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Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields

Mitya Boyarchenko, Sergei Levendorski (2006)

Annales de l’institut Fourier

We present a pair of conjectural formulas that compute the leading term of the spectral asymptotics of a Schrödinger operator on L 2 ( ℝ n ) with quasi-homogeneous polynomial magnetic and electric fields. The construction is based on the orbit method due to Kirillov. It makes sense for any nilpotent Lie algebra and is related to the geometry of coadjoint orbits, as well as to the growth properties of certain “algebraic integrals,” studied by Nilsson. By using the direct variational method, we prove that the...

Bilinear operators associated with Schrödinger operators

Chin-Cheng Lin, Ying-Chieh Lin, Heping Liu, Yu Liu (2011)

Studia Mathematica

Let L = -Δ + V be a Schrödinger operator in ℝ d and H ¹ L ( ℝ d ) be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by T ± ( f , g ) ( x ) = ( T ₁ f ) ( x ) ( T ₂ g ) ( x ) ± ( T ₂ f ) ( x ) ( T ₁ g ) ( x ) , where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T¯ is bounded from L p ( ℝ d ) × L q ( ℝ d ) to H ¹ L ( ℝ d ) for 1 < p,q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.

Bilinear virial identities and applications

Fabrice Planchon, Luis Vega (2009)

Annales scientifiques de l'École Normale Supérieure

We prove bilinear virial identities for the nonlinear Schrödinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries.

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