The energy-critical quantum harmonic oscillator

Abstract

Abstract. We consider the energy critical nonlinear Schrödinger equation in dimensions d ≥ 3 with a harmonic oscillator potential V (x) = 1 2 |x| 2 . When the nonlinearity is defocusing, we prove global wellposedness for all initial data in the energy space Σ, consisting of all functions u 0 such that both ∇u 0 and xu 0 belong to L 2 . This result extends a theorem of , which treats the radial case. For the focusing problem, we obtain global wellposedness for all data satisfying an analogue of the usual size restriction in terms of the ground state W . The proof uses the concentration compactness variant of the induction on energy paradigm. In particular, we develop a linear profile decomposition adapted to the propagator exp[it(|x| 2 )] for bounded sequences in Σ.

Hermite Multipliers on Modulation Spaces

“…It is well known that the harmonic oscillator H = −∆ + |x| 2 appears in various applications. We refer to the recent article [5] and the reference therein for details.…”

The energy-critical NLS with inverse-square potential

“…This paper is not the first to implement the credo that to treat critical dispersive equations with broken symmetries, one must first solve the limiting problems where the symmetries are restored. Previous examples include [15,16,17,18,19,24,26,27,31,36]. This is the first model to be studied, however, which retains scale invariance, but loses space translation invariance.…”

Energy-critical NLS with potentials of quadratic growth

“…where V = V (x) is a real-valued potential The equation is defocusing or focusing if µ = 1 or µ = −1, respectively. In a recent work [13], we studied large-data global wellposedness of the Cauchy problem with the harmonic oscillator potential V (x) = 1 2 |x| 2 , for which Σ :=Ḣ 1 ∩ |x| −1 L 2 , the weighted Sobolev space with norm f 2 Σ := ∇f 2 L 2 + xf 2 L 2 < ∞, is precisely the function space associated with the conserved energy E(u(t)) = This note extends the previous results to a wider class of potentials that grow approximately quadratically. More precisely, we assume that V is smooth and satisfies ∂ α x V ∈ L ∞ for all |α| ≥ 2, (1.2)…”

“…To prove this one need only make notational changes to the discussion in [13,Section 7], and we refer the reader to there for details.…”