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Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry
We consider an invariant quantum Hamiltonian H = − Δ LB + V in the L 2 space based on a Riemannian manifold M ̃ with a countable discrete symmetry group Γ . Typically, M ̃ is the universal covering space of a multiply connected Riemannian manifold M and Γ is the fundamental group of M . On the one h...
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| Published in: | Journal of mathematical physics 2008-03, Vol.49 (3), p.033518-033518-16 |
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| container_end_page | 033518-16 |
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| container_title | Journal of mathematical physics |
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| creator | KOCABOVA, P STOVICEK, P |
| description | We consider an invariant quantum Hamiltonian
H
=
−
Δ
LB
+
V
in the
L
2
space based on a Riemannian manifold
M
̃
with a countable discrete symmetry group
Γ
. Typically,
M
̃
is the universal covering space of a multiply connected Riemannian manifold
M
and
Γ
is the fundamental group of
M
. On the one hand, following the basic step of the Bloch analysis, one decomposes the
L
2
space over
M
̃
into a direct integral of Hilbert spaces formed by equivariant functions on
M
̃
. The Hamiltonian
H
decomposes correspondingly, with each component
H
Λ
being defined by a quasiperiodic boundary condition. The quasiperiodic boundary conditions are in turn determined by irreducible unitary representations
Λ
of
Γ
. On the other hand, fixing a quasiperiodic boundary condition (i.e., a unitary representation
Λ
of
Γ
) one can express the corresponding propagator in terms of the propagator associated with the Hamiltonian
H
. We discuss these procedures in detail and show that in a sense they are mutually inverse. |
| doi_str_mv | 10.1063/1.2898484 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_pasca</sourceid><recordid>TN_cdi_pascalfrancis_primary_20277660</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1456857801</sourcerecordid><originalsourceid>FETCH-LOGICAL-c474t-c7f3a33756f10d88b04c50df5a572ee8ecef85415f32ea37f97f66d544dcde3b3</originalsourceid><addsrcrecordid>eNqNkU9rlDEQh0NR6Nr20G8QLB4svDV_32QvQi21CgVB7KmHkCYTm_Jusk1Sy_bTN-su6kXxNHN45mF-MwgdUnJCycjf0ROm51posYNmlOj5oEapX6AZIYwNTGi9i17VekcIpVqIGbq-gATFTvEJPP4wZXeLbbLTqsbaG4-XJS_td9tyqTgn_DXCwqYUbcK9xpAnX_FjbH0K-1hdgQa4rhYLaGW1j14GO1U42NY9dPXx_NvZp-Hyy8Xns9PLwQkl2uBU4JZzJcdAidf6hggniQ_SSsUANDgIWgoqA2dguQpzFcbRSyG888Bv-B462nhzbdFUFxu4W5dTAtcMo7RnF7pTrzdUj3T_ALWZu_xQetbaGTlSzdgaeruBXMm1FghmWeLClpWhxKwPbKjZHrizb7ZCW52dQrHJxfprgBGm1DiSzr3fcOvNbIs5_V36xzfMz28Ym7rg-L8F_4J_5PIbNEsf-DPtlKw5</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>215618228</pqid></control><display><type>article</type><title>Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry</title><source>American Institute of Physics (AIP) Publications</source><source>American Institute of Physics:Jisc Collections:Transitional Journals Agreement 2021-23 (Reading list)</source><creator>KOCABOVA, P ; STOVICEK, P</creator><creatorcontrib>KOCABOVA, P ; STOVICEK, P</creatorcontrib><description>We consider an invariant quantum Hamiltonian
H
=
−
Δ
LB
+
V
in the
L
2
space based on a Riemannian manifold
M
̃
with a countable discrete symmetry group
Γ
. Typically,
M
̃
is the universal covering space of a multiply connected Riemannian manifold
M
and
Γ
is the fundamental group of
M
. On the one hand, following the basic step of the Bloch analysis, one decomposes the
L
2
space over
M
̃
into a direct integral of Hilbert spaces formed by equivariant functions on
M
̃
. The Hamiltonian
H
decomposes correspondingly, with each component
H
Λ
being defined by a quasiperiodic boundary condition. The quasiperiodic boundary conditions are in turn determined by irreducible unitary representations
Λ
of
Γ
. On the other hand, fixing a quasiperiodic boundary condition (i.e., a unitary representation
Λ
of
Γ
) one can express the corresponding propagator in terms of the propagator associated with the Hamiltonian
H
