe and in which the European contries have the leading position. Thin branched structures provide models for a fundamental research in nano-structures as provided in FP7 NMP Work Programme on Nanosciences, Nanotechnologies, Materials and New Production Technologies.

One of the main goals of the project was the question whether thin branched structures and their properties can approximately be described by the pure one-dimenional limit, the underlying network structure. We successfully answered a longstanding open problem in Exner-Post [1], namely how general couplings at the vertices (the nods of the network) can be approximated on thin branched structures by suitably scaled Schrödinger operators. Moreover, an approach where the approximating operator is a pure Laplacian (no potential), was elaborated by Kuchment-Post and is almost finished; a draft version is available upon request [5]. This approach also uses ideas from operator theory (Dirichlet-to-Neumann operators); and the PI profited much from the expertise of the Cardiff Analysis group. The PI together with J. Behrndt (Graz, Austria) also elaborated a similar approach for the Dirichlet-to-Neumann operator: Roughly speaking, we resolve the non-uniqueness of the inverse problems for networks (graphs) (``Can one hear the shape of a network'') by ``blowing up'' the network to a thin branched structure, see [6]. In summary, we provided a large class of solvable models for thin branched structures; including the zero-thickness limit; one of the goals of the project.

Another goal of the project was to develop tools from operator theory in order to treat branched structures in a systematic way. The principal investigator (PI) introduced in [3] a new approach (``boundary pairs'') extending known concepts (``boundary triples'') in an easy way to the relevant structures (partial differential operators), and the PI profitted very much from the expertise of the Cardiff analysis group (Brown, Marletta, Schmidt) on this approach (see also [10]). This approach is a basis for a systematic analysis of operators on branched structures, another goal of the project.

This analysis is partially included in [3].

The third goal of the project, the decomposition into building blocks according to a network and the use of ideas from discrete models for networks was addressed by the PI during several visits to the research group in Jena, Germany, (Lenz, Haeseler, Keller, [9]) and during various discussions with M.~Brown (Cardiff). Before addressing this goal more deaply, the PI realised that the profound analysis of the concept of ``boundary pairs'' in [3] was necessary. The project deepened the collaboration between the groups in Jena and Cardiff.

The PI profitted very much from the expertise of the Analysis research group in Cardiff, including the expertise on boundary triples (Brown, Marletta, Schmidt, Strauss). Moreover he established new future research directions in the area of homogenisation (Cherednichenko) and the possible application of operator-theoretic methods developed by the PI to this area.

The PI profitted also very much from the possibility of meeting research groups all over Europe: The grant allowed him to continue his collaboration with the Berlin Analysis group (Batu Güneysu, Jörn Müller, HU Berlin, [2], [8]), the collaboration with R. Hempel and R. Weder [4], the project on discrete branched structures with F. Lledó (Madrid) [7]. He also established new connections with Laughbourough (A. Strohmayer), UC London (L. Parnovski), Bristol (Ram Band) and Glasgow (Matthias Langer). Moreover, the attendence of several conferences resulted in an invitation from C. Anné for a research stay in Nantes (France) in 2013. He also was invited to give a short course in a summer school in Tunesia, establishing new scientific contacts.

The PI also organised a mini-workhop and a conference on related topics (``Boundary value problems and Spectral Geometry'', MF Oberwolfach, Germany; ``Dynamical systems on random graphs'', CIEM Castro Urdiales, Spain), establishing new and deepening existing collaborations on the European and international level.

List of publications and projects in preparation (author list in alphabetic order, as usual in Mathematics), for the list of

peer-reviewed publications, see next section)

Preprints

[3] Boundary pairs associated with quadratic forms (Olaf Post), submitted to Mathematische Nachrichten (47 pages)

[4] On open scattering channels for manifolds with ends (Rainer Hempel (Braunschweig, Germany), Olaf Post and Ricardo Weder (Mexico)), submitted to Journal of Functional Analysis (38 pages)

draft versions available on request:

[5] Generating vertex conditions by geometry (Peter Kuchment (Texas A&M) and Olaf Post) (ca. 32 pages)

[6] Inverse problems on thin branched manifolds and convergence of the Dirichlet-to-Neumann maps (Jussi Behrndt (TU Graz, Austria) and Olaf Post) (ca. 34 pages)

further projects:

[7] Spectral gaps for discrete magnetic periodic Laplacians (Fernando Lledó (Madrid) and Olaf Post)

[8] Boundary pairs and the Dirichlet-to-Neumann operator on non-compact warped products (Jörn Müller (HU Berlin), Olaf Post)

[9] Purely absolutely continuous spectrum for random metric tree graphs (Sebastian Haeseler, Matthias Keller, Daniel Lenz (Jena); Olaf Post)

[10] Boundary triples associated with quadratic forms (Olaf Post)

Physics during recent years. One reason is the increasing technological feasability of manufacturing nano-structures at an atomic level, making a detailed quantum mechanical analysis of the problem necessary. Thin branched structures occur e.g. in micro-electronics or in opto-electronics; and a theoretical analysis is necessary in order to understand the behaviour of such media or to engeneer materials with certain properties. One is for example interested whether a material conducts or transmits light or not, e.g. semi-conductors or photonic crystals.

