Spectral Asymptotics for Schrödinger Operators with Periodic Point Interactions
Spectrum of the second-order differential operator with periodic point interactions in L2(R) is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms...
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| Published in | Journal of mathematical analysis and applications Vol. 266; no. 1; pp. 127 - 148 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.02.2002
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| Abstract | Spectrum of the second-order differential operator with periodic point interactions in L2(R) is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely. It is proven that the spectrum of the operators with anomalous spectral asymptotics (when the ratio between the lengths of the bands and gaps tends to zero at infinity) can be approximated by standard periodic “weighted” operators with step-wise density functions. It is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal (generalized) resolvent of the periodic “Schrödinger operator” with certain energy-dependent boundary conditions. The operator acting in an extended Hilbert space such that its resolvent restricted to L2(R) coincides with the formal resolvent is constructed explicitly. |
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| AbstractList | Spectrum of the second-order differential operator with periodic point interactions in L2(R) is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely. It is proven that the spectrum of the operators with anomalous spectral asymptotics (when the ratio between the lengths of the bands and gaps tends to zero at infinity) can be approximated by standard periodic “weighted” operators with step-wise density functions. It is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal (generalized) resolvent of the periodic “Schrödinger operator” with certain energy-dependent boundary conditions. The operator acting in an extended Hilbert space such that its resolvent restricted to L2(R) coincides with the formal resolvent is constructed explicitly. |
| Author | Kurasov, P. Larson, J. |
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| Cites_doi | 10.1088/0305-4470/26/2/025 10.1006/jmaa.1996.0256 10.1006/jfan.1999.3479 10.1006/jmaa.1998.5922 10.1002/andp.19874990503 10.1006/jmaa.1997.5497 10.1098/rspa.1931.0019 10.1090/S0002-9939-98-04291-9 10.1103/PhysRevLett.72.896 10.1006/jfan.1999.3406 10.1016/0034-4877(86)90045-5 10.1007/BF01597402 10.1016/0022-247X(88)90003-0 10.1007/BF00398401 10.1006/jfan.1993.1006 10.1088/0305-4470/20/15/030 |
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| Keywords | point interactions self-adjoint extensions spectral asymptotics |
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| References_xml | – year: 1988 ident: RF3 publication-title: Solvable Models in Quantum Mechanics contributor: fullname: Holden – volume: 26 start-page: 427 year: 1993 end-page: 432 ident: RF7 article-title: Four-parameter point-interaction in 1D quantum systems publication-title: J. Phys. A contributor: fullname: Carreau – volume: 74 start-page: 82 year: 1988 end-page: 93 ident: RF21 article-title: An electron in a homogeneous crystal of point-like atoms with internal structure. II publication-title: Teoret. Mat. Fiz. contributor: fullname: Pavlov – volume: 362 start-page: 28 year: 1985 end-page: 50 ident: RF15 article-title: One-dimensional Schrödinger operators with interactions singular on a discrete set publication-title: J. Reine Angew. Math. contributor: fullname: Kirsch – volume: 108 start-page: 203 year: 1999 end-page: 208 ident: RF8 article-title: , Mathematical results in quantum mechanics (Prague, 1998) publication-title: Oper. Theory Adv. 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Physik doi: 10.1002/andp.19874990503 contributor: fullname: Šeba – ident: 10.1006/jmaa.2001.7716_RF11 – volume: 212 start-page: 263 year: 1997 ident: 10.1006/jmaa.2001.7716_RF17 article-title: Some examples in one-dimensional “geometric” scattering on manifolds publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.1997.5497 contributor: fullname: Kiselev – year: 1988 ident: 10.1006/jmaa.2001.7716_RF10 contributor: fullname: Demkov – volume: 130 start-page: 499 year: 1931 ident: 10.1006/jmaa.2001.7716_RF20 article-title: Quantum mechanics of electrons in crystal lattices publication-title: Proc. Roy. Soc. London doi: 10.1098/rspa.1931.0019 contributor: fullname: Kronig – volume: 126 start-page: 1673 year: 1998 ident: 10.1006/jmaa.2001.7716_RF6 