An Improved Combes-Thomas Estimate of Magnetic Schrödinger Operators

0.31 MB·English

by  Zhongwei Shen

An Improved Combes-Thomas Estimate of Magnetic
Schr¨odinger Operators
Zhongwei Shen
∗
2
Department of Mathematics and Statistics
1
0 Auburn University
2
Auburn, AL 36849
l
u
J USA

6
1 July 17, 2012

]
h
p
-
h
Abstract
t
a
m
In the present paper, we prove an improved Combes-Thomas estimate, i.e., the
[ Combes-Thomas estimate in trace-class norms, for magnetic Schro¨dinger operators

1 under general assumptions. In particular, we allow unbounded potentials. We also
v show that for any function in the Schwartz space on the reals the operator kernel de-
2
cays, in trace-class norms, faster than any polynomial.
8
7 Keywords: magnetic Schro¨dinger operator, Combes-Thomas estimate, trace ideal esti-
3
mate, operator kernel estimate.
.
7 2010 Mathematics Subject Classification: Primary 81Q10, 47F05; Secondary 35P05.
0
2
1
Contents
:
v
i
X
1 Introduction 2
r
a
2 Standing Notations 4
3 Semigroup and Trace Ideal Estimates 5
4 The Combes-Thomas Estimate in Trace Ideals 8
5 The Operator Kernel Estimate in Trace Ideals 16
A Sectorial Form and m-Sectorial Operator 20
B Justification of (37) 21
∗Email: [email protected]
1
C The Helffer-Sj¨ostrand Formula 23
1 Introduction
ThispaperisconcernedwiththesocalledCombes-ThomasestimateofthefollowingSchro¨dinger
operator with magnetic field
1
H (A,V) = ( i A(x))2 +V(x) on Λ, (1)
Λ
2 − ∇−
where i = √ 1 is the imaginary unit, = (∂ ,∂ ,...,∂ ) is the gradient, A is the vector
− ∇ x1 x2 xd
potential giving rise to the magnetic field A, V is the electric potential and Λ Rd is
∇× ⊂
the configuration space with dimension d. This operator is used to characterize a spinless
particle subject to a scaler potential and a magnetic filed in non-relativistic quantum physics
[20, 21, 40].
As it is known, the Combes-Thomas estimate plays an important role in the theory of
Schro¨dinger operators, magnetic Schro¨dinger operators, classical wave operators, etc. in ran-
dom media. It was invented by Combes and Thomas [9] to study the asymptotic behavior of
eigenfunctions for multi-particle Schro¨inger operators. Later, Fr¨olich and Spencer [15] used
it to study the localization for the multidimensional discrete Anderson model. Meanwhile,
the Combes-Thomas estimate, as well as Wegner estimate [42] and Lifshitz tail [31], became
important ingredients in multiscale analysis. Specifically, the initial scale estimate in multi-
scale analysis for localization near the bottom of the spectrum is successful because of the
Combes-Thomas estimate. See [1, 4, 7, 13, 14, 17, 18, 19, 25, 27, 28, 29, 35, 38] and refer-
ences therein for further applications. Moreover, a stronger version of the Combes-Thomas
estimate, i.e., the estimate in trace-class norms, is very useful. In [8] and [26], such estimates
have been applied to study the regularity of the integrated density of states, a concept of
great physical significance [32]. See [3, 30] for other applications.
Since the pioneering work of Combes and Thomas [9], the Combes-Thomas estimate in
operator norm has been well studied (see [1, 13, 14, 28, 35, 38] and reference therein). We
point out the work of Germinet and Klein [18]. They proved a Combes-Thomas estimate, in
operator norm, with explicit bound of general Schro¨dinger operators including Schro¨dinger
operator, magnetic Schro¨dinger operator, acoustic operator, Maxwell operator and so on.
