# On the concentration-compactness phenomenon for the first Schrodinger eigenvalue

## Related Articles

- On the spectral estimates for the SchrÃ¶dinger operator on â„¤ d, d â‰½ 3. Rozenblum, Grigori; Solomyak, Michael // Journal of Mathematical Sciences;May2009, Vol. 159 Issue 2, p241
For the discrete SchrÃ¶dinger operator we obtain sharp estimates for the number of negative eigenvalues. Bibliography: 19 titles.

- Eigenvalue problem for SchrÃ¶dingerâ€™s equation with repulsive potential. Matsumoto, S.; Kakazu, K.; Nagamine, T. // Journal of Mathematical Physics;Jan1986, Vol. 27 Issue 1, p232
SchrÃ¶dinger's operator â€” (ℏÂ²/2m){dÂ²/drÂ² + (2/r)d/dr} + V(r) is studied, and what happens when V(r) approaches â€” âˆž rapidly as r â†’ âˆž is shown. The cases in which V(r) âˆ¼ - Î²r[sup Î´] (Î² > 0, Î´ > 2) as r â†’ âˆž are covered....

- SchrÃ¶dinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds. Jensen, Arne; Nenciu, Gheorghe // Proceedings of the Indian Academy of Sciences: Mathematical Scie;Nov2006, Vol. 116 Issue 4, p375
We consider SchrÃ¶dinger operators H = -dÂ²/drÂ² + V on LÂ²([0,âˆž)) with the Dirichlet boundary condition. The potential V may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of H is classified, and asymptotic expansions of the resolvent...

- Stability for inverse resonance problem. Korotyaev, Evgeni // IMRN: International Mathematics Research Notices;2004, Vol. 2004 Issue 73, p3927
For the SchrÃ¶dinger operator on the half line, we show that if Ï°0={Ï°0}1âˆž is a sequence of zeroes (eigenvalues and resonances) of the Jost function for some real compactly supported potential q0 and Ï°âˆ’Ï°0âˆˆâ„“Îµ2 for some Îµ > 1, then Ï° is the sequence of...

- A SPECTRAL EXPANSION FOR SCHRÃ–DINGER OPERATOR. Bascanbaz-Tunca, Gulen // Proyecciones - Journal of Mathematics;2006, Vol. 25 Issue 1, p63
In this paper we consider the SchrÃ¶dinger operator L generated in LÂ² (R+) by yâ€³ + q (x) y = Âµy, x âˆˆ R+ := [0, âˆž) subject to the boundary condition yâ€² (0) - hy (0) = 0, where, q is a complex valued function summable in [0, âˆž and h â‰ 0 is a complex...

- The Spectrum of the Two-Dimensional Periodic SchrÃ¶dinger Operator. Danilov, L. I. // Theoretical & Mathematical Physics;Mar2003, Vol. 134 Issue 3, p392
The absence of eigenvalues (of infinite multiplicity) for the two-dimensional periodic SchrÃ¶dinger operator with a variable metric is proved. The method of proof does not use the change of variables reducing the metric to a scalar form.

- Concentration of Eigenvalues for Skew-Shift SchrÃ¶dinger Operators. Krüger, Helge // Journal of Statistical Physics;Dec2012, Vol. 149 Issue 6, p1096
I analyze the microscopic behavior of the eigenvalues of skew-shift SchrÃ¶dinger operators, and show that their statistics must resemble the one of the Anderson model rather than the one of quasi-periodic SchrÃ¶dinger operators.

- SchrÃ¶dinger Operator Levels for a Crystal Film with a Nonlocal Potential. Smetanina, M. S.; Chuburin, Yu. P. // Theoretical & Mathematical Physics;Aug2004, Vol. 140 Issue 2, p1146
For a crystal film, we consider the SchrÃ¶dinger operator defined on Bloch functions (with respect to two variables) in a cell. The potential is the sum of two small terms: a function decreasing with respect to the third variable and an operator of rank one. We prove the existence of two...

- GORDON TYPE THEOREM FOR MEASURE PERTURBATION. SEIFERT, CHRISTIAN // Electronic Journal of Differential Equations;2011, Vol. 2011, Special section p1
Generalizing the concept of Gordon potentials to measures we prove a version of Gordon's theorem for measures as potentials and show absence of eigenvalues for these one-dimensional SchrÃ¶dinger operators.