. We discuss these procedures in detail and show that in a sense they are mutually inverse.</description><identifier>ISSN: 0022-2488</identifier><identifier>EISSN: 1089-7658</identifier><identifier>DOI: 10.1063/1.2898484</identifier><identifier>CODEN: JMAPAQ</identifier><language>eng</language><publisher>Melville, NY: American Institute of Physics</publisher><subject>BOUNDARY CONDITIONS ; BOUNDARY-VALUE PROBLEMS ; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS ; Combinatorics ; Exact sciences and technology ; FUNCTIONS ; Geometry ; HAMILTONIANS ; HILBERT SPACE ; INTEGRALS ; Mathematical methods in physics ; Mathematics ; Physics ; PROPAGATOR ; QUANTUM MECHANICS ; Quantum theory ; RIEMANN SPACE ; Sciences and techniques of general use ; SYMMETRY ; SYMMETRY GROUPS ; Topological manifolds</subject><ispartof>Journal of mathematical physics, 2008-03, Vol.49 (3), p.033518-033518-16</ispartof><rights>American Institute of Physics</rights><rights>2008 American Institute of Physics</rights><rights>2008 INIST-CNRS</rights><rights>Copyright American Institute of Physics Mar 2008</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c474t-c7f3a33756f10d88b04c50df5a572ee8ecef85415f32ea37f97f66d544dcde3b3</citedby><cites>FETCH-LOGICAL-c474t-c7f3a33756f10d88b04c50df5a572ee8ecef85415f32ea37f97f66d544dcde3b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://pubs.aip.org/jmp/article-lookup/doi/10.1063/1.2898484$$EHTML$$P50$$Gscitation$$H</linktohtml><link.rule.ids>230,314,780,782,784,795,885,27924,27925,76383</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=20277660$$DView record in Pascal Francis$$Hfree_for_read</backlink><backlink>$$Uhttps://www.osti.gov/biblio/21100248$$D View this record in Osti.gov$$Hfree_for_read</backlink></links><search><creatorcontrib>KOCABOVA, P</creatorcontrib><creatorcontrib>STOVICEK, P</creatorcontrib><title>Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry</title><title>Journal of mathematical physics</title><description>We consider an invariant quantum Hamiltonian
H
=
−
Δ
LB
+
V
in the
L
2
space based on a Riemannian manifold
M
̃
with a countable discrete symmetry group
Γ
. Typically,
M
̃
is the universal covering space of a multiply connected Riemannian manifold
M
and
Γ
is the fundamental group of
M
. On the one hand, following the basic step of the Bloch analysis, one decomposes the
L
2
space over
M
̃
into a direct integral of Hilbert spaces formed by equivariant functions on
M
̃
. The Hamiltonian
H
decomposes correspondingly, with each component
H
Λ
being defined by a quasiperiodic boundary condition. The quasiperiodic boundary conditions are in turn determined by irreducible unitary representations
Λ
of
Γ
. On the other hand, fixing a quasiperiodic boundary condition (i.e., a unitary representation
Λ
of
Γ
) one can express the corresponding propagator in terms of the propagator associated with the Hamiltonian
H
. We discuss these procedures in detail and show that in a sense they are mutually inverse.</description><subject>BOUNDARY CONDITIONS</subject><subject>BOUNDARY-VALUE PROBLEMS</subject><subject>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</subject><subject>Combinatorics</subject><subject>Exact sciences and technology</subject><subject>FUNCTIONS</subject><subject>Geometry</subject><subject>HAMILTONIANS</subject><subject>HILBERT SPACE</subject><subject>INTEGRALS</subject><subject>Mathematical methods in physics</subject><subject>Mathematics</subject><subject>Physics</subject><subject>PROPAGATOR</subject><subject>QUANTUM MECHANICS</subject><subject>Quantum theory</subject><subject>RIEMANN SPACE</subject><subject>Sciences and techniques of general use</subject><subject>SYMMETRY</subject><subject>SYMMETRY GROUPS</subject><subject>Topological manifolds</subject><issn>0022-2488</issn><issn>1089-7658</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2008</creationdate><recordtype>article</recordtype><recordid>eNqNkU9rlDEQh0NR6Nr20G8QLB4svDV_32QvQi21CgVB7KmHkCYTm_Jusk1Sy_bTN-su6kXxNHN45mF-MwgdUnJCycjf0ROm51posYNmlOj5oEapX6AZIYwNTGi9i17VekcIpVqIGbq-gATFTvEJPP4wZXeLbbLTqsbaG4-XJS_td9tyqTgn_DXCwqYUbcK9xpAnX_FjbH0K-1hdgQa4rhYLaGW1j14GO1U42NY9dPXx_NvZp-Hyy8Xns9PLwQkl2uBU4JZzJcdAidf6hggniQ_SSsUANDgIWgoqA2dguQpzFcbRSyG888Bv-B462nhzbdFUFxu4W5dTAtcMo7RnF7pTrzdUj3T_ALWZu_xQetbaGTlSzdgaeruBXMm1FghmWeLClpWhxKwPbKjZHrizb7ZCW52dQrHJxfprgBGm1DiSzr3fcOvNbIs5_V36xzfMz28Ym7rg-L8F_4J_5PIbNEsf-DPtlKw5</recordid><startdate>20080301</startdate><enddate>20080301</enddate><creator>KOCABOVA, P</creator><creator>STOVICEK, P</creator><general>American