Abstract
We show that the positive supersymmetric matrix-valued differential operator H = p x 2 + p y 2 + x 2 y 2 + xσ3 + yσ1 has no zero modes, i.e., Hψ = 0 implies ψ = 0. The result depends on a virial type argument for the corresponding supercharge. The model may be regarded as a simple relative of dimensional reductions of supersymmetric Yang–Mills theories.
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References
de Wit, B., Lüscher, M. and Nicolai, H.: The supermembrane is unstable, Nuclear Phys. B 320 (1989), 135–159.
Koubek, A.: Das Potential x 2 y 2 und seine supersymmetrische Erweiterung, diploma thesis, Universität Hamburg (1990).
Banks, T., Fischler, W., Shenker, S. H. and Susskind, L.: M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997), 5112–5128.
Simon, B.: Some quantum operators with discrete spectrum but classically continuous spectrum, Ann. Phys. 146 (1983), 209–220.
Fröhlich, J., Graf, G. M., Hasler, D, Hoppe, J. and Yau, S.-T.: Asymptotic form of zero energy wave functions in supersymmetric matrix models, Nuclear Phys B 567 (2000), 231–248.
Fröhlich, J. and Hoppe, J.: On zero-mass ground states in super-membrane matrix models, Comm. Math. Phys. 191 (1998), 613–626.
Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York, 1978.
Kato, T.: Perturbation Theory for Linear Operators, Springer, New York, 1966.
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Graf, G.M., Hasler, D. & Hoppe, J. No Zero Energy States for the Supersymmetric x 2 y 2 Potential. Letters in Mathematical Physics 60, 191–196 (2002). https://doi.org/10.1023/A:1016106107913
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DOI: https://doi.org/10.1023/A:1016106107913