An optimal semiclassical bound on commutators of spectral projections with position and momentum operators
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An optimal semiclassical bound on commutators of spectral projections with position and momentum operators Søren Fournais1
· Søren Mikkelsen2
Received: 24 January 2020 / Revised: 31 July 2020 / Accepted: 27 August 2020 © Springer Nature B.V. 2020
Abstract We prove an optimal semiclassical bound on the trace norm of the following commutators [1(−∞,0] (H ), x], [1(−∞,0] (H ), −i∇] and [1(−∞,0] (H ), eit,x ], where H is a Schrödinger operator with a semiclassical parameter , x is the position operator, − i∇ is the momentum operator, and t in Rd is a parameter. These bounds are in the non-interacting setting the ones introduced as an assumption by N. Benedikter, M. Porta and B. Schlein in a study of the mean-field evolution of a fermionic system. Keywords Optimal semiclassics · Weyl law · Commutator estimates Mathematics Subject Classification 81Q20
1 Introduction and main result We consider a Schrödinger operator H = −2 + V acting in L 2 (Rd ) with d ≥ 2. Here is the Laplacian acting in L 2 (Rd ) and V is a real valued function. We will be interested in the trace norms of commutators: [1(−∞,0] (H ), x j ]1 ,
[1(−∞,0] (H ), Q j ]1
and
[1(−∞,0] (H ), eit,x ]1 ,
where Q j = −i∂x j , x j is the position operator for j ∈ {1, . . . , d}, and t is a parameter in Rd . Moreover 1 A denotes the characteristic function of a set A and ·1 denotes the trace norm. The main theorem will be the bound for the first two commutators and the bound on the last will follow as a corollary.
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Søren Fournais [email protected] Søren Mikkelsen [email protected]
1
Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark
2
School of Mathematics, University of Bristol, Bristol BS8 1UG, UK
123
S. Fournais, S. Mikkelsen
Let us specify the assumptions on the function V for which we study the operator H . Assumption 1.1 Let V : Rd → R be a function for which there exists an open set ΩV ⊂ Rd and ε > 0 such that 1. V is in C ∞ (ΩV ). 2. There exists an open bounded set Ωε such that Ω ε ⊂ ΩV such that V ≥ ε for all x ∈ Ωεc . 3. V 1ΩVc is an element of L 1loc (Rd ). The assumption of smoothness in the set ΩV is needed in order to use the theory of pseudo-differential operators. The second assumption is needed to ensure that we have non-continuous spectrum in (− ∞, 0] and enable us to localise the operator. The last assumption is just to ensure that we can define the operator H by a Friedrichs extension of the associated form. Among the allowed potentials we especially have the physically interesting case of a smooth, radial, confining potential. We can now state our main theorem: Theorem 1.2 Let H = −2 + V be a Schrödinger operator acting in L 2 (Rd ) with d ≥ 2, where V satisfies Assumption 1.1 and let Q j = −i∂x j for j ∈ {1, . . . , d}. Furthermore, let 0 be a strictly positive number. Then the following bounds hold [1(−∞,0] (H ), x j ]1 ≤ C1−d and [1(−∞,0] (H ), Q j ]1 ≤ C1−d , (1.1) for all in (0, 0 ], where C is a positive constant. From Theorem
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