{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:36:53Z","timestamp":1787323013206,"version":"build-2736575974"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>We consider the existence of localized modes corresponding to eigenvalues of the periodic Schr\u00f6dinger operator $-\\partial_x^2+V(x)$ with an interface. The interface is modeled by a jump either in the value or the derivative of $V(x)$ and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be different. As we show, eigenvalues can occur only in spectral gaps. We pose the eigenvalue problem as a $C^1$ gluing problem for the fundamental solutions (Bloch functions) of the second order ODEs on each side of the interface. The problem is thus reduced to finding matchings of the ratio functions $R_\\pm=\\frac{\\psi'_\\pm(0)}{\\psi_\\pm(0)}$, where $\\psi_\\pm$ are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Pr\u00fcfer transformation. The limit values of $R_\\pm$ at band edges depend on the ordering of Dirichlet and Neumann eigenvalues at gap edges. We show that the ordering can be determined in the first two gaps via variational analysis for potentials satisfying certain monotonicity conditions. Numerical computations of interface eigenvalues are presented to corroborate the analysis.<\/jats:p>","DOI":"10.1137\/080743366","type":"journal-article","created":{"date-parts":[[2009,11,19]],"date-time":"2009-11-19T01:52:12Z","timestamp":1258595532000},"page":"1967-1993","source":"Crossref","is-referenced-by-count":11,"title":["Localized Modes of the Linear Periodic Schr\u00f6dinger Operator with a Nonlocal Perturbation"],"prefix":"10.1137","volume":"41","author":[{"given":"Tom\u00e1\u0161","family":"Dohnal","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Plum","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wolfgang","family":"Reichel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,11,18]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01217808"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.4171\/JEMS\/104"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1134\/S1064562407020202"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.82.2022"},{"key":"R5","unstructured":"E. A. Coddington and N. Levinson,\n                      Theory of Ordinary Differential Equations\n                      , McGraw-Hill, New York, 1955."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01211761"},{"key":"R7","unstructured":"M. S. P. Eastham,\n                      Spectral Theory of Periodic Differential Equations\n                      , Scottish Academic Press, Edinburgh, London, 1973."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02566350"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02731494"},{"key":"R10","unstructured":"G. H. Hardy, J. E. Littlewood, and G. P\u00f3lya,\n                      Inequalities\n                      , 2nd ed., Cambridge University Press, Cambridge, UK, 1952."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"B. Kawohl,\n                      Rearrangements and Convexity of Level Sets in PDE\n                      , Lecture Notes in Math. 1150, Springer-Verlag, Berlin, 1985.","DOI":"10.1007\/BFb0075060"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1515\/rose.1998.6.3.241"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005529"},{"key":"R14","doi-asserted-by":"crossref","first-page":"73","DOI":"10.3233\/ASY-2005-716","volume":"45","author":"Korotyaev E.","year":"2005","journal-title":"Asymptot. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0921-7134","issn-type":"print"},{"key":"R15","unstructured":"W. Magnus and S. Winkler,\n                      Hill's Equation\n                      , Interscience, New York, 1966; Dover Publications, Inc., New York, 1979 (corrected reprint)."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0039-6028(92)91112-O"},{"key":"R17","first-page":"451","volume":"13","author":"Pitaevskii L. P.","year":"1961","journal-title":"Z. Eksper. Teoret. Fiz., 40 (1961), pp. 646\u2013651 (in Russian); Soviet Phys. JETP"},{"key":"R18","unstructured":"M. Reed and B. Simon,\n                      Methods of Modern Mathematical Physics.\n                      IV\n                      . Analysis of Operators\n                      , Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1364\/OE.15.004663"},{"key":"R20","first-page":"733","volume":"1","author":"Tamm I.","year":"1932","journal-title":"Phys. Z. Sowjetunion","ISSN":"https:\/\/id.crossref.org\/issn\/0369-9811","issn-type":"print"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/s00023-002-8621-x"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"W. Walter,\n                      Ordinary Differential Equations\n                      , Grad. Texts in Math. 182, Springer-Verlag, New York, 1998. Translated from the sixth German (1996) edition by Russell Thompson, Readings in Mathematics.","DOI":"10.1007\/978-1-4612-0601-9"},{"key":"R23","unstructured":"V. A. \u017deludev,\n                      The eigenvalues of a perturbed Schr\u00f6dinger operator with periodic potential\n                      , in Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems, Izdat. Leningrad. Univ., Leningrad, 1967, pp. 108\u2013123 (in Russian)."}],"container-title":["SIAM Journal on Mathematical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080743366","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:36:43Z","timestamp":1787319403000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080743366"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":23,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080743366"],"URL":"https:\/\/doi.org\/10.1137\/080743366","relation":{},"ISSN":["0036-1410","1095-7154"],"issn-type":[{"value":"0036-1410","type":"print"},{"value":"1095-7154","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}