However, most existing results abouttheCombes-Thomas estimate intrace-class norms were
proven, more or less, under additional assumptions. For instance, Barbaroux, Combes and
Hislop proved in [3] the estimate under the assumption of some sort of analyticity. Klopp’s
result, obtained in [30], for Schro¨dinger operators without magnetic fields was proven under
the assumption of the boundedness of the potential. Results about the Combes-Thomas
estimate in trace-class norms under general assumptions are unknown so far.
The main goal of the current paper is to obtain the Combes-Thomas estimate of (1)
and the associated operator kernel estimate in trace-class norms under general assumptions,
which allow the potential to be unbounded. We first prove an improved Combes-Thomas es-
timate, i.e., the Combes-Thomas estimate in trace-class norms, for the magnetic Schro¨dinger
operator (1) under general assumptions. Based on the improved Combes-Thomas estimate,
we also show that for any function in the Schwartz space on the reals the operator kernel
decays, in trace-class norms, faster than any polynomial.
2
To be more specific, we assume that the magnetic vector potential A (Rd) is Rd-
loc
∈ H
valued, the electric potential V (Rd) is real-valued and the dimension d 2. The
notations (Rd) and (Rd) f∈orKsp±aces are explained in Section 2. Let Λ ≥Rd be an
loc
H K± ⊂
open set. We assume that Λ is bounded with sufficiently smooth boundary if it is not the
whole space. The self-adjoint realization of H (A,V) on L2(Λ) is still denoted by H (A,V).
Λ Λ
If Λ = Rd, then H (A,V) is nothing but the localized operator with homogeneous Dirichlet
Λ
6
boundary on ∂Λ. These self-adjoint operators are constructed via sesquilinear forms. In
Section 3, we will recall the constructions done in [5].
Our first purpose is to study the Combes-Thomas estimate in trace class norms, i.e., the
trace ideal estimate of the operators
χ (H (A,V) z) nχ , β,γ Rd,
β Λ − γ
− ∈
where χ is the characteristic function of the unit cube centered at β Rd and z
β
∈ ∈
ρ(H (A,V)), the resolvent set of H (A,V). More precisely, we want to obtain the ex-
Λ Λ
ponential decay of χ (H (A,V) z) nχ in terms of β γ for suitable n and p, where
k β Λ − − γkJp | − |
is the p-th von Neumann-Schatten norm reviewed in Section 2. Following the defini-
ktio·nkJipn [18], the family of operators χ (H (A,V) z) nχ is also called the operator
β Λ − γ β,γ Rd
kernel of the bounded operator (H {(A,V) z) n.−In gener}al,∈if f is a bounded Borel func-
Λ −
−
tion on σ(H (A,V)), the spectrum of H (A,V), then the family χ f(H (A,V))χ
Λ Λ β Λ γ β,γ Rd
{ } ∈
is called the operator kernel of the bounded linear operator f(H (A,V)). Our first main
Λ
result regarding the Combes-Thomas estimate is roughly stated as follows (see Theorem 4.5
and Theorem 4.6 for details).
Theorem 1.1. Let A (Rd), V (Rd) and Λ Rd open. Suppose p > d with
n N and n 1. Fo∈r aHnlyocz ρ(H∈(AK,±V)), the reso⊂lvent set of H (A,V), ther2en exist
Λ Λ
∈ ≥ ∈
constants C = C(p,z,n) > 0 and a = a (z) > 0 such that
0 0
χ (H (A,V) z) nχ Ce a0β γ , β,γ Rd. (2)
k β Λ − − γkJp ≤ − | − | ∀ ∈
In this paper, we also study operator kernel estimate in trace-class norms. That is, we
prove the polynomial decay of the operators
χ f(H (A,V))χ , β,γ Rd
β Λ γ
∈
in trace-class norms in terms of β γ , where f belongs to the Schwartz space (R) reviewed
| − | S
in Section 2. The main result related to operator kernel estimate is roughly stated as follows
(see Theorem 5.2 for details).