The research topics have a close relationship to some areas of nano-electronics, which is one of the priority research themes in

Europe and in which the European contries have the leading position. Thin branched structures provide models for a fundamental research in nano-structures as provided in FP7 NMP Work Programme on Nanosciences, Nanotechnologies, Materials and New Production Technologies.

One of the main goals of the project was the question whether thin branched structures and their properties can approximately be described by the pure one-dimenional limit, the underlying network structure. We successfully answered a longstanding open problem in Exner-Post [1], namely how general couplings at the vertices (the nods of the network) can be approximated on thin branched structures by suitably scaled Schrödinger operators. Moreover, an approach where the approximating operator is a pure Laplacian (no potential), was elaborated by Kuchment-Post and is almost finished; a draft version is available upon request [5]. This approach also uses ideas from operator theory (Dirichlet-to-Neumann operators); and the PI profited much from the expertise of the Cardiff Analysis group. The PI together with J. Behrndt (Graz, Austria) also elaborated a similar approach for the Dirichlet-to-Neumann operator: Roughly speaking, we resolve the non-uniqueness of the inverse problems for networks (graphs) (``Can one hear the shape of a network'') by ``blowing up'' the network to a thin branched structure, see [6]. In summary, we provided a large class of solvable models for thin branched structures; including the zero-thickness limit; one of the goals of the project.

Another goal of the project was to develop tools from operator theory in order to treat branched structures in a systematic way. The principal investigator (PI) introduced in [3] a new approach (``boundary pairs'') extending known concepts (``boundary triples'') in an easy way to the relevant structures (partial differential operators), and the PI profitted very much from the expertise of the Cardiff analysis group (Brown, Marletta, Schmidt) on this approach (see also [10]). This approach is a basis for a systematic analysis of operators on branched structures, another goal of the project.

This analysis is partially included in [3].

The third goal of the project, the decomposition into building blocks according to a network and the use of ideas from discrete models for networks was addressed by the PI during several visits to the research group in Jena, Germany, (Lenz, Haeseler, Keller, [9]) and during various discussions with M.~Brown (Cardiff). Before addressing this goal more deaply, the PI realised that the profound analysis of the concept of ``boundary pairs'' in [3] was necessary. The project deepened the collaboration between the groups in Jena and Cardiff.

The PI profitted very much from the expertise of the Analysis research group in Cardiff, including the expertise on boundary triples (Brown, Marletta, Schmidt, Strauss). Moreover he established new future research directions in the area of homogenisation (Cherednichenko) and the possible application of operator-theoretic methods developed by the PI to this area.

The PI profitted also very much from the possibility of meeting research groups all over Europe: The grant allowed him to continue his collaboration with the Berlin Analysis group (Batu Güneysu, Jörn Müller, HU Berlin, [2], [8]), the collaboration with R. Hempel and R. Weder [4], the project on discrete branched structures with F. Lledó (Madrid) [7]. He also established new connections with Laughbourough (A. Strohmayer), UC London (L. Parnovski), Bristol (Ram Band) and Glasgow (Matthias Langer). Moreover, the attendence of several conferences resulted in an invitation from C. Anné for a research stay in Nantes (France) in 2013. He also was invited to give a short course in a summer school in Tunesia, establishing new scientific contacts.

The PI also organised a mini-workhop and a conference on related topics (``Boundary value problems and Spectral Geometry'', MF Oberwolfach, Germany; ``Dynamical systems on random graphs'', CIEM Castro Urdiales, Spain), establishing new and deepening existing collaborations on the European and international level.

List of publications and projects in preparation (author list in alphabetic order, as usual in Mathematics), for the list of

peer-reviewed publications, see next section)

Preprints

[3] Boundary pairs associated with quadratic forms (Olaf Post), submitted to Mathematische Nachrichten (47 pages)

[4] On open scattering channels for manifolds with ends (Rainer Hempel (Braunschweig, Germany), Olaf Post and Ricardo Weder (Mexico)), submitted to Journal of Functional Analysis (38 pages)

draft versions available on request:

[5] Generating vertex conditions by geometry (Peter Kuchment (Texas A&M) and Olaf Post) (ca. 32 pages)

[6] Inverse problems on thin branched manifolds and convergence of the Dirichlet-to-Neumann maps (Jussi Behrndt (TU Graz, Austria) and Olaf Post) (ca. 34 pages)

further projects:

[7] Spectral gaps for discrete magnetic periodic Laplacians (Fernando Lledó (Madrid) and Olaf Post)

[8] Boundary pairs and the Dirichlet-to-Neumann operator on non-compact warped products (Jörn Müller (HU Berlin), Olaf Post)

[9] Purely absolutely continuous spectrum for random metric tree graphs (Sebastian Haeseler, Matthias Keller, Daniel Lenz (Jena); Olaf Post)

[10] Boundary triples associated with quadratic forms (Olaf Post)