On the Magnetic Pekar Functional and the Existence of Bipolarons

Abstract

First, this paper proves the existence of a minimizer for the Pekar
functional including a constant magnetic field and possibly some additional
local fields that are energy reducing. Second, the existence of the
aforementioned minimizer is used to establish the binding of polarons in the
model of Pekar-Tomasevich including external fields.

Full text

Page 1

arXiv:1111.1624v1 [math-ph] 7 Nov 2011
On the Magnetic Pekar Functional and the Existence of
Bipolarons
M. Griesemer, F. Hantsch and D. Wellig
Universit¨ at Stuttgart, Fachbereich Mathematik
70550 Stuttgart, Germany
Abstract
First, this paper proves the existence of a minimizer for the Pekar functional
including a constant magnetic field and possibly some additional local fields that
are energy reducing. Second, the existence of the aforementioned minimizer is used
to establish the binding of polarons in the model of Pekar-Tomasevich including
external fields.
1 Introduction
The Pekar functional including external electric and magnetic potentials is given by
??
|DAϕ|2+ V |ϕ|2?
dx −
?
|ϕ(x)|2|ϕ(y)|2
|x − y|
dxdy(1)
where DA:= −i∇ + A and ϕ ∈ H1
scalar and vector potentials associated with the external electric and magnetic fields
−∇V and curlA. Since ϕ denotes the wave function of a quantum particle (electron)
we impose the constraint that
?
The functional (1) arises e.g. in the study of the ground state energy of the polaron
[5, 11] and in the analysis of a self-gravitating quantum particle [14]. Depending on
the context, the Euler-Lagrange equation associated with (1), (2) is called Choquard
equation or Schr¨ odinger-Newton equation. The time-dependent version of the Euler-
Lagrange equation describes the dynamics of interacting many-boson systems in the
mean field limit [6]. We are interested in the question whether the functional (1)
subject to (2) has a minimizer, and we shall give a positive answer for a class of
potentials including all previously considered cases. Second, we shall use the existence
of a minimizer to prove binding of polarons in the model of Pekar and Tomasevich with
an external magnetic field.
In the case A = 0 and V = 0 it is a well-known result, due to Lieb [10], that the
Pekar functional (1), (2) possesses a unique, rotationally symmetric minimizer, which
moreover can be chosen pointwise positive. For the existence part a second proof has
A(R3). The letters V and A denote (real-valued)
|ϕ|2dx = 1. (2)
1

Page 2

been given by Lions as an application of his concentration compactness principle [12].
Lions also considered the case of non-vanishing V ≤ 0. In this paper we establish
existence of a minimizer for constant magnetic fields and vanishing V , as well as for
certain local perturbations of this field configuration. For example, if curlA is constant,
V (x) = −|x|−1, then (1) has a minimizer as well. More generally, the Pekar functional
has a minimizer for any local perturbation of the fields A(x) = (B ∧ x)/2, V = 0 that
leads to a reduction of the energy. We give examples of non-linear vector potentials for
which this trapping assumption is satisfied.
In the second part of the paper we address the question of binding of two polarons
subject to given electromagnetic fields A,V in the model of Pekar and Tomasevich.
For A = 0,V = 0 this question has been studied by Miyao, Spohn and by Lewin and
answered in the affirmative for admissible values of the electron-electron repulsion close
to the critical one [13, 9]. In fact, Lewin proved the binding of any given number of
polarons by establishing a Van der Waals type interaction between two polaron clusters.
This method makes use of a spherical invariance which is broken by the presence of
a magnetic field. We here describe a much softer argument to explain the binding of
two polarons that works for any given A,V and requires nothing but the existence of
a minimizer for (1), (2). This argument is based on the observation that the product
ψ ⊗ ψ of two copies of a minimizer ψ of (1), (2) does not solve the Euler-Lagrange
equation of the Pekar-Tomasevich functional and hence cannot be a minimizer of this
functional. This argument does not depend on the presence of external fields and
seems to be novel. It can be extended to multipolaron systems, and this will be done
in subsequent work.
In a companion paper we derive estimates on the ground state energy of the Fr¨ ohlich
polaron subject to electromagnetic fields A,V in the limit of strong electron-phonon
coupling, α → ∞. For fields A,V that are suitably rescaled with α, it turns out that
this ground state energy is correctly given by α2times the minimum of (1), (2) up
to errors of smaller order. In view of the results of the present paper the binding of
Fr¨ ohlich polarons subject to strong external fields and large α will follow. In the case
A = 0, V = 0 a similar result has previously been established by Miyao and Spohn on
the bases of [5, 11, 10]. In the physical literature the existence of Fr¨ ohlich bipolarons
in the presence of magnetic fields is studied e.g. in [2].
Solutions to the Choquard equation with magnetic field have very recently been
studied in [4, 3]. In [3] infinitely many solutions are found whose symmetry corresponds
to the symmetry of A. Constant magnetic fields seem to be excluded, however. The
constrained minimization problem (1), (2) with non-vanishing magnetic field does not
seem to have been studied yet. Nevertheless, as our methods are not new, we would
not be surprised if some of our results on the existence of a minimizer for (1),(2) with
A ?= 0 could be inferred from existing results in the literature.
Section 2 is devoted to the problem of existence of minimizers for (1), (2); in Sec-
tion 3 the binding of polarons is established. There is an appendix where technical
auxiliaries are collected.
2

