Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart
We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Frontiers in Physical Chemistry and Chemical P...
Gespeichert in:
Veröffentlicht in: | arXiv.org 2017-03 |
---|---|
Hauptverfasser: | , , , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
container_end_page | |
---|---|
container_issue | |
container_start_page | |
container_title | arXiv.org |
container_volume | |
creator | Becker, Simon Mirahmadi, Marjan Schmidt, Burkhard Schatz, Konrad Friedrich, Bretislav |
description | We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Frontiers in Physical Chemistry and Chemical Physics 2, 1-16, (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -- forty in total to be exact -- analytic solutions. Furthermore, an analogous analysis of the hyperbolic counterpart of the planar pendulum, the Razavy problem, which was shown to be also C-QES [American Journal of Physics 48, 285 (1980)], confirmed that it is anti-isospectral with the pendular eigenproblem. Of key importance for both eigenproblems proved to be the topological index \(\kappa\), as it determines the loci of the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless interaction parameters \(\eta\) and \(\zeta\). It also encapsulates the conditions under which analytic solutions to the two eigenproblems obtain and provides the number of analytic solutions. At a given \(\kappa\), the anti-isospectrality occurs for single states only (i.e., not for doublets), like C-QES holds solely for integer values of \(\kappa\), and only occurs for the lowest eigenvalues of the pendular and Razavy Hamiltonians, with the order of the eigenvalues reversed for the latter. For all other states, the pendular and Razavy spectra become in fact qualitatively different, as higher pendular states appear as doublets whereas all higher Razavy states are singlets. |
doi_str_mv | 10.48550/arxiv.1702.08733 |
format | Article |
fullrecord | <record><control><sourceid>proquest_arxiv</sourceid><recordid>TN_cdi_arxiv_primary_1702_08733</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2075934623</sourcerecordid><originalsourceid>FETCH-LOGICAL-a523-a54866a61040ff72bbf8ed8d09f7b8797d151eecffe46d6a7e6d6fafa251c29c3</originalsourceid><addsrcrecordid>eNotkMtqwzAQRUWh0JDmA7qqoWuneliWvCyhLwh0k70Z2xJRUCxHj5D8fZXH5s4Mc7iLg9ALwctKco7fwZ_McUkEpkssBWMPaEYZI6WsKH1CixB2GGNaC8o5m6G0cuNgonEj2OKQIJhSnaCPRXD2CJ2xJp4Lp4u4VZf3GNO-mCyM4ItJjUOy-YZxuCAmhrxGU5rgwqT66HPl9jwp3zlr-qJ3aYzKT-DjM3rUYINa3Occbb4-N6ufcv33_bv6WJfAKctRybqGmuAKay1o12mpBjngRotOikYMhBOleq1VVQ81CJVTgwbKSU-bns3R66326qSdvNmDP7cXN-3VTSbebsTk3SGpENudSz67CC3FgjesqrO8f4F5atk</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2075934623</pqid></control><display><type>article</type><title>Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart</title><source>arXiv.org</source><source>Open Access: Freely Accessible Journals by multiple vendors</source><creator>Becker, Simon ; Mirahmadi, Marjan ; Schmidt, Burkhard ; Schatz, Konrad ; Friedrich, Bretislav</creator><creatorcontrib>Becker, Simon ; Mirahmadi, Marjan ; Schmidt, Burkhard ; Schatz, Konrad ; Friedrich, Bretislav</creatorcontrib><description>We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Frontiers in Physical Chemistry and Chemical Physics 2, 1-16, (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -- forty in total to be exact -- analytic solutions. Furthermore, an analogous analysis of the hyperbolic counterpart of the planar pendulum, the Razavy problem, which was shown to be also C-QES [American Journal of Physics 48, 285 (1980)], confirmed that it is anti-isospectral with the pendular eigenproblem. Of key importance for both eigenproblems proved to be the topological index \(\kappa\), as it determines the loci of the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless interaction parameters \(\eta\) and \(\zeta\). It also encapsulates the conditions under which analytic solutions to the two eigenproblems obtain and provides the number of analytic solutions. At a given \(\kappa\), the anti-isospectrality occurs for single states only (i.e., not for doublets), like C-QES holds solely for integer values of \(\kappa\), and only occurs for the lowest eigenvalues of the pendular and Razavy Hamiltonians, with the order of the eigenvalues reversed for the latter. For all other states, the pendular and Razavy spectra become in fact qualitatively different, as higher pendular states appear as doublets whereas all higher Razavy states are singlets.