Theorem 1.2. Let A (Rd), V (Rd) and Λ Rd open. Suppose p > d. Then,
for any f (R) and a∈nyHkloc Z with k∈ K1±, there exists⊂a constant C = C(p,k,f)2> 0 such
∈ S ∈ ≥
that
χ f(H (A,V))χ C β γ k, β,γ Rd. (3)
k β Λ γkJp ≤ | − |− ∀ ∈
Estimates like (3), with A being Zd-period, V being bounded and f being a smooth
function with compact support, have been used, as a technical tool, to study the regularity
of integrated density of states. For instance, Combes, Hislop and Klopp [8, Eq.(2.30)] utilize
3
the polynomial decay of any order to prove the convergence of some series, which leads to
an expected estimate. It should be pointed out that Germinet and Klein proved in [18] for
slowly decreasing smooth functions (see Appendix C for the definiton) the operator kernels
for general Schro¨dinger operators decay, in the operator norm, faster than any polynomial.
Their result was then used as a crucial ingredient in their following paper [19].
The rest of the paper is organized as follows. In Section 2, we collect the notations
used in this paper. In Section 3, we study trace ideal estimates of operators of the form
gf(H (A,V)) for suitable f and g. Such estimates, with g being characteristic functions of
Λ
unit cubes and f being integer powers of the resolvent of H (A,V), are used as technical
Λ
tools in the proof of (2). Section 4 is devoted to the study of the Combes-Thomas estimate
in trace-class norms. That is, we prove Theorem 1.1. In Section 5, we study the operator
kernel estimate in trace-class norms and prove Theorem 1.2.
2 Standing Notations
In this section, we collect the notations which will be used in the sequel.
The configuration space Λ is an open set of Rd. We assume that Λ is bounded with
sufficiently smoothboundaryunless itisthewhole space. Wealsoassume thatthedimension
d 2 since, by gauge transform, vector potentials in one spatial dimension are of no physical
≥
interest.
We denote by χ the characteristic function of the unit cube centered at β Rd. If the
β
∈
configuration space in question is Λ(= Rd), then χ should be understood as χ χ , where
β β Λ
6
χ is the characteristic function of Λ. Generally speaking, if a function is defined on Λ, then
Λ
we consider it as a function defined on Rd by zero extension on Rd Λ.
\
The Banach space of p-th Lebesgue integrable functions on Λ is
Lp(Λ) = φ measurable on Λ φ < ,
p
k k ∞
1 (cid:8) (cid:12) (cid:9)
where φ = φ(x) pdx p if p [1, ) and φ (cid:12) = ess sup φ(x) . When p = 2,
L2(Λ) iks akpHilbertΛs|pace|with inner p∈rodu∞ct k k∞ x∈Λ| |
(cid:0)R (cid:1)
φ,ψ = φ¯(x)ψ(x)dx.
h i
ZΛ
Moreover, φ = φ,φ . As a convention, we simply write as .
2 2
k k h i k·k k·k
If L : Lp(Λ) Lq(Λ) is a bounded linear operator, the operator norm is defined by
→p
L := sup Lφ .
p,q q
k k k k
φ p=1
k k
If p = q = 2, we simply write as .
2,2
k·k k·k
Although we use the same notation for both the norm of a function in L2(Λ) and
k ·k
the norm of an operator on L2(Λ), it should not give rise to any confusion. Similarly, we do
not distinguish the notations for norms corresponding to different configuration spaces.
For any p [1, ), the Banach space (also an operator ideal) is defined by
p
∈ ∞ J
= C : L2(Λ) L2(Λ) linear and bounded C < ,
Jp → k kJp ∞
(cid:8) (cid:12) (cid:9)
4 (cid:12)
1
where C = Tr C p p < is the p-th von Neumann-Schatten norm of C. See [33, 36]
k kJp | | ∞
for more details. We here single out the space (also called the space of Hilbert-Schmidt
2
(cid:0) (cid:1) J
operators) for the following important property (see [34, Theorem VI.23]): a bounded linear
operator K on L2(Λ) belongs to if and only if it is an integral operator with some integral
2
J
1
kernel k(x,y) being in L2(Λ Λ). In this case, K = k(x,y) 2dxdy 2. We will
use this property in Section 3×. k kJ2 Λ×Λ| |
(cid:0)R (cid:1)
Let g(x) = ln x if d = 2 and g(x) = x 2 d if d 3. We say a function V (Rd),
−
− | | | | ≥ ∈ K
the Kato class, if
lim sup g(x y) V(y) dy = 0.
