On the concentration-compactness phenomenon for the first Schrodinger eigenvalue

Kokarev, Gerasim
May 2010
Calculus of Variations & Partial Differential Equations;May2010, Vol. 38 Issue 1/2, p29
Academic Journal
We study a variational problem for the first eigenvalue ?1(V) of the Schrodinger operator (-?g + V) on closed Riemannian surfaces. More precisely, we explore concentration- compactness properties of sequences formed by ?1-extremal potentials.


Related Articles

  • On the spectral estimates for the Schrödinger operator on ℤ d, d ≽ 3. Rozenblum, Grigori; Solomyak, Michael // Journal of Mathematical Sciences;May2009, Vol. 159 Issue 2, p241 

    For the discrete Schrödinger operator we obtain sharp estimates for the number of negative eigenvalues. Bibliography: 19 titles.

  • Eigenvalue problem for Schrödinger’s equation with repulsive potential. Matsumoto, S.; Kakazu, K.; Nagamine, T. // Journal of Mathematical Physics;Jan1986, Vol. 27 Issue 1, p232 

    Schrödinger's operator — (ℏ²/2m){d²/dr² + (2/r)d/dr} + V(r) is studied, and what happens when V(r) approaches — ∞ rapidly as r → ∞ is shown. The cases in which V(r) ∼ - βr[sup δ] (β > 0, δ > 2) as r → ∞ are covered....

  • Schrödinger 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 Schrödinger operators H = -d²/dr² + V on L²([0,∞)) 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 Schrödinger operator on the half line, we show that if ϰ0={ϰ0}1∞ is a sequence of zeroes (eigenvalues and resonances) of the Jost function for some real compactly supported potential q0 and ϰ−ϰ0∈ℓε2 for some ε > 1, then ϰ is the sequence of...

  • A SPECTRAL EXPANSION FOR SCHRÖDINGER OPERATOR. Bascanbaz-Tunca, Gulen // Proyecciones - Journal of Mathematics;2006, Vol. 25 Issue 1, p63 

    In this paper we consider the Schrödinger operator L generated in L² (R+) by y″ + q (x) y = µy, x ∈ R+ := [0, ∞) subject to the boundary condition y′ (0) - hy (0) = 0, where, q is a complex valued function summable in [0, ∞ and h ≠ 0 is a complex...

  • The Spectrum of the Two-Dimensional Periodic Schrödinger 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 Schrödinger 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 Schrödinger Operators. Krüger, Helge // Journal of Statistical Physics;Dec2012, Vol. 149 Issue 6, p1096 

    I analyze the microscopic behavior of the eigenvalues of skew-shift Schrödinger operators, and show that their statistics must resemble the one of the Anderson model rather than the one of quasi-periodic Schrödinger operators.

  • Schrödinger 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 Schrödinger 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 Schrödinger operators.


Read the Article


Sign out of this library

Other Topics