{"id":2282,"date":"2020-01-07T23:19:00","date_gmt":"2020-01-07T23:19:00","guid":{"rendered":"http:\/\/lptms.u-psud.fr\/christophe_texier\/?page_id=2282"},"modified":"2020-01-07T23:23:41","modified_gmt":"2020-01-07T23:23:41","slug":"susy-breaking-by-a-random-potential","status":"publish","type":"page","link":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/some-properties-of-the-one-dimensional-schrodinger-operator-with-random-potential\/susy-breaking-by-a-random-potential\/","title":{"rendered":"Disordered supersymmetric quantum mechanics"},"content":{"rendered":"<h4>Cf. the review on SUSY QM : <a href=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/recherche\/a-review-article\/\">page on review articles<\/a><\/h4>\n<h3>How to break supersymmetry with a random scalar potential<\/h3>\n<p>Cf. paragraph on<a href=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/recherche\/a-review-article\/\"> supersymmetric quantum mechanics<\/a><\/p>\n<ul>\n<li>Christian Hagendorf and Christophe Texier,<br \/>\n<strong>Breaking of supersymmetry in one-dimensional a random Hamiltonian<\/strong><br \/>\n<a href=\"http:\/\/iopscience.iop.org\/1751-8121\/41\/40\/405302\">J. Phys. A: Math. Theor. <strong>41<\/strong>, 405302 (2008).<\/a> (32pp)<br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/0805.2883\">cond-mat arXiv:0805.2883.<\/a><\/li>\n<li>Christophe Texier and Christian Hagendorf,<br \/>\n<strong>One-dimensional classical diffusion in a random force field with weakly concentrated absorbers<\/strong><br \/>\n<a href=\"http:\/\/epljournal.edpsciences.org\/index.php?option=com_article&amp;access=standard&amp;Itemid=129&amp;url=\/articles\/epl\/abs\/2009\/09\/epl11811\/epl11811.html\">Europhys. Lett. <strong>86<\/strong>, 37011 (2009). <\/a>(6pp)<br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/0902.2698\">cond-mat arXiv:0902.2698.<\/a><\/li>\n<li>Aur\u00e9lien Grabsch, Christophe Texier and Yves Tourigny,<br \/>\n<strong>One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers<\/strong><br \/>\n<a href=\"http:\/\/link.springer.com\/article\/10.1007%2Fs10955-014-0957-3\">J. Stat. Phys. <strong>155<\/strong>, 237-276 (2014)<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1310.6519\">cond-mat arXiv:1310.6519.<\/a><\/li>\n<\/ul>\n<p style=\"padding-left: 60px\"><span style=\"color: #ff0000\"><strong>See also<\/strong><\/span> the page on <a href=\"http:\/\/lptms.u-psud.fr\/christophe_texier\/recherche\/classical-diffusion-in-random-environment\/\"><strong>Classical diffusion in a random force field (Sinai problem)<\/strong><\/a><\/p>\n<h3>One-dimensional supersymmetric Hamiltonian with L\u00e9vy noises<\/h3>\n<p>In this work we consider Schr\u00f6dinger supersymmetric Hamiltonians when the function \u03c6(x) is a L\u00e9vy noise, i.e. when \u03a6(x)=\u222b<sub>0<\/sub><sup>x<\/sup> dx&rsquo; \u03c6(x&rsquo;) is a <a href=\"http:\/\/en.wikipedia.org\/wiki\/L%C3%A9vy_process\">L\u00e9vy process<\/a> (a random process generalising the Brownian motion). The case of <em>subordinators<\/em> is considered (non decreasing processes). We have discovered a new exact model for a L\u00e9vy process with singular L\u00e9vy measure m(dy). Moreover, we have provided a general discussion of low-energy spectral properties for arbitrary subordinators. (i) For regular L\u00e9vy measure we show that the main exponential behaviour of the integrated density of states is N(E) \u223c exp[-\u03c0\u03c1\/\u221aE] where \u03c1=\u222b<sub>0<\/sub><sup>\u221e<\/sup>m(dy). (ii) For singular L\u00e9vy measures, \u222b<sub>0<\/sub><sup>\u221e<\/sup>m(dy)=\u221e, we obtain N(E) \u223c exp[-C E<sup>-\u03b7\/2<\/sup>] where the exponent is related to the singularity of the L\u00e9vy measure by \u03b7=1\/(1-\u03b1), where m(dy) \u221d y<sup>-1-\u03b1<\/sup>dy when y \u2192 0+ (for 0&lt;\u03b1&lt;1).<\/p>\n<ul>\n<li>Alain Comtet, Christophe Texier and Yves Tourigny,<br \/>\n<strong>Supersymmetric quantum mechanics with L\u00e9vy disorder in one-dimension<\/strong><br \/>\n<a href=\"http:\/\/www.springerlink.com\/content\/076k25851080j222\/\">J. Stat. Phys. <strong>145<\/strong>, 1291-1323 (2011)<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1105.5506\">math-ph arXiv:1105.5506<\/a><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Cf. the review on SUSY QM : page on review articles How to break supersymmetry with a random scalar potential Cf. paragraph on supersymmetric quantum mechanics Christian Hagendorf and Christophe Texier, Breaking of supersymmetry in one-dimensional a random Hamiltonian J. &hellip; <a href=\"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/some-properties-of-the-one-dimensional-schrodinger-operator-with-random-potential\/susy-breaking-by-a-random-potential\/\">Continuer la lecture <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"parent":42,"menu_order":13,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2282","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/2282","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\/7"}],"replies":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/comments?post=2282"}],"version-history":[{"count":4,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/2282\/revisions"}],"predecessor-version":[{"id":2287,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/2282\/revisions\/2287"}],"up":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/42"}],"wp:attachment":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/media?parent=2282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}