ǫ↓0 x RdZx y ǫ − | |
∈ | − |≤
A function V is said to be in the local Kato class (Rd) if Vχ (Rd) for all compact
loc K
K ∈ K
set K Rd, where χ is the characteristic function of K. We refer to [39] for equivalent
K
⊂
definitions from the viewpoint of probability theory.
Let V defined on Rd be real-valued. We say that V is Kato decomposable, in symbols
V (Rd), if the positive part V is in (Rd) and the negative part V is in (Rd).
+ loc
∈AKC±d-valued function A is said to be Kin the class (Rd) if its squared−norm KA A and
H ·
its divergence A, considered as a distribution on C (Rd), are both in the Kato class
∇ · 0∞
(Rd). It is said to be in the class (Rd) if both A A and A are in the local Kato
loc
K H · ∇ ·
class (Rd). We refer the reader to [2, 5, 6, 10] for further remarks about these spaces.
loc
K
The Schwartz space (R) consists of those C (R) functions which, together with alltheir
∞
S
derivatives, vanish at infinity faster than any power of x . More precisely, for any N Z,
| | ∈
N 0 and any r Z, r 0, we define for f C (R)
∞
≥ ∈ ≥ ∈
f = sup(1+ x )N f(r)(x) ,
N,r
k k x R | | | |
∈
then
(R) = f C (R) f < for all N,r .
∞ N,r
S { ∈ |k k ∞ }
See Folland [16] for more discussions about the Schwartz space.
3 Semigroup and Trace Ideal Estimates
Inthissection, asapreparationforprovingTheorem1.1andTheorem1.2,westudyestimates
of operators of the form gf(H (A,V)) in trace-class norms for suitable f and g.
Λ
The self-adjoint realization of H (A,V) on L2(Λ), still denoted by H (A,V), is defined
Λ Λ
via sesquilinear forms as follows (see [5]): the sesquilinear form
hA,V+ :C (Λ) C (Λ) C,
Λ 0∞ × 0∞ →
d
1
(ψ,φ) hA,V+(ψ,φ) := V ψ, V φ + ( i∂ A )ψ,( i∂ A )φ
7→ Λ + + 2 − j − j − j − j
j=1
(cid:10)p p (cid:11) X(cid:10) (cid:11)
is densely defined in L2(Λ), nonnegative and closable, where , denotes the usual inner
h· ·i
product on L2(Λ). Its closure is still denoted by hA,V+ with form domain (hA,V+), which is
Λ Q Λ
5
the completion of C (Λ) with respect to the norm
0∞
φ = φ 2 +hA,V+(φ,φ),
k khA,V+ k k Λ
Λ
q
where = is the norm on L2(Λ) associated with , as mentioned in Section 2. We
2
k·k k·k h· ·i
denotebyH (A,V )theassociated self-adjointoperator. SinceV (Rd)isinfinitesimally
Λ +
− ∈ K
form-bounded with respective to H (A,0)( H (A,V )), i.e., there exist Θ (0,1) (can
Λ Λ + 1
≤ ∈
be taken to be arbitrarily small) and Θ 0 depending on Θ so that
2 1
≥
φ,V φ Θ hA,0(φ,φ)+Θ φ 2, φ (hA,0), (4)
h − i ≤ 1 Λ 2k k ∈ Q Λ
KLMNtheorem(see[34,TheoremX.17])yieldsthat,with (hA,V) = (hA,V+),thesesquilin-
Q Λ Q Λ
ear form
hA,V : (hA,V) (hA,V) C,
Λ Q Λ ×Q Λ → (5)
(ψ,φ) hA,V(ψ,φ) := hA,V+(ψ,φ) V ψ, V φ
7→ Λ Λ − − −
is closed and bounded from below and has C (Λ) as a fo(cid:10)rpm corep. The(cid:11)associated semi-
0∞
bounded self-adjoint operator is denoted by H (A,V).
