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Discrete vs. Continuous in the Semiclassical Limit: Bottom of the Spectrum for Periodic Potentials.

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Title: Discrete vs. Continuous in the Semiclassical Limit: Bottom of the Spectrum for Periodic Potentials.
Authors: Becker, Simon1 (AUTHOR) simon.becker@math.ethz.ch, Wittsten, Jens2 (AUTHOR) jens.wittsten@hb.se, Zworski, Maciej3 (AUTHOR) zworski@math.berkeley.edu
Source: SIAM Journal on Mathematical Analysis. 2026, Vol. 58 Issue 1, p162-181. 20p.
Subjects: Semiclassical limits, Schrödinger operator, Eigenanalysis, Numerical analysis, Quantization (Physics)
Abstract: We compare the bottom of the spectrum of discrete and continuous Schrödinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the semiclassical limit, and we provide an explicit rate of convergence. We demonstrate the optimality of our results by using Bohr–Sommerfeld quantization conditions for potentials exhibiting nondegenerate wells and by numerical experiments for more general potentials. We also investigate the dependence of the spectrum of the discrete semiclassical Schrödinger operator on the semiclassical parameter \(h\) and show that it can be discontinuous. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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  Data: Discrete vs. Continuous in the Semiclassical Limit: Bottom of the Spectrum for Periodic Potentials.
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  Data: <searchLink fieldCode="DE" term="%22Semiclassical+limits%22">Semiclassical limits</searchLink><br /><searchLink fieldCode="DE" term="%22Schrödinger+operator%22">Schrödinger operator</searchLink><br /><searchLink fieldCode="DE" term="%22Eigenanalysis%22">Eigenanalysis</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Quantization+%28Physics%29%22">Quantization (Physics)</searchLink>
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  Data: We compare the bottom of the spectrum of discrete and continuous Schrödinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the semiclassical limit, and we provide an explicit rate of convergence. We demonstrate the optimality of our results by using Bohr–Sommerfeld quantization conditions for potentials exhibiting nondegenerate wells and by numerical experiments for more general potentials. We also investigate the dependence of the spectrum of the discrete semiclassical Schrödinger operator on the semiclassical parameter \(h\) and show that it can be discontinuous. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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        Type: general
      – SubjectFull: Schrödinger operator
        Type: general
      – SubjectFull: Eigenanalysis
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      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Quantization (Physics)
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      – TitleFull: Discrete vs. Continuous in the Semiclassical Limit: Bottom of the Spectrum for Periodic Potentials.
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              Text: 2026
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