Proceedings of the 3th Asian Conference on
Nonlinear Analysis and Optimization
(Matsue, Japan, 2012), 43–55
43
FORMALLY SELF-ADJOINT SCHERÖDINGER OPERATORS
KIYOKO FURUYA
Abstract. A Schrödinger operator, (not self-adjoint but) formally selfadjoint, generates a (not unitary but) contraction semigroup. Our class of
potentials U in Schrödinger equation is wide enough : the real measurable
potential U should be locally essentially bounded except a closed set of
measure zero.
1. Introduction
We shall construct a family of unique solutions to the Schrödinger equation
in RN
(1.1)
h
∂
ih2
u(t, x) =
△u(t, x) − iU (x)u(t, x),
∂t
2m
u(0, x) = φ(x),
N
for U ∈ L∞
loc (R \N ; R) where N is a closed set of measure 0. Here h and m
are positive constants.
We define a sequence of functions {Un }n∈N as follows:


if
n < U (x),
 n
(1.2) Un (x) =
U (x) if −n ≤ U (x) ≤ n
f or n ∈ N


−n
if
U (x) < −n
Then we shall approximate the potential U by Un . The unique solution obtained by this approximation seems natural for the theory of path integrals.
The physical meaning of the solution by Nelson[11] is unclear to the author.
Definition 1.1. Let B be a densely defined operator on Hilbert space H. Then
(i) B is essentially self-adjoint if and only if it has a unique self-adjoint extension, necessarily its closure B̄,
(ii) B is formally self-adjoint if ⟨Bφ, ψ⟩ = ⟨φ, Bψ⟩ for all φ and ψ in H.
2010 Mathematics Subject Classification. Primary 47D06; Secondary 81S40(58D30).
Key words and phrases. Schrödinger operator, Trotter-Kato Theorem, weak convergence,
contraction C0 -semigroups.
44
KIYOKO FURUYA
We consider a closed extension of (not necessarily essentially self-adjoint but)
formally self-adjoint operator iA ≡ −(h2 /2m)△ + U on C0∞ (RN \N ). Here
C0∞ (E) denote the set of all infinitely differentiable functions with compact
support in E. The semigroup of our solution family, which is obtained by
the approximation (1.2), is not necessarily a group of unitary operators but a
semigroup of contractions. Our result improves one of the Nelson [11]’s, which
says the contraction semigroup of his solution family exists (not for all but) for
a. e. m > 0 and for any U ∈ C(RN \N0 ; R), where N0 is a closed subset of
capacity 0. In these half a century, no essential progress in this direction has
been obtained as far as the author knows. Our results is closely related to the
theory of path integrals (see Furuya [4]).
2. Schrödinger equation
For simplicity we consider the following normalized equation :
(2.1)
∂
u(t, x) = i△u(t, x) − iU (x)u(t, x), u(0, x) = φ(x) for φ ∈ H (2) (RN ; C),
∂t
where H (2) (RN ; C) denote the Sobolev space of L2 -functions with first and
second distributional derivatives also in L2 on RN to C.
If △ − U is essentially self-adjoint, the operator family {T (t)} defined by )
T (t)φ = u(t) is uniquely extended to a group of unitary operators from L2 (RN ; C
to L2 (RN ; C). Let N be a fixed closed subset of RN of measure 0 and D = {D}
be the maximum family such that each element D ⊂ D ⊂ RN \N is a finite
∪ union ofNconnected bounded open sets. The family D = {D} satisfies
D∈D D = R \N . We denote the restriction of f to D by f |D .
We use the following notation
{ }
N
(2.2) L∞
(R
\N
;
R)
=
f f (x) ∈ R, x ∈ RN , f |D ∈ L∞ (D; R), ∀D ∈ D .
loc
N
Let U ∈ L∞
loc (R \N ; R). We assume for any neighbourhood of any point of
N , U is not essentially bounded. By this assumption, U uniquely determines
N in the following sense :
∩
N
(2.3)
N =
{Nν U ∈ L∞
loc (R \Nν ; R)}.
ν
Let
(2.4)
Bn = { x ∈ RN − n < U (x) < n }
f or
n ∈ N.
We have Bm ⊃ Bn for m > n and
(2.5)
for any D ∈ D, there exists Bn such that D ⊂ D ⊂ Bn .
FORMALLY SELF-ADJOINT SCHERÖDINGER OPERATORS
45
(Strictly speaking, {D\Bn is not necessarily
empty, but a null set.) We de}
note Un (x) = min n, max{−n, U (x)} . Thus Un ∈ L∞ (RN ; R). For U ∈
N
L∞
loc (R \N ; R) we consider the approximative equation
d
un (t) = An un (t), where An = i(△ − Un ).
dt
In this case the operator −iAn is essentially self-adjoint. Hence the semigroup
{Tn (t)} generated by −iAn is the family of solutions to (2.6) and is a group of
unitary operators : ∥Tn (t)φ∥ = ∥φ∥ for t ∈ R and φ ∈ L2 (RN ; C).