Λ
The main result of this section is stated as follows. Let
E = the infimum of the L2(Rd)-spectrum of H (0,V). (6)
0 Rd
Theorem 3.1. Let A (Rd), V (Rd) and Λ Rd open. Suppose p 2. Let f be
loc
∈ H ∈ K± ⊂ ≥
a Borel function satisfying
f(λ) C(1+ λ ) α, λ σ(H (A,V)), (7)
− Λ
| | ≤ | | ∈
for α > d . Then gf(H (A,V)) is in with
2p Λ Jp
gf(H (A,V)) C g (H (A,V) λ )αf(H (A,V))
k Λ kJp ≤ α,p,λ0k kpk Λ − 0 Λ k
whenever g Lp(Λ), where λ < E and C > 0 depends only on α, p and λ .
∈ 0 0 α,p,λ0 0
To prove the above theorem, we first present some lemmas. We begin with the celebrated
Feynman-Kac-Itˆo formula proven by Broderix, Hundertmark and Leschke (See [24, 37, 39]
and references therein for earlier versions).
Lemma 3.2 ([5]). Let A (Rd), V (Rd) and Λ Rd open. For any φ L2(Λ)
loc
∈ H ∈ K± ⊂ ∈
and t 0, there holds
≥
e−tHΛ(A,V)φ (x) = Ex e−Stω(A,V)ΞΛ,t(ω)φ(ω(t)) for a.e. x Λ, (8)
∈
where (cid:0) (cid:1) (cid:8) (cid:9)
t i t t
Sω(A,V) = i A(ω(s))dω(s)+ ( A)(ω(s))ds+ V(ω(s))ds,
t 2 ∇·
Z0 Z0 Z0
E denotes the expectation for the Brownian motion starting at x and Ξ is the charac-
x Λ,t
{·}
teristic function of the set ω ω(s) Λ for all s [0,t] .
{ | ∈ ∈ }
6
As consequences of (8), we get the so called diamagnetic inequality
e tHΛ(A,V)φ e tHΛ(0,V) φ , t 0, (9)
− −
≤ | | ≥
the monotonicity of semigro(cid:12)up for vanis(cid:12)hing magnetic field in the sense that for Λ Λ
′
(cid:12) (cid:12) ⊂
e−tHΛ(0,V)χΛφ e−tHΛ′(0,V)φ, φ 0, t 0
≤ ≥ ≥
and then the Lp-smoothing of semigroups: for 1 p q , there exist constant C > 0
≤ ≤ ≤ ∞
and E such that
e−tHΛ(A,V) e−tHΛ(0,V) e−tHRd(0,V) Ct−γeEt, (10)
p,q ≤ p,q ≤ p,q ≤
where γ = d(1 (cid:13) 1). We r(cid:13)emark(cid:13)that E can(cid:13)be ch(cid:13)osen such th(cid:13)at E < E (See [5, 35]).
2 p (cid:13)− q (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) − 0
We extend [35, Theorem B.2.1] to the magnetic case.
Lemma 3.3. Let A (Rd), V (Rd) and Λ Rd open. Let α > 0 and 1 p q
loc
∈ H ∈ K± ⊂ ≤ ≤ ≤
satisfy
∞ 1 1 2α
< . (11)
p − q d
Then (H (A,V) z) α is bounded from Lp(Λ) to Lq(Λ) whenever the real part z < E .
Λ − 0
− ℜ
Proof. It follows from the formula
(H (A,V) z) α = c ∞e tHΛ(A,V)etztα 1dt (12)
Λ − α − −
−
Z0
and (10), where the assumption (11) is applied to insure the convergence of the integral in
(12).
