Lieb–Thirring inequalities on the half-line with critical exponent

Creators: Ekholm, Tomas; Frank, Rupert

Abstract

We consider the operator -d^2/dr^2 - V in L_2(R_+) with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound tr (-d^2)/dr^2 -V)^y_- ≤ Cy,ɑʃ_(R+) (v(r)-1/_(4r)^2)^(y+(1+ɑ)/2_(r^ɑ dr) for any ɑ Є [0,1] and y ≥ (1 - ɑ)/2. This includes a Lieb–Thirring inequality in the critical endpoint case.

Additional Information

© 2008 European Mathematical Society. Received December 6, 2006 and in revised form January 22, 2007. This work was partially supported by FCT Portugal, post-doc grant SFRH/BPD/23820/2005, (T.E.), by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT) (R.F.), as well as through the ESF Scientific Programme in Spectral Theory and Partial Differential Equations (SPECT). The authors are grateful to the American Institute of Mathematics for the invitation to the workshop Low Eigenvalues of Laplace and Schrödinger Operators where this problem was brought up. R.F. would like to thank E.H. Lieb and R. Seiringer for the hospitality at Princeton University and for helpful discussions. Remarks by A. Hansson and A. Laptev are gratefully acknowledged.

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