Institute of Physics</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7U5</scope><scope>8FD</scope><scope>H8D</scope><scope>JQ2</scope><scope>L7M</scope><scope>OTOTI</scope></search><sort><creationdate>20080301</creationdate><title>Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry</title><author>KOCABOVA, P ; STOVICEK, P</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c474t-c7f3a33756f10d88b04c50df5a572ee8ecef85415f32ea37f97f66d544dcde3b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2008</creationdate><topic>BOUNDARY CONDITIONS</topic><topic>BOUNDARY-VALUE PROBLEMS</topic><topic>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</topic><topic>Combinatorics</topic><topic>Exact sciences and technology</topic><topic>FUNCTIONS</topic><topic>Geometry</topic><topic>HAMILTONIANS</topic><topic>HILBERT SPACE</topic><topic>INTEGRALS</topic><topic>Mathematical methods in physics</topic><topic>Mathematics</topic><topic>Physics</topic><topic>PROPAGATOR</topic><topic>QUANTUM MECHANICS</topic><topic>Quantum theory</topic><topic>RIEMANN SPACE</topic><topic>Sciences and techniques of general use</topic><topic>SYMMETRY</topic><topic>SYMMETRY GROUPS</topic><topic>Topological manifolds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>KOCABOVA, P</creatorcontrib><creatorcontrib>STOVICEK, P</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>OSTI.GOV</collection><jtitle>Journal of mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>KOCABOVA, P</au><au>STOVICEK, P</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry</atitle><jtitle>Journal of mathematical physics</jtitle><date>2008-03-01</date><risdate>2008</risdate><volume>49</volume><issue>3</issue><spage>033518</spage><epage>033518-16</epage><pages>033518-033518-16</pages><issn>0022-2488</issn><eissn>1089-7658</eissn><coden>JMAPAQ</coden><abstract>We consider an invariant quantum Hamiltonian
H
=
−
Δ
LB
+
V
in the
L
2
space based on a Riemannian manifold
M
̃
with a countable discrete symmetry group
Γ
. Typically,
M
̃
is the universal covering space of a multiply connected Riemannian manifold
M
and
Γ
is the fundamental group of
M
. On the one hand, following the basic step of the Bloch analysis, one decomposes the
L
2
space over
M
̃
into a direct integral of Hilbert spaces formed by equivariant functions on
M
̃
. The Hamiltonian
H
decomposes correspondingly, with each component
H
Λ
being defined by a quasiperiodic boundary condition. The quasiperiodic boundary conditions are in turn determined by irreducible unitary representations
Λ
of
Γ
. On the other hand, fixing a quasiperiodic boundary condition (i.e., a unitary representation
Λ
of
Γ
) one can express the corresponding propagator in terms of the propagator associated with the Hamiltonian
H
. We discuss these procedures in detail and show that in a sense they are mutually inverse.</abstract><cop>Melville, NY</cop><pub>American Institute of Physics</pub><doi>10.1063/1.2898484</doi><tpages>16</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-2488 |
| ispartof | Journal of mathematical physics, 2008-03, Vol.49 (3), p.033518-033518-16 |
| issn | 0022-2488 1089-7658 |
| language | eng |
| recordid | cdi_pascalfrancis_primary_20277660 |
| source | American Institute of Physics (AIP) Publications; American Institute of Physics:Jisc Collections:Transitional Journals Agreement 2021-23 (Reading list) |
| subjects | BOUNDARY CONDITIONS BOUNDARY-VALUE PROBLEMS CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS Combinatorics Exact sciences and technology FUNCTIONS Geometry HAMILTONIANS HILBERT SPACE INTEGRALS Mathematical methods in physics Mathematics Physics PROPAGATOR QUANTUM MECHANICS Quantum theory RIEMANN SPACE Sciences and techniques of general use SYMMETRY SYMMETRY GROUPS Topological manifolds |
| title | Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry |
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