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The negative discrete spectrum of the operator (-τ)^{$l$}-α$V$in$L$_{2}($R$^{$d$}) for$d$even and 2$l$≥$d$in$L$_{2}($R$^{$d$}) for$d$even and 2$l$≥$d$

Mikhail Sh. Birman Department of Physics, St. Petersburg University Ari Laptev Department of Mathematics, Royal Institute of Technology Michael Solomyak Department of Theoretical Mathematics, The Weizmann Institute of Science

TBD mathscidoc:1701.332866

Arkiv for Matematik, 35, (1), 87-126, 1995.11
We study the asymptotic behaviour of$N$(α)—the number of negative eigenvalues of the operator (-τ)^{$l$}-α$V$in$L$_{2}($R$^{$d$}) for an even$d$and$2l≥d$. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases where$N$(α) is of the Weyl-type,$N$(α) is of the Weyl-order but not the Weyl-type coefficient and finally where$N$(α)=$O$(α^{q}) with$q>d/2l$.
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@inproceedings{mikhail1995the,
  title={The negative discrete spectrum of the operator (-τ)^{$l$}-α$V$in$L$_{2}($R$^{$d$}) for$d$even and 2$l$≥$d$in$L$_{2}($R$^{$d$}) for$d$even and 2$l$≥$d$},
  author={Mikhail Sh. Birman, Ari Laptev, and Michael Solomyak},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203549828765675},
  booktitle={Arkiv for Matematik},
  volume={35},
  number={1},
  pages={87-126},
  year={1995},
}
Mikhail Sh. Birman, Ari Laptev, and Michael Solomyak. The negative discrete spectrum of the operator (-τ)^{$l$}-α$V$in$L$_{2}($R$^{$d$}) for$d$even and 2$l$≥$d$in$L$_{2}($R$^{$d$}) for$d$even and 2$l$≥$d$. 1995. Vol. 35. In Arkiv for Matematik. pp.87-126. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203549828765675.
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