As a consequence of Lemma 3.3, we have
Lemma 3.4. Let A (Rd), V (Rd) and Λ Rd open. Let α > 0 and 1 p
loc
∈ H ∈ K± ⊂ ≤ ≤
2 q satisfy (11). For any Borel function f satisfying (7), the operator f(H (A,V))
Λ
≤ ≤ ∞
is bounded from Lp(Λ) to Lq(Λ) with
f(H (A,V)) C (H (A,V) λ )αf(H (A,V)) , (13)
k Λ kp,q ≤ p,q,α,λ0k Λ − 0 Λ k
where λ < E and C > 0 depends only on p, q, α and λ .
0 0 p,q,α,λ0 0
Proof. It follows from the arguments in [35, Theorem B.2.3].
Wenextdiscussthetraceidealestimateofoperatorsoftheformgf(H (A,V))forsuitable
Λ
f and g. We start with recalling a result of Dunford and Pettis (See [10, 35, 41] for abstract
versions).
Lemma 3.5. Let (M,µ) be a separable measurable space. If L is a bounded linear operator
from Lp(M) to L (M) with 1 p < , then there is a measurable function k( , ) on M M
∞
≤ ∞ · · ×
such that L is an integral operator with integral kernel k( , ) and
· ·
1
sup k(x,y) p′dµ(y) p′ = L < ,
p,
x∈M(cid:18)ZM | | (cid:19) k k ∞ ∞
where p = p is the conjugate exponent of p.
′ p 1
−
7
We are now ready to prove Theorem 3.1.
Proof of Theorem 3.1. By complex interpolation (see [36, Theorem 2.9]), it suffices to prove
the result in the case p = 2, which we show now. For p = 2 and q = , we have d 1 1 =
∞ 2 p − q
d < α by assumption, i.e., (11) is satisfied, and thus, Lemma 3.4 implies that f(H (A,V))
4 (cid:0)Λ (cid:1)
is bounded from L2(Λ) to L (Λ). By Lemma 3.5, f(H (A,V)) is an integral operator with
∞ Λ
kernel kA,V(x,y) satisfying
Λ
sup kA,V(x,y) 2dy = f(H (A,V)) 2 < .
Λ k Λ k2, ∞
x∈ΛZΛ ∞
(cid:12) (cid:12)
Thus, gf(H (A,V)) is an in(cid:12)tegral oper(cid:12)ator on L2(Λ) with kernel g(x)kA,V(x,y). Moreover,
Λ Λ
g(x)kA,V(x,y) 2dxdy g 2sup kA,V(x,y) 2dy = g 2 f(H (A,V)) 2 ,
ZZΛ×Λ(cid:12) Λ (cid:12) ≤ k k2 x∈ΛZΛ(cid:12) Λ (cid:12) k k2k Λ k2,∞
which impl(cid:12)ies that gf(H (cid:12)(A,V)) is a Hilbert-S(cid:12)chmidt op(cid:12)erator as mentioned in Section 2,
Λ
i.e., in , with -norm bounded by g f(H (A,V)) . The expected bound is given
2 2 2 Λ 2,
J J k k k k ∞
by (13). This completes the proof.
We remark that results obtained in this section are well-known for Schro¨dinger operators
without magnetic fields. See [2, 35] and references therein. It should be pointed out that
the result of Theorem 3.1 in the case H (0,V) was proven in [35, Theorem B.9.3] for any
Rd
p 1. To prove the result for p [1,2), it was first shown that gf(H (0,V)) for
Rd 1
≥ ∈ ∈ J
g ℓ1(L2(Rd)), the Birman-Solomjak space, then proceeded to complex interpolation. The
∈
proof relies on the translation invariance of the free Laplacian (see [35, Theorem B.9.2]
and [36, Theorem 4.5] for instance), which, however, is not true for magnetic Schro¨dinger
operators. This prevents us from obtaining the result for p [1,2).
