{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T12:21:58Z","timestamp":1769516518164,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,9]],"date-time":"2019-03-09T00:00:00Z","timestamp":1552089600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider a compact metric graph of size    \u03b5    and attach to it several edges (leads) of length of order one (or of infinite length). As    \u03b5    goes to zero, the graph     G \u03b5     obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On     G \u03b5     we define an Hamiltonian     H \u03b5    , properly scaled with the parameter    \u03b5   . We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates     H \u03b5     (in a suitable norm resolvent sense) as     \u03b5 \u2192 0    . The effective Hamiltonian depends on the spectral properties of an auxiliary    \u03b5   -independent Hamiltonian defined on the compact graph obtained by setting     \u03b5 = 1    . If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit     \u03b5 \u2192 0    , the leads are decoupled.<\/jats:p>","DOI":"10.3390\/sym11030359","type":"journal-article","created":{"date-parts":[[2019,3,12]],"date-time":"2019-03-12T03:49:31Z","timestamp":1552362571000},"page":"359","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7256-4265","authenticated-orcid":false,"given":"Claudio","family":"Cacciapuoti","sequence":"first","affiliation":[{"name":"Dipartimento di Scienza e Alta Tecnologia, Sezione di Matematica, Universit\u00e0 dell\u2019Insubria, Via Valleggio 11, 22100 Como, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1088\/0305-4470\/32\/4\/006","article-title":"Kirchhoff\u2019s rule for quantum wires","volume":"32","author":"Kostrykin","year":"1999","journal-title":"J. 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