Improvement of Weyl’s Inequality
Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, an...
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| Autor*in: |
Marmorino, M.G. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2005 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, Inc. 2005 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of mathematical chemistry - Kluwer Academic Publishers-Plenum Publishers, 1987, 38(2005), 4 vom: Nov., Seite 415-424 |
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| Übergeordnetes Werk: |
volume:38 ; year:2005 ; number:4 ; month:11 ; pages:415-424 |
| Links: |
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| DOI / URN: |
10.1007/s10910-004-6893-8 |
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OLC2060407354 |
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| 520 | |a Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. | ||
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10.1007/s10910-004-6893-8 doi (DE-627)OLC2060407354 (DE-He213)s10910-004-6893-8-p DE-627 ger DE-627 rakwb eng 510 540 VZ Marmorino, M.G. verfasserin aut Improvement of Weyl’s Inequality 2005 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, Inc. 2005 Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. Weyl lower bound inequality energy Enthalten in Journal of mathematical chemistry Kluwer Academic Publishers-Plenum Publishers, 1987 38(2005), 4 vom: Nov., Seite 415-424 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:38 year:2005 number:4 month:11 pages:415-424 https://doi.org/10.1007/s10910-004-6893-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2409 GBV_ILN_4012 AR 38 2005 4 11 415-424 |
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10.1007/s10910-004-6893-8 doi (DE-627)OLC2060407354 (DE-He213)s10910-004-6893-8-p DE-627 ger DE-627 rakwb eng 510 540 VZ Marmorino, M.G. verfasserin aut Improvement of Weyl’s Inequality 2005 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, Inc. 2005 Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. Weyl lower bound inequality energy Enthalten in Journal of mathematical chemistry Kluwer Academic Publishers-Plenum Publishers, 1987 38(2005), 4 vom: Nov., Seite 415-424 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:38 year:2005 number:4 month:11 pages:415-424 https://doi.org/10.1007/s10910-004-6893-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2409 GBV_ILN_4012 AR 38 2005 4 11 415-424 |
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10.1007/s10910-004-6893-8 doi (DE-627)OLC2060407354 (DE-He213)s10910-004-6893-8-p DE-627 ger DE-627 rakwb eng 510 540 VZ Marmorino, M.G. verfasserin aut Improvement of Weyl’s Inequality 2005 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, Inc. 2005 Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. Weyl lower bound inequality energy Enthalten in Journal of mathematical chemistry Kluwer Academic Publishers-Plenum Publishers, 1987 38(2005), 4 vom: Nov., Seite 415-424 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:38 year:2005 number:4 month:11 pages:415-424 https://doi.org/10.1007/s10910-004-6893-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2409 GBV_ILN_4012 AR 38 2005 4 11 415-424 |
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10.1007/s10910-004-6893-8 doi (DE-627)OLC2060407354 (DE-He213)s10910-004-6893-8-p DE-627 ger DE-627 rakwb eng 510 540 VZ Marmorino, M.G. verfasserin aut Improvement of Weyl’s Inequality 2005 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, Inc. 2005 Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. Weyl lower bound inequality energy Enthalten in Journal of mathematical chemistry Kluwer Academic Publishers-Plenum Publishers, 1987 38(2005), 4 vom: Nov., Seite 415-424 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:38 year:2005 number:4 month:11 pages:415-424 https://doi.org/10.1007/s10910-004-6893-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2409 GBV_ILN_4012 AR 38 2005 4 11 415-424 |
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10.1007/s10910-004-6893-8 doi (DE-627)OLC2060407354 (DE-He213)s10910-004-6893-8-p DE-627 ger DE-627 rakwb eng 510 540 VZ Marmorino, M.G. verfasserin aut Improvement of Weyl’s Inequality 2005 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, Inc. 2005 Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. Weyl lower bound inequality energy Enthalten in Journal of mathematical chemistry Kluwer Academic Publishers-Plenum Publishers, 1987 38(2005), 4 vom: Nov., Seite 415-424 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:38 year:2005 number:4 month:11 pages:415-424 https://doi.org/10.1007/s10910-004-6893-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2409 GBV_ILN_4012 AR 38 2005 4 11 415-424 |
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Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. © Springer Science+Business Media, Inc. 2005 |
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Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. © Springer Science+Business Media, Inc. 2005 |
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Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized. © Springer Science+Business Media, Inc. 2005 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">OLC2060407354</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230508112344.0</controlfield><controlfield tag="007">tu</controlfield><controlfield tag="008">200819s2005 xx ||||| 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10910-004-6893-8</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)OLC2060407354</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-He213)s10910-004-6893-8-p</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="a">540</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Marmorino, M.G.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Improvement of Weyl’s Inequality</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2005</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Springer Science+Business Media, Inc. 2005</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Weyl</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">lower bound</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">inequality</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">energy</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Journal of mathematical chemistry</subfield><subfield code="d">Kluwer Academic Publishers-Plenum Publishers, 1987</subfield><subfield code="g">38(2005), 4 vom: Nov., Seite 415-424</subfield><subfield code="w">(DE-627)129246441</subfield><subfield code="w">(DE-600)59132-4</subfield><subfield code="w">(DE-576)27906036X</subfield><subfield code="x">0259-9791</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:38</subfield><subfield code="g">year:2005</subfield><subfield code="g">number:4</subfield><subfield code="g">month:11</subfield><subfield code="g">pages:415-424</subfield></datafield><datafield tag="856" ind1="4" ind2="1"><subfield code="u">https://doi.org/10.1007/s10910-004-6893-8</subfield><subfield code="z">lizenzpflichtig</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_OLC</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-CHE</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-MAT</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-PHA</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-DE-84</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OPC-MAT</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_11</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_40</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_70</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_2409</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4012</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">38</subfield><subfield code="j">2005</subfield><subfield code="e">4</subfield><subfield code="c">11</subfield><subfield code="h">415-424</subfield></datafield></record></collection>
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