∈
4 The Combes-Thomas Estimate in Trace Ideals
Inthissection, westudytheimprovedCombes-Thomasestimate, i.e., thetraceidealestimate
of the operators
χ (H (A,V) z) nχ for β,γ Rd,
β Λ − γ
− ∈
where χ is the characteristic function of the unit cube centered at β. More precisely, we
β
want to obtain the exponential decay of χ (H (A,V) z) 1χ in terms of β γ . The
k β Λ − − γkJp | − |
main result is stated in Theorem 1.1. Since we also consider localized operators, χ should
β
be understood as χ χ if the operators is restricted to Λ as it is mentioned in Section 2,
β Λ
where χ is the characteristic function of the domain Λ. The basic tools we use here are
Λ
sectorial formandm-sectorial operatorreviewed inAppendix A. Wealso employ theclassical
argument of Combes and Thomas developed in [9].
First of all, we establish some results by applying the theory of sectorial form and m-
sectorial operator. For this purpose, we first define auxiliary sesquilinear forms with associ-
ated operators formally given by
Ha(A,V) = eaxH (A,V)e ax, a Rd, (14)
Λ · Λ − · ∈
8
where eax and e ax are multiplicative operators. Note that the operator Ha(A,V) is not
· − · Λ
self-adjoint unless a = 0. First, we denote by D the closure of √2( i A) on C (Λ),
A,Λ 2 − ∇− 0∞
so H (A,0) = D D . This can be seen by sesquilinear forms. Moreover, the domain of
Λ A∗,Λ A,Λ
D , denoted by D(D ), is the form domain, denoted by (hA,0), of the sesquiliner form
A,Λ A,Λ Q Λ
associated with the lower bounded self-adjoint operator H (A,0). For a Λ, we define
Λ
∈
D (a) = eaxD e ax and D (a) = eaxD e ax.
A,Λ · A,Λ − · A∗,Λ · A∗,Λ − ·
It’s easy to see that
√2
D (a) = D +i a, on D(D ),
A,Λ A,Λ A,Λ
2
(15)
√2
D (a) = D +i a, on D(D )
A∗,Λ A∗,Λ 2 A∗,Λ
and they are closed, densely defined operators. Note that (D (a)) = D (a). Next, we
A,Λ ∗ 6 A∗,Λ
define the sesquilinear form hA,0(a) on D(D ) = (hA,0) by
Λ A,Λ Q Λ
hA,0(a)(ψ,φ) = (D (a)) ψ,D (a)φ . (16)
Λ A∗,Λ ∗ A,Λ
Obviously, hA,0(0) hA,0. Finally, we defi(cid:10)ne the sesquilinear form(cid:11) hA,V(a) on (hA,V+) by
Λ ≡ Λ Λ Q Λ
hA,V(a)(ψ,φ) = hA,0(a)(ψ,φ)+ V ψ, V φ V ψ, V φ . (17)
Λ Λ + + − − −
(cid:10)p p (cid:11) (cid:10)p p (cid:11)
For notational simplicity, we let
2Θ s 1 1
c = 2 + +s , s > 0 (18)
s
1 Θ 2 s
− 1 (cid:18) (cid:19)
where Θ , Θ are given in (4). For a > 0, let
1 2 0
2s
Ξ (s) = , Ξ (s,a ) = c a2. (19)
1 1 Θ 2 0 s 0
1
−
We will write Ξ (s) and Ξ (s,a ) as Ξ and Ξ , respectively, in the sequel.
1 2 0 1 2
We next prove several lemmas related to Ha(A,V). Our first lemma is about the relation
Λ
between hA,V(a) and Ha(A,V).
Λ Λ
Lemma 4.1. Let A (Rd), V (Rd) and Λ Rd open. The sesquilinear form
loc
hA,V(a) defined in (1∈7)His a closed s∈ecKto±rial form asso⊂ciated with the unique m-sectorial
Λ
operator Ha(A,V) given by (14).