Page 3

Acknowledgments. Fabian Hantsch is supported by the Studienstiftung des Deutschen
Volkes, David Wellig has been supported by a stipend of the Landesgraduiertenf¨ orderung
of Baden-W¨ urttemberg.
2 The Magnetic Pekar Functional
This section contains all our results on the existence of a minimizer for the Pekar
functional, as well as the main parts of the proofs. Some technical auxiliaries have
been deferred to the appendix.
The minimal assumptions that we shall make throughout the paper, are that A,V
are real-valued with Ak,V ∈ L2
to −∆, V ≪ −∆. This means that for every ε > 0 there exists Cε∈ R such that
loc(R3) and that V is infinitesimally small with respect
?V ϕ? ≤ ε?∆ϕ? + Cε?ϕ?
for all ϕ ∈ C∞
V that admits a decomposition V = V1+ V2with V1∈ L2(R3) and V2∈ L∞(R3) is
infinitesimally small w.r.t. −∆.
We define DA:= −i∇ + A and
0(R3). Here and henceforth ? · ? denotes an L2-norm. Every potential
H1
A(R3) =?ϕ ∈ L2(R3) | DAϕ ∈ L2(R3;C3)?.
A:= ?DAϕ?2+ ?ϕ?2this space is complete and C∞
is dense. This means that the quadratic form ?DAϕ,DAϕ? is closed on H1
that C∞
denoted D2
A.
We define the Pekar functional EA,V(ϕ) by the expression (1). For the domain of
this functional we take?ϕ ∈ H1
hard to see, using the Hardy and the diamagnetic inequalities, that EA,Vis bounded
below and that every minimizing sequence is bounded in H1
set
CA,V(λ) := inf?EA,V(ϕ)??ϕ ∈ H1
establish a few general properties of the Pekar functional (1) and its lower bounds (3).
To this end, and for use throughout the paper, we introduce the following notation:
Equipped with the norm ?ϕ?2
0(R3)
A(R3) and
0(R3) is a core. The unique self-adjoint operator associated with this form is
A(R3)|?|ϕ|2dx = 1?unless explicitly stated otherwise.
In particular, by a minimizer of EA,Vwe mean a vector ϕ from this domain. It is not
A(R3), see Lemma A.2. We
A(R3), ?ϕ?2= λ?
(3)
where λ > 0. As a preparation for the proofs of the theorems of this section we first
Vϕ(x) :=
?
|ϕ(y)|2
|x − y|dy,D(ρ) :=
?
ρ(x)ρ(y)
|x − y|
dxdy,
where usually ρ = ρϕ:= |ϕ|2.
Lemma 2.1. Under the above minimal assumptions on V,A, the following is true:
(i) If EA,V(ϕn) → CA,V(1) and ϕn→ ϕ as n → ∞, then EA,V(ϕ) = CA,V(1) and
ϕn→ ϕ in H1
A(R3).
3

