{"id":62,"date":"2010-03-30T12:23:41","date_gmt":"2010-03-30T12:23:41","guid":{"rendered":"http:\/\/lptms.u-psud.fr\/christophe_texier\/?page_id=62"},"modified":"2018-09-07T09:37:21","modified_gmt":"2018-09-07T09:37:21","slug":"spectral-determinants-of-metric-graphs-glued-at-one-point","status":"publish","type":"page","link":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/metric-graphs-spectrum-and-scattering-2\/spectral-determinants-of-metric-graphs-glued-at-one-point\/","title":{"rendered":"Functional determinants"},"content":{"rendered":"<p><span style=\"font-size: medium;color: #008000\"><strong>Spectral determinants (functional determinants) on metric graphs<\/strong><\/span><\/p>\n<p>The spectral determinant is a compact object encoding information on the spectrum of a linear operator (the Laplace or the Schr\u00f6dinger operator, of interest here):<\/p>\n<p style=\"padding-left: 60px\"><span style=\"font-size: medium\"><em>S<\/em>(\u03b3)=det(\u03b3-\u0394+<em>V<\/em>(<em>x<\/em>))=\u220f(\u03b3+<em>E<sub>n<\/sub><\/em>)<\/span><\/p>\n<p>where \u03b3 is a (spectral) parameter and {<em>E<sub>n<\/sub><\/em>} is the spectrum of eigenvalues of the operator -\u0394+<em>V<\/em>(<em>x<\/em>) (the product runs over all eigenstates ; it is a formal writing since the set of eigenstates is infinite). Such objects are also denoted \u00ab\u00a0functional determinants\u00a0\u00bb in the mathematical literature.<\/p>\n<p>A trace formula for the partition function (heat kernel) of the Laplace operator on a metric graph was obtained by J.-P. Roth in 1983. In our article we have established the relation between the Roth&rsquo;s trace formula and a recent result by M. Pascaud &amp; G. Montambaux expressing the spectral determinant in term of the determinant of a matrix of finite size coupling vertices of the graph.<\/p>\n<ul>\n<li>Eric Akkermans, Alain Comtet, Jean Desbois, Gilles Montambaux and Christophe Texier,<br \/>\n<strong>Spectral determinant on quantum graphs<\/strong><br \/>\n<a href=\"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0003491600960561\">Ann. Phys. (N.Y.) <strong>284<\/strong>, 10-51 (2000)<\/a><br \/>\n<a href=\"http:\/\/xxx.lanl.gov\/abs\/cond-mat\/9911183\">cond-mat\/9911183.<\/a><\/li>\n<\/ul>\n<p><strong>Spectral determinants of metric graphs glued at one point<\/strong><\/p>\n<p>The spectral determinant is a compact object that can be easily and systematically constructed for metric graphs in terms of their topological properties. This short paper shows that when two graphs G<sub>1<\/sub> and G<sub>2<\/sub> are glued at one point in order to form a larger graph G, Spectral determinant of G can be related to spectral determinants of G<sub>1<\/sub> and G<sub>2<\/sub>. Therefore spectrum of G can be related to the spectra of G<sub>1<\/sub> and G<sub>2<\/sub> (plus some information on the connectivity of the vertex where graphs are attached).<\/p>\n<ul>\n<li>Christophe Texier,<br \/>\n<strong>On the spectrum of the Laplace operator of metric graphs attached at a vertex &#8212; Spectral determinant approach<\/strong><br \/>\n<a href=\"http:\/\/www.iop.org\/EJ\/abstract\/1751-8121\/41\/8\/085207\">J. Phys. A : Math. Theor. <strong>41<\/strong>, 085207 (2008).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/0706.0120\">math-ph arXiv:0706.0120.<\/a><\/li>\n<\/ul>\n<p><strong>Zeta regularisation of spectral determinants on metric graphs<\/strong><\/p>\n<p>The question of the regularisation of the spectral determinant and its precise prefactor is discussed.<\/p>\n<ul>\n<li>Christophe Texier,<br \/>\n<strong>\u03b6-regularised spectral determinants on metric graphs<\/strong><br \/>\n<a href=\"http:\/\/iopscience.iop.org\/1751-8121\/43\/42\/425203\">J. Phys. A : Math. Theor. <strong>43<\/strong>, 425203 (2010).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1006.2163\">math-ph arXiv:1006.2163<\/a><\/li>\n<li>Jon M. Harrison, Klaus Kirsten and Christophe Texier,<br \/>\n<strong>Spectral determinants and zeta functions of Schr\u00f6dinger operators on metric graphs<\/strong><br \/>\n<a href=\"http:\/\/iopscience.iop.org\/1751-8121\/45\/12\/125206\">J. Phys. A : Math. Theor. <strong>45<\/strong>, 125206 (2012).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1111.0643\">math-ph arXiv:1111.0643<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Spectral determinants (functional determinants) on metric graphs The spectral determinant is a compact object encoding information on the spectrum of a linear operator (the Laplace or the Schr\u00f6dinger operator, of interest here): S(\u03b3)=det(\u03b3-\u0394+V(x))=\u220f(\u03b3+En) where \u03b3 is a (spectral) parameter and &hellip; <a href=\"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/metric-graphs-spectrum-and-scattering-2\/spectral-determinants-of-metric-graphs-glued-at-one-point\/\">Continuer la lecture <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":363,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-62","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/62","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/comments?post=62"}],"version-history":[{"count":25,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/62\/revisions"}],"predecessor-version":[{"id":1994,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/62\/revisions\/1994"}],"up":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/363"}],"wp:attachment":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/media?parent=62"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}