Index theory for non-compact quantum graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914419634601984 |
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| author | Garrisi, Daniele Portaluri, Alessandro Wu, Li |
| author_facet | Garrisi, Daniele Portaluri, Alessandro Wu, Li |
| contents | We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating the negative directions of the Lagrangian action with the total multiplicity of conjugate instants along the edges.
These results extend classical tools in global analysis and symplectic geometry to graph based models, with applications to nonlinear wave equations such as the nonlinear Schroedinger equation. The spectral flow formula is proved by constructing a Lagrangian intersection theory in the Gelfand-Robbin quotients of the second variation of the action. This approach also recovers, in a unified way, the known formulas for heteroclinic, halfclinic, homoclinic, and bounded orbits of (non)autonomous Lagrangian systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_09749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Index theory for non-compact quantum graphs Garrisi, Daniele Portaluri, Alessandro Wu, Li Functional Analysis Classical Analysis and ODEs Dynamical Systems Symplectic Geometry Spectral Theory 37E25, 58J30, 58J50, 34B45, 34L40, 35Q55 We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating the negative directions of the Lagrangian action with the total multiplicity of conjugate instants along the edges. These results extend classical tools in global analysis and symplectic geometry to graph based models, with applications to nonlinear wave equations such as the nonlinear Schroedinger equation. The spectral flow formula is proved by constructing a Lagrangian intersection theory in the Gelfand-Robbin quotients of the second variation of the action. This approach also recovers, in a unified way, the known formulas for heteroclinic, halfclinic, homoclinic, and bounded orbits of (non)autonomous Lagrangian systems. |
| title | Index theory for non-compact quantum graphs |
| topic | Functional Analysis Classical Analysis and ODEs Dynamical Systems Symplectic Geometry Spectral Theory 37E25, 58J30, 58J50, 34B45, 34L40, 35Q55 |
| url | https://arxiv.org/abs/2509.09749 |