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Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators

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Abstract

In this paper, we consider the Schrödinger equation, \(Hu = - {u^"} + \left({V\left(x \right) + {V_0}\left(x \right)} \right)u = Eu,\) where V0(x) is 1-periodic and V(x) is a decaying perturbation. By Floquet theory, the spectrum of H0 = − ∇2 + V0 is purely absolutely continuous and consists of a union of closed intervals (often referred to as spectral bands). Given any finite set of points \(\left\{{{E_j}} \right\}_{j = 1}^N\) in any spectral band of H0 obeying a mild non-resonance condition, we construct smooth functions \(V\left(x \right) = {{O\left(1 \right)} \over {1 + \left| x \right|}}\) such that H = H0 + V has eigenvalues \(\left\{{{E_j}} \right\}_{j = 1}^N\). Given any countable set of points {Ej} in any spectral band of Ho obeying the same non-resonance condition, and any function h(x) > 0 going to infinity arbitrarily slowly, we construct smooth functions \(\left| {V\left(x \right)} \right| \le {{h\left(x \right)} \over {1 + \left| x \right|}}\) such that H = H0 + V has eigenvalues {Ej}. On the other hand, we show that there is no eigenvalue of H = H0 + V embedded in the spectral bands if \(V\left(x \right) = {{o\left(1 \right)} \over {1 + \left| x \right|}}\) as x goes to infinity. We prove also an analogous result for Jacobi operators.

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Acknowledgments

W. L. was supported by NSF DMS-1700314/2015683, DMS-2000345, the AMS-Simons Travel Grant 2016–2018 and the Southeastern Conference (SEC) Faculty Travel Grant 2020–2021. D. O. was supported by a grant from the Fundamental Research Grant Scheme of the Malaysian Ministry of Education (Grant No. FRGS/1/2018/STG06/XMU/02/1) and two Xiamen University Malaysia Research Funds (Grant No. XMUMRF/2018-C1/IMAT/0001 and XMUMRF/2020-C5/IMAT/0011).

The authors also wish to thank Jake Fillman, Svetlana Jitomirskaya, Milivoje Lukic, Christian Remling, and the anonymous referee for helpful conversations and comments.

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  1. Wencai Liu

    Present address: Current address Department of Mathematics, Texas A M University, College Station, TX, 77843-3368, USA

Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697-3875, USA

    Wencai Liu

  2. Department of Mathematics, Xiamen University Malaysia Jalan Sunsuria, Bandar Sunsuria, Sepang, 43900, Selangor, Malaysia

    Darren C. Ong

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  1. Wencai Liu
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  2. Darren C. Ong
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Correspondence to Darren C. Ong.

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Liu, W., Ong, D.C. Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators. JAMA 141, 625–661 (2020). https://doi.org/10.1007/s11854-020-0111-x

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  • DOI: https://doi.org/10.1007/s11854-020-0111-x

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