</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.1702.08733</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Eigenvalues ; Exact solutions ; Hamiltonian functions ; Interaction parameters ; Intersections ; Mathematical analysis ; Matrix methods ; Organic chemistry ; Pendulums ; Physical chemistry ; Physics - Quantum Physics ; Symmetry ; Topology</subject><ispartof>arXiv.org, 2017-03</ispartof><rights>2017. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,784,885,27925</link.rule.ids><backlink>$$Uhttps://doi.org/10.1140/epjd/e2017-80134-6$$DView published paper (Access to full text may be restricted)$$Hfree_for_read</backlink><backlink>$$Uhttps://doi.org/10.48550/arXiv.1702.08733$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Becker, Simon</creatorcontrib><creatorcontrib>Mirahmadi, Marjan</creatorcontrib><creatorcontrib>Schmidt, Burkhard</creatorcontrib><creatorcontrib>Schatz, Konrad</creatorcontrib><creatorcontrib>Friedrich, Bretislav</creatorcontrib><title>Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart</title><title>arXiv.org</title><description>We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Frontiers in Physical Chemistry and Chemical Physics 2, 1-16, (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -- forty in total to be exact -- analytic solutions. Furthermore, an analogous analysis of the hyperbolic counterpart of the planar pendulum, the Razavy problem, which was shown to be also C-QES [American Journal of Physics 48, 285 (1980)], confirmed that it is anti-isospectral with the pendular eigenproblem. Of key importance for both eigenproblems proved to be the topological index \(\kappa\), as it determines the loci of the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless interaction parameters \(\eta\) and \(\zeta\). It also encapsulates the conditions under which analytic solutions to the two eigenproblems obtain and provides the number of analytic solutions. At a given \(\kappa\), the anti-isospectrality occurs for single states only (i.e., not for doublets), like C-QES holds solely for integer values of \(\kappa\), and only occurs for the lowest eigenvalues of the pendular and Razavy Hamiltonians, with the order of the eigenvalues reversed for the latter. For all other states, the pendular and Razavy spectra become in fact qualitatively different, as higher pendular states appear as doublets whereas all higher Razavy states are singlets.</description><subject>Eigenvalues</subject><subject>Exact solutions</subject><subject>Hamiltonian functions</subject><subject>Interaction parameters</subject><subject>Intersections</subject><subject>Mathematical analysis</subject><subject>Matrix methods</subject><subject>Organic chemistry</subject><subject>Pendulums</subject><subject>Physical chemistry</subject><subject>Physics - Quantum Physics</subject><subject>Symmetry</subject><subject>Topology</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GOX</sourceid><recordid>eNotkMtqwzAQRUWh0JDmA7qqoWuneliWvCyhLwh0k70Z2xJRUCxHj5D8fZXH5s4Mc7iLg9ALwctKco7fwZ_McUkEpkssBWMPaEYZI6WsKH1CixB2GGNaC8o5m6G0cuNgonEj2OKQIJhSnaCPRXD2CJ2xJp4Lp4u4VZf3GNO-mCyM4ItJjUOy-YZxuCAmhrxGU5rgwqT66HPl9jwp3zlr-qJ3aYzKT-DjM3rUYINa3Occbb4-N6ufcv33_bv6WJfAKctRybqGmuAKay1o12mpBjngRotOikYMhBOleq1VVQ81CJVTgwbKSU-bns3R66326qSdvNmDP7cXN-3VTSbebsTk3SGpENudSz67CC3FgjesqrO8f4F5atk</recordid><startdate>20170327</startdate><enddate>20170327</enddate><creator>Becker, Simon</creator><creator>Mirahmadi, Marjan</creator><creator>Schmidt, Burkhard</creator><creator>Schatz, Konrad</creator><creator>Friedrich, Bretislav</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>GOX</scope></search><sort><creationdate>20170327</creationdate><title>Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart</title><author>Becker, Simon ; Mirahmadi, Marjan ; Schmidt, Burkhard ; Schatz, Konrad ; Friedrich, Bretislav</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a523-a54866a61040ff72bbf8ed8d09f7b8797d151eecffe46d6a7e6d6fafa251c29c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Eigenvalues</topic><topic>Exact solutions</topic><topic>Hamiltonian functions</topic><topic>Interaction parameters</topic><topic>Intersections</topic><topic>Mathematical analysis</topic><topic>Matrix methods</topic><topic>Organic chemistry</topic><topic>Pendulums</topic><topic>Physical chemistry</topic><topic>Physics - Quantum Physics</topic><topic>Symmetry</topic><topic>Topology</topic><toplevel>online_resources</toplevel><creatorcontrib>Becker, Simon</creatorcontrib><creatorcontrib>Mirahmadi, Marjan</creatorcontrib><creatorcontrib>Schmidt, Burkhard</creatorcontrib><creatorcontrib>Schatz, Konrad</creatorcontrib><creatorcontrib>Friedrich, Bretislav</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>ProQuest Central Essentials</collection><collection>AUTh Library subscriptions: ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>ProQuest - Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering collection</collection><collection>arXiv.org</collection><jtitle>arXiv.org</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Becker, Simon</au><au>Mirahmadi, Marjan</au><au>Schmidt, Burkhard</au><au>Schatz, Konrad</au><au>Friedrich, Bretislav</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart</atitle><jtitle>arXiv.org</jtitle><date>2017-03-27</date><risdate>2017</risdate><eissn>2331-8422</eissn><abstract>We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Frontiers in Physical Chemistry and Chemical Physics 2, 1-16, (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -- forty in total to be exact -- analytic solutions. Furthermore, an analogous analysis of the hyperbolic counterpart of the planar pendulum, the Razavy problem, which was shown to be also C-QES [American Journal of Physics 48, 285 (1980)], confirmed that it is anti-isospectral with the pendular eigenproblem. Of key importance for both eigenproblems proved to be the topological index \(\kappa\), as it determines the loci of the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless interaction parameters \(\eta\) and \(\zeta\). It also encapsulates the conditions under which analytic solutions to the two eigenproblems obtain and provides the number of analytic solutions. At a given \(\kappa\), the anti-isospectrality occurs for single states only (i.e., not for doublets), like C-QES holds solely for integer values of \(\kappa\), and only occurs for the lowest eigenvalues of the pendular and Razavy Hamiltonians, with the order of the eigenvalues reversed for the latter. For all other states, the pendular and Razavy spectra become in fact qualitatively different, as higher pendular states appear as doublets whereas all higher Razavy states are singlets.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><doi>10.48550/arxiv.1702.08733</doi><oa>free_for_read</oa></addata></record> |
fulltext | fulltext |
identifier | EISSN: 2331-8422 |
ispartof | arXiv.org, 2017-03 |
issn | 2331-8422 |
language | eng |
recordid | cdi_arxiv_primary_1702_08733 |
source | arXiv.org; Open Access: Freely Accessible Journals by multiple vendors |
subjects | Eigenvalues Exact solutions Hamiltonian functions Interaction parameters Intersections Mathematical analysis Matrix methods Organic chemistry Pendulums Physical chemistry Physics - Quantum Physics Symmetry Topology |
title | Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart |
url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-25T08%3A07%3A22IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_arxiv&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Conditional%20quasi-exact%20solvability%20of%20the%20quantum%20planar%20pendulum%20and%20of%20its%20anti-isospectral%20hyperbolic%20counterpart&rft.jtitle=arXiv.org&rft.au=Becker,%20Simon&rft.date=2017-03-27&rft.eissn=2331-8422&rft_id=info:doi/10.48550/arxiv.1702.08733&rft_dat=%3Cproquest_arxiv%3E2075934623%3C/proquest_arxiv%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2075934623&rft_id=info:pmid/&rfr_iscdi=true |