Λ
Proof. By (5), (15), (16) and (17), we have for any φ (hA,V),
∈ Q Λ
hA,V(a)(φ,φ) hA,V(φ,φ) = hA,0(a)(φ,φ) hA,0(φ,φ)
Λ − Λ Λ − Λ
1
(cid:12) (cid:12) (cid:12)√2 φ,a D φ + a(cid:12)2 φ 2
(cid:12) (cid:12) ≤ (cid:12) |ℜh · A,Λ i| 2| (cid:12)| k k
9
so that
1
hA,V(a)(φ,φ) hA,V(φ,φ) 2 4 a 2 φ 2 D φ 2 + a 4 φ 4,
Λ − Λ ≤ | | k k k A,V k 2| | k k
which implies(cid:12)that for any s > 0, (cid:12)
(cid:12) (cid:12)
1
1 2
hA,V(a)(φ,φ) hA,V(φ,φ) a φ 4 φ 2 D φ 2 + a 2 φ 2
Λ − Λ ≤ | |k k k k k A,V k 2| | k k
(cid:18) (cid:19)
(cid:12) (cid:12) 1 s 1
(cid:12) (cid:12) a 2 φ 2 + 4 φ 2 D φ 2 + a 2 φ 2 (20)
A,V
≤ 2s| | k k 2 k k k k 2| | k k
(cid:18) (cid:19)
1 s
= 2shA,0(φ,φ)+ + a 2 φ 2,
Λ 2s 2 | | k k
(cid:18) (cid:19)
since hA,0(φ,φ) = D φ 2. Thanks to (4) and (5),
Λ k A,V k
hA,V (1 Θ )hA,0 Θ on (hA,V) (hA,0) .
Λ ≥ − 1 Λ − 2 Q Λ ⊂ Q Λ
(cid:0) (cid:1)
This, together with (20), implies that
hA,V(a)(φ,φ) hA,V(φ,φ) Ξ hA,V(φ,φ)+Ξ φ 2, φ (hA,V), (21)
Λ − Λ ≤ 1 Λ 2k k ∈ Q Λ
(cid:12) (cid:12)
where Ξ1 an(cid:12)d Ξ2 are given in (19) wit(cid:12)h a0 replaced by a .
To apply Theorem A.1, we choose s 0, 1 Θ1 s|o|that Ξ = 2s < 1. Since hA,V
∈ −2 1 1 Θ1 Λ
is symmetric, closed and bounded from below, Theorem A.1 says tha−t hA,V(a) is a closed
(cid:0) (cid:1) Λ
sectorial form defined on (hA,V). Theorem A.2 then guarantees that there exists a unique
m-sectorial operator, denoQtedΛby Ha(A,V), associated to hA,V(a).
Λ Λ
The next lemma gives an operator equality connecting Ha(A,V) and H (A,V).
Λ Λ
Lemma 4.2. Let A (Rd), V (Rd) and Λ Rd open. Suppose s 0, 1 Θ1 so
that Ξ < 1. Let the m∈-Hselcotcorial oper∈atKor±Ha(A,V) be⊂as in Lemma 4.1. Let∈ −2
1 Λ (cid:0) (cid:1)
H˜ (A,V) = H (A,V)+Ξ 1Ξ , (22)
Λ Λ −1 2
where Ξ and Ξ are given in (19) with a replaced by a . Then H˜ (A,V) is nonnegative
1 2 0 Λ
| |
and there exists a bounded linear operator B from L2(Λ) to itself with B 2Ξ such that
1
k k ≤
Ha(A,V) = H (A,V)+ H˜ (A,V)B H˜ (A,V). (23)
Λ Λ Λ Λ
q q
Proof. Set
h¯A,V(a) = hA,V(a) hA,V on (hA,V),
Λ Λ − Λ Q Λ (24)
h˜A,V = hA,V +Ξ 1Ξ on (hA,V).
Λ Λ −1 2 Q Λ
Then (21) can be rewritten as
h¯A,V(a)(φ,φ) Ξ h˜A,V(φ,φ), φ (hA,V),
Λ ≤ 1 Λ ∈ Q Λ
(cid:12) (cid:12)
(cid:12) (cid:12)
10