{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T11:06:39Z","timestamp":1769511999413,"version":"3.49.0"},"reference-count":0,"publisher":"Wiley","issue":"27","license":[{"start":{"date-parts":[[2004,6,30]],"date-time":"2004-06-30T00:00:00Z","timestamp":1088553600000},"content-version":"vor","delay-in-days":181,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>We investigate the spectrum of the differential operator\n<jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic><\/jats:sub> defined by the Klein\u2010Gordon <jats:italic>s<\/jats:italic>\u2010wave equation\n<jats:italic>y<\/jats:italic><jats:sup>\u2033<\/jats:sup> + (<jats:italic>\u03bb<\/jats:italic> \u2212 <jats:italic>q<\/jats:italic>(<jats:italic>x<\/jats:italic>))\u2009<jats:sup>2<\/jats:sup><jats:italic>y<\/jats:italic> = 0, <jats:italic>x<\/jats:italic> \u2208 <jats:italic>\u211d<\/jats:italic><jats:sub>+<\/jats:sub> = [0, <jats:italic>\u221e<\/jats:italic>),\nsubject to the spectral parameter\u2010dependent boundary condition\n<jats:italic>y<\/jats:italic><jats:sup>\u2032<\/jats:sup>(0) \u2212 (<jats:italic>a<\/jats:italic><jats:italic>\u03bb<\/jats:italic> + <jats:italic>b<\/jats:italic>)<jats:italic>y<\/jats:italic>(0) = 0 in the space <jats:italic>L<\/jats:italic><jats:sup>2<\/jats:sup>(<jats:italic>\u211d<\/jats:italic><jats:sub>+<\/jats:sub>), where <jats:italic>a<\/jats:italic> \u2260 \u00b1<jats:italic>i<\/jats:italic>, <jats:italic>b<\/jats:italic> are complex\nconstants, <jats:italic>q<\/jats:italic> is a complex\u2010valued function. Discussing the\nspectrum, we prove that <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic><\/jats:sub> has a finite number of\neigenvalues and spectral singularities with finite multiplicities\nif the conditions lim<jats:sub><jats:italic>x<\/jats:italic>\u2192<jats:italic>\u221e<\/jats:italic><\/jats:sub><jats:italic>q<\/jats:italic>(<jats:italic>x<\/jats:italic>) = 0, ,\n<jats:italic>\u03f5<\/jats:italic> &gt; 0, hold. Finally we show the properties of the\nprincipal functions corresponding to the spectral singularities.<\/jats:p>","DOI":"10.1155\/s0161171204203088","type":"journal-article","created":{"date-parts":[[2004,7,1]],"date-time":"2004-07-01T10:16:50Z","timestamp":1088677010000},"page":"1437-1445","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Spectral properties of the Klein\u2010Gordon <i>s<\/i>\u2010wave equation with spectral parameter\u2010dependent boundary condition"],"prefix":"10.1155","volume":"2004","author":[{"given":"G\u00fclen","family":"Ba\u015fcanbaz-Tunca","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2004,6,30]]},"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2004\/640620.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/S0161171204203088","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T14:15:03Z","timestamp":1719497703000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/S0161171204203088"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":0,"journal-issue":{"issue":"27","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1155\/S0161171204203088"],"URL":"https:\/\/doi.org\/10.1155\/s0161171204203088","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,1]]},"assertion":[{"value":"2002-03-12","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2004-06-30","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}