Page 4

(ii) If EA,V(ϕ) = CA,V(1), then ϕ is an eigenvector of D2
the lowest eigenvalue of this operator, which is CA,V(1) − D(ρϕ).
A+V −2Vϕassociated with
(iii) The map λ ?→ CA,V(λ) is continuous.
(iv) If liminfn→∞D(ρϕn) > 0 for every (normalized) minimizing sequence of EA,V,
then for all λ ∈ (0,1),
CA,V(1) < CA,V(λ) + CA,V(1 − λ).
Proof. (i) Since (ϕn) is bounded in H1
H1
follows that EA,V(ϕ) = CA,V(1) = limn→∞EA,V(ϕn) and, using Lemma A.2 again,
that ?DAϕn?2→ ?DAϕ?2. This proves (i).
(ii) We claim that
A(R3) and ϕn → ϕ we see that ϕn ⇀ ϕ in
A(R3), and hence that EA,V(ϕ) ≤ liminfn→∞EA,V(ϕn), by Lemma A.2, (ii).It
EA,V(ψ) ≤?ψ,(D2
A(R3). This follows from 0 ≤ D(ρϕ− ρψ) = D(ρϕ) + D(ρψ) −
2?ψ,Vϕψ?. If ϕ is a minimizer of EA,V, then it follows from (4) that for every normalized
ψ ∈ H1
CA,V(1) ≤?ψ,(D2
with equality if ψ = ϕ. This proves part (ii).
(iii) Clearly for all λ > 0,
A+ V − 2Vϕ)ψ?+ D(ρϕ) (4)
for any given ψ ∈ H1
A(R3),
A+ V − 2Vϕ)ψ?+ D(ρϕ)
CA,V(λ) = λ · inf??DAϕ?2+ ?ϕ,V ϕ? − λD(ρϕ)???ϕ? = 1?.
We see that g(λ) = CA,V(λ)/λ is the infimum of linear functions of λ. It follows that
g is concave and hence continuous.
(iv) It suffices to show that
(5)
CA,V(λ) > λCA,V(1) for allλ ∈ (0,1). (6)
Then CA,V(1 − λ) > (1 − λ)CA,V(1) and the asserted inequality follows. Since, by
(5), CA,V(λ) ≥ λCA,V(1), it remains to exclude equality. Again by (5), the equality
CA,V(λ) = λCA,V(1) would imply the existence of a normalized sequence (ϕn) with
?DAϕn?2+ ?ϕn,V ϕn?− λD(ρϕn) → CA,V(1). A fortiori, this sequence would be mini-
mizing for EA,Vand D(ρϕn) → 0, in contradiction with the assumption.
Lemma 2.2. If A is linear with B = curlA, then
(i) C0,0(1) ≤ CA,0(1) ≤ C0,0(1) + |B|, and C0,0(1) < 0.
(ii) If (ϕn) is a minimizing sequence for EA,0then liminfn→∞D(ρϕn) > 0.
4

Page 5

Proof. The inequality C0,0(1) ≤ CA,0(1) follows from the diamagnetic inequality, and
C0,0(1) < 0 follows from a simple variational argument. By combining (4) with the
enhanced binding inequality of Lieb [1], we conclude that, for ϕ ∈ H1(R3) with ?ϕ? = 1,
CA,0(1)≤inf σ(D2
A− 2Vϕ) + D(ρϕ)
inf σ(−∆ − 2Vϕ) + D(ρϕ) + |B|≤
≤?ϕ,(−∆ − 2Vϕ)ϕ? + D(ρϕ) + |B|
E0,0(ϕ) + |B|.=
To prove (ii), suppose that D(ρϕn) → 0 as n → ∞ for some minimizing sequence (ϕn)
of EA,0. Then
CA,0(1) = lim
n→∞EA,0(ϕn) = lim
n→∞?DAϕn?2≥ |B|, (7)
which is in contradiction with the fact that CA,0(1) ≤ C0,0(1) + |B| < |B|, by (i).
Theorem 2.3. Suppose that A is linear.
?|ϕ|2dx = 1 such that
and every minimizing sequence for EA,0has a subsequence that converges to a minimizer
after suitable translations and phase shifts.
Then there exists a ϕ ∈ H1
A(R3) with
EA,0(ϕ) = CA,0(1),
Remark. The Pekar functional EA,0with a linear vector potential A is invariant under
magnetic translations ψ ?→ ψv, v ∈ R3, where
ψv(x) := e−iχ(x)ψ(x − v),χ(x) := A(v) · x,v ∈ R3. (8)
This means that minimizing sequences will in general not be relatively compact. By
the concentration compactness principle every minimizing sequence has a subsequence
that becomes relatively compact upon suitable translations of the type (8).
Proof. Let (ϕn) be a minimizing sequence for EA,0and let (ϕnk) be the subsequence
given by Lemma A.1. We shall exclude vanishing and dichotomy in order to conclude
compactness of the sequence of suitably shifted functions. In the following we use ρn
as a short hand for ρϕn.
Vanishing does not occur. We show that vanishing implies D(ρnk) → 0 as k → ∞,
which contradicts Lemma 2.2 (ii). To this end we use that D(ρϕ) =?Vϕρϕdx ≤ ?Vϕ?∞
Hardy inequalities,
where ϕ ∈ L2(R3) is normalized. For every R > 0, by the H¨ older and the magnetic
|Vϕnk(x)| ≤
?
BR(x)
|ϕnk(y)|2
|x − y|
dy +1
R
≤ 2?DAϕnk?
??
BR(x)
|ϕnk(y)|2dy
?1/2
+1
R.
Since supk?DAϕnk? < ∞, vanishing implies ?Vϕnk?∞→ 0 and D(ρnk) → 0 as k → ∞.
5