On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity
We consider the Schrödinger operator on the plane with bounded potential , where is a real potential, and are compactly supported complex potentials, and is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator consists of a pair of isolated eige...
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| Veröffentlicht in: | Differential equations 2023-02, Vol.59 (2), p.278-282 |
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| creator | Borisov, D. I. Zezyulin, D. A. |
| description | We consider the Schrödinger operator on the plane with bounded potential
, where
is a real potential,
and
are compactly supported complex potentials, and
is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator
consists of a pair of isolated eigenvalues and the essential spectrum of the operator
has a virtual level at its lower edge and a spectral singularity inside. Additionally, we assume that there is a certain superposition of eigenvalues of the operator
with the virtual level and spectral singularity of the operator
; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones. |
| doi_str_mv | 10.1134/S0012266123020118 |
| format | Article |
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, where
is a real potential,
and
are compactly supported complex potentials, and
is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator
consists of a pair of isolated eigenvalues and the essential spectrum of the operator
has a virtual level at its lower edge and a spectral singularity inside. Additionally, we assume that there is a certain superposition of eigenvalues of the operator
with the virtual level and spectral singularity of the operator
; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones.</description><identifier>ISSN: 0012-2661</identifier><identifier>EISSN: 1608-3083</identifier><identifier>DOI: 10.1134/S0012266123020118</identifier><language>eng</language><publisher>Moscow: Pleiades Publishing</publisher><subject>Bifurcations ; Difference and Functional Equations ; Differential equations ; Eigenvalues ; Mathematics ; Mathematics and Statistics ; Operators (mathematics) ; Ordinary Differential Equations ; Partial Differential Equations ; Perturbation ; Short Communications ; Singularities</subject><ispartof>Differential equations, 2023-02, Vol.59 (2), p.278-282</ispartof><rights>Pleiades Publishing, Ltd. 2023</rights><rights>COPYRIGHT 2023 Springer</rights><rights>Pleiades Publishing, Ltd. 2023.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c355t-114958c395d6de7a34d27511675ba5be1401f512fcfe09cfc8ae964655c5b3b63</citedby><cites>FETCH-LOGICAL-c355t-114958c395d6de7a34d27511675ba5be1401f512fcfe09cfc8ae964655c5b3b63</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1134/S0012266123020118$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1134/S0012266123020118$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>315,781,785,27926,27927,41490,42559,51321</link.rule.ids></links><search><creatorcontrib>Borisov, D. I.</creatorcontrib><creatorcontrib>Zezyulin, D. A.</creatorcontrib><title>On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity</title><title>Differential equations</title><addtitle>Diff Equat</addtitle><description>We consider the Schrödinger operator on the plane with bounded potential
, where
is a real potential,
and
are compactly supported complex potentials, and
is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator
consists of a pair of isolated eigenvalues and the essential spectrum of the operator
has a virtual level at its lower edge and a spectral singularity inside. Additionally, we assume that there is a certain superposition of eigenvalues of the operator
with the virtual level and spectral singularity of the operator
; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones.</description><subject>Bifurcations</subject><subject>Difference and Functional Equations</subject><subject>Differential equations</subject><subject>Eigenvalues</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operators (mathematics)</subject><subject>Ordinary Differential Equations</subject><subject>Partial Differential Equations</subject><subject>Perturbation</subject><subject>Short Communications</subject><subject>Singularities</subject><issn>0012-2661</issn><issn>1608-3083</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp1kF9LwzAUxYMoOKcfwLeCz525SZO2j3PMPzDYw6avJU2TLaNLZtIi-_amVPBB5D5cbs75JScXoXvAMwCaPW4wBkI4B0IxwQDFBZoAx0VKcUEv0WSQ00G_RjchHDDGZQ5sgj7WNun2KnkyuvdSdMbZxOlku_cq7F3bhGEaDMsQlO2MaJPNScnO98fky3T7RPzMg2Dsrm-FN935Fl1p0QZ199On6P15uV28pqv1y9tivkolZaxLAbKSFZKWrOGNygXNGpIzAJ6zWrBaQYZBMyBaaoVLqWUhVMkzzphkNa05naKH8d6Td5-9Cl11cL238cmKFECHX0MRXbPRtROtqozVLuaVsRp1NNJZpU08n-dZCcBwjiMAIyC9C8ErXZ28OQp_rgBXw76rP_uODBmZEL12p_xvlP-hb_J7gEo</recordid><startdate>20230201</startdate><enddate>20230201</enddate><creator>Borisov, D. I.</creator><creator>Zezyulin, D. A.</creator><general>Pleiades Publishing</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20230201</creationdate><title>On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity</title><author>Borisov, D. I. ; Zezyulin, D. A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c355t-114958c395d6de7a34d27511675ba5be1401f512fcfe09cfc8ae964655c5b3b63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Bifurcations</topic><topic>Difference and Functional Equations</topic><topic>Differential equations</topic><topic>Eigenvalues</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operators (mathematics)</topic><topic>Ordinary Differential Equations</topic><topic>Partial Differential Equations</topic><topic>Perturbation</topic><topic>Short Communications</topic><topic>Singularities</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Borisov, D. I.</creatorcontrib><creatorcontrib>Zezyulin, D. A.</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Differential equations</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Borisov, D. I.</au><au>Zezyulin, D. A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity</atitle><jtitle>Differential equations</jtitle><stitle>Diff Equat</stitle><date>2023-02-01</date><risdate>2023</risdate><volume>59</volume><issue>2</issue><spage>278</spage><epage>282</epage><pages>278-282</pages><issn>0012-2661</issn><eissn>1608-3083</eissn><abstract>We consider the Schrödinger operator on the plane with bounded potential
, where
is a real potential,
and
are compactly supported complex potentials, and
is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator
consists of a pair of isolated eigenvalues and the essential spectrum of the operator
has a virtual level at its lower edge and a spectral singularity inside. Additionally, we assume that there is a certain superposition of eigenvalues of the operator
with the virtual level and spectral singularity of the operator
; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones.</abstract><cop>Moscow</cop><pub>Pleiades Publishing</pub><doi>10.1134/S0012266123020118</doi><tpages>5</tpages></addata></record> |
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| ispartof | Differential equations, 2023-02, Vol.59 (2), p.278-282 |
| issn | 0012-2661 1608-3083 |
| language | eng |
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| source | SpringerNature Journals |
| subjects | Bifurcations Difference and Functional Equations Differential equations Eigenvalues Mathematics Mathematics and Statistics Operators (mathematics) Ordinary Differential Equations Partial Differential Equations Perturbation Short Communications Singularities |
| title | On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity |
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