article-title: Finite rank singular perturbations and distributions with discontinuous test functions publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-98-04291-9 contributor: fullname: Boman – volume: 72 start-page: 896 year: 1994 ident: 10.1006/jmaa.2001.7716_RF5 article-title: Periodic Schrödinger operators with large gaps and Wannier–Stark ladders publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.72.896 contributor: fullname: Avron – volume: 165 start-page: 150 year: 1999 ident: 10.1006/jmaa.2001.7716_RF25 article-title: Common eigenvalue problem and periodic Schrödinger operators publication-title: J. Funct. Anal. doi: 10.1006/jfan.1999.3406 contributor: fullname: Mikhailets – volume: 24 start-page: 111 year: 1986 ident: 10.1006/jmaa.2001.7716_RF28 article-title: Some remarks on the δ′-interaction in one dimension publication-title: Rep. Math. Phys. doi: 10.1016/0034-4877(86)90045-5 contributor: fullname: Šeba – volume: 74 start-page: 82 year: 1988 ident: 10.1006/jmaa.2001.7716_RF21 article-title: An electron in a homogeneous crystal of point-like atoms with internal structure. II publication-title: Teoret. Mat. Fiz. contributor: fullname: Kurasov – volume: 36 start-page: 667 year: 1986 ident: 10.1006/jmaa.2001.7716_RF29 article-title: The generalized point interaction in one dimension publication-title: Czechoslovak J. Phys. doi: 10.1007/BF01597402 contributor: fullname: Šeba – volume: I, II year: 1981 ident: 10.1006/jmaa.2001.7716_RF1 contributor: fullname: Akhiezer – volume: 134 start-page: 9 year: 1988 ident: 10.1006/jmaa.2001.7716_RF14 article-title: On energy gaps in a new type of analytically solvable model in quantum mechanics publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(88)90003-0 contributor: fullname: Gesztesy – volume: 21 start-page: 320 year: 1957 ident: 10.1006/jmaa.2001.7716_RF19 article-title: On the characteristic function A(λ) of a linear canonical system of differential equations of second order with periodic coefficients publication-title: Prikl. Mat. Mehk. contributor: fullname: Kreǐn – volume: 42 start-page: 99 year: 1987 ident: 10.1006/jmaa.2001.7716_RF26 article-title: The theory of extensions, and explicitly solvable models publication-title: Uspekhi Mat. Nauk contributor: fullname: Pavlov – volume: 108 start-page: 203 year: 1999 ident: 10.1006/jmaa.2001.7716_RF8 article-title: Some aspects of generalized contact interaction in one-dimensional quantum mechanics, Mathematical results in quantum mechanics (Prague, 1998) publication-title: Oper. Theory Adv. Appl. contributor: fullname: Cheon – volume: 25 start-page: 287 year: 1992 ident: 10.1006/jmaa.2001.7716_RF22 article-title: Zero-range potentials with internal structures and the inverse scattering problem publication-title: Lett. Math. Phys. doi: 10.1007/BF00398401 contributor: fullname: Kurasov – volume: 111 start-page: 97 year: 1993 ident: 10.1006/jmaa.2001.7716_RF9 article-title: A new class of point interactions in one dimension publication-title: J. Funct. Anal. doi: 10.1006/jfan.1993.1006 contributor: fullname: Chernoff – volume: 20 start-page: 5157 year: 1987 ident: 10.1006/jmaa.2001.7716_RF13 article-title: A new class of solvable models in quantum mechanics describing point interactions on the line publication-title: J. Phys. A doi: 10.1088/0305-4470/20/15/030 contributor: fullname: Gesztesy – volume: 12 start-page: 101 year: 1984 ident: 10.1006/jmaa.2001.7716_RF2 article-title: On point interactions in one dimension publication-title: J. Operator Theory contributor: fullname: Albeverio – ident: 10.1006/jmaa.2001.7716_RF12 – volume: 136 start-page: 163 year: 1988 ident: 10.1006/jmaa.2001.7716_RF27 article-title: Boundary conditions on thin manifolds and the semiboundedness of the three-body Schrödinger operator with point potential publication-title: Mat. Sb. (N.S.) contributor: fullname: Pavlov – ident: 10.1006/jmaa.2001.7716_RF31 |
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| Snippet | Spectrum of the second-order differential operator with periodic point interactions in L2(R) is investigated. Classes of unitary equivalent operators of this... |
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| SubjectTerms | Exact sciences and technology Matematik Mathematics Natural Sciences Naturvetenskap point interactions Sciences and techniques of general use self-adjoint extensions spectral asymptotics |
| Title | Spectral Asymptotics for Schrödinger Operators with Periodic Point Interactions |
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