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Spectral Estimates for Two-Dimensional Schrödinger Operators with Application to Quantum Layers

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Abstract

A logarithmic type Lieb-Thirring inequality for two-dimensional Schrödinger operators is established. The result is applied to prove spectral estimates on trapped modes in quantum layers.

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References

  1. Abramowitz M. and Stegun I.A. (1964). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC

    MATH  Google Scholar 

  2. Birman M.S. and Laptev A. (1996). The negative discrete spectrum of a two-dimensional Schrödinger operator. Comm. Pure and Appl. Math. XLIX: 967–997

    Article  MathSciNet  Google Scholar 

  3. Chadan K., Khuri N.N., Martin A. and Wu T.T. (2003). Bound states in one and two spatial dimensions. J. Math. Phys. 44: 406–422

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Cwikel M. (1977). Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. of Math. 106: 93–100

    Article  MathSciNet  Google Scholar 

  5. Ekholm, T., Frank, R.L.: Lieb-Thirring inequalities on the half-line with critical exponent.http://arxiv.org/list/math.SP/0611247, 2006

  6. Exner, P., Weidl, T.: Lieb-Thirring inequalities on trappedmodes in quantum wires. XIIIth International Congress on Mathematical Physics (London, 2000), Boston, MA: Int. Press, 2001, pp. 437–443

  7. Laptev, A., Netrusov, Y.: On the negative eigenvalues of aclass of Schrödinger operators, In: Diff. operators andspectral theory. Am. Math. Soc. Transl. 2, 189, 173–186(1999)

  8. Laptev A. (2000). The negative spectrum of a class of two-dimensional Schrödinger operators with spherically symmetric potentials. Func. Anal. Appl. 34: 305–307

    Article  MATH  MathSciNet  Google Scholar 

  9. Lieb E. (1976). Bound states of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc. 82: 751–753

    Article  MATH  MathSciNet  Google Scholar 

  10. Lieb, E., Thirring, W.: Inequalities for the moments of theeigenvalues of the Schrödinger Hamiltonian and their relation toSobolev inequalities. tudies in Mathematical Physics.Princeton, NJ: Princeton University Press, 1976, pp. 269–303

  11. Newton R.G. (1983). Bounds on the number of bound states for the Schrödinger equation in one and two dimensions. J. Op. Theory 10: 119–125

    MATH  Google Scholar 

  12. Rosenblum, G.V.: Distribution of the discrete spectrum ofsingular differential operators (in Russian), Izv. Vassh.Ucheb. Zaved. Matematika 1, 75–86 (1976); English transl.Soviet Math. 20, 63–71 (1976)

  13. Simon B. (1976). The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Ann. of Phys. 97: 279–288

    Article  MATH  ADS  Google Scholar 

  14. Solomyak M. (1994). Piecewise-polynomial approximation of functions from H l((0,1)d),  2l = d and applications to the spectral theory of the Schrödinger operator. Israel J. of Math. 86: 253–275

    MATH  MathSciNet  Google Scholar 

  15. Stoiciu M. (2003). An estimate for the number of bound states of the Schrödinger operator in two dimensions. Proc. of AMS 132: 1143–1151

    Article  MathSciNet  Google Scholar 

  16. Weidl T. (1996). On the Lieb-Thirring constants L γ,1 for γ ≥ 1/2. Commun. Math. Phys. 178(1): 135–146

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Analysis, Dynamics and Modeling, Universität Stuttgart, PF-80 11 40, D-70569, Stuttgart, Germany

    Hynek Kovařík, Semjon Vugalter & Timo Weidl

Authors
  1. Hynek Kovařík
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  2. Semjon Vugalter
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  3. Timo Weidl
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Correspondence to Hynek Kovařík.

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Communicated by B. Simon

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Kovařík, H., Vugalter, S. & Weidl, T. Spectral Estimates for Two-Dimensional Schrödinger Operators with Application to Quantum Layers. Commun. Math. Phys. 275, 827–838 (2007). https://doi.org/10.1007/s00220-007-0318-z

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  • DOI: https://doi.org/10.1007/s00220-007-0318-z

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