(2.6)
The main theorem in this paper is the following :
N
Theorem 2.1. For any U ∈ L∞
loc (R \N , R), there exists a closed extension
2
N
of (i△ − iU )|C0∞ (RN \N ) in L (R ; C) to L2 (RN ; C) which generates a unique
contraction C0 -semigroup {T (t) t ≥ 0} such that
(2.7)
T (t)φ = w- lim Tn (t)φ
n→∞
f or
all
φ ∈ L2 (RN ; C),
where Tn (t)φ is the solution to (2.6) and w-lim means the weak convergence.
For the proof of existence of {T (t)}, we use a theory of abstract semigroups,
which will be give in the section 4. The proof of uniqueness is given in the
section 6.2.
3. Preliminaries
In this section we begin by introducing some terminology and notation and
present those aspects of the basic theory which are required in subsequent
sections.
3.1. Filter.
Definition 3.1. Given a set E, a partial ordering ⊂ can be defined on the
powerset P(E) by subset inclusion. Define a filter F on E as a subset of P(E)
with the following properties:
(i)
∅ ̸∈ F.(The empty set is not∩
in F.)
(ii) If A ∈ F and B ∈ F , then A B ∈ F . (F is closed under finite meets.)
(iii) If A ∈ F and A ⊂ B, then B ∈ F . (Therefore E ∈ F .)
Definition 3.2. Let B is a subset of P(E). B is called filter base on E if and
only if
(i) The intersection of any two sets of B contains a set of B,
(ii) B is non-empty and the empty set is not in B
46
KIYOKO FURUYA
Definition 3.3. To say that filter base B converges to x, denoted B → x,
means that for every neighourhood U of x, there is a B ∈ B such that B ⊂ U .
In this case, x is called a limit of B and B is called a convergent filter base.
Lemma 3.4. X is a Hausdorff space if and only if every filter base on X has
at most one limit.
Definition 3.5. A filter F in a topological spacec is called ultra filter if having
the property that no other filter exists in the space having among its subsets
all the subsets in the given filter.
For details concerning the filter, we refer to Bourbak[1].
3.2. Compact open topology.
Definition 3.6. A linear topological space X is called a locally convex linear
topological space, or, in short, a locally convex space, if and only if its open
sets ∋ 0 contains a convex, balanced and absorbing open set.
Let X and X ′ be two linear spaces over the complex field C and a scalar
product ⟨x, x′ ⟩ ∈ C for x ∈ X and x′ ∈ X ′ be defined.
Definition 3.7. Let X be topological vector space. The weak topology on
X, denote by σ(X, X ′ ), is the weakest topology such that all elements of X ′
remains continuous.
Definition 3.8. The strong topology β of X ′ is the topology of uniform convergence on every σ(X, X ′ )-bounded set in X. Xβ′ denotes (X ′ )β .
Definition 3.9. τ0 is the
locally convex topology on X, defined by the seminorm system P = {pγ pγ (f ) = supg∈Cγ |⟨f, g⟩|, Cγ ∈ C}, where C = {Cγ }
denotes the family of the compact subsets of Xβ′ . Equivalently, Uτ0 = {Up }p∈P ,
where Up = {x ∈ X p(x) < 1}, is a fundamental system of τ0 -neighbourhoods
of zero. τ0 is called the compact open topology.
In the case of Banach space J. Dieudonné has proved the following theorem.
Theorem 3.10 (Dieudonné[2]). The bounded weak ∗ topology in a Banach
space is identical with the compact open tpoplogy
We denote by X
set in Xβ′ .
′
∗
the space of linear functionals bounded on every bounded
Proposition 3.11 (Kōmura and Furuya[9] Proposition 1). Let X τ0 be the
completion of the space Xτ0 . Then we have:
′
(Xβ′ )′ ⊂ X τ0 ⊂ X ∗ .
FORMALLY SELF-ADJOINT SCHERÖDINGER OPERATORS
47
Lemma 3.12 (Kōmura and Furuya[9] Lemma 5). Let x′′ ∈ X ′′ . x′′ ∈ X τ0 if
and only if x′′ is σ(X ′ , X)-continuous on every τ0 -equi-continuous set {Upo |Up ∈
Uτ0 }. Here Upo is a polar set of Up .
Corollary 3.13. If X is a Banach space, we have X ′′ = X τ0 .
4. Existence of weak limit of unitary groups in abstract case
(H, ∥ · ∥), or simply H , denotes a Hilbert space with norm ∥ · ∥ . Instead of
the convergence of subsequences we use the convergence of filters. We consider
an infinite semi-ordered index set A = {α}. We assume that there exists an
ultra-filter Φ of infinite subsets of A satisfying
(4.1)
∀ϕ ∈ Φ, ∀α ∈ A, ∃α′ ∈ ϕ : α′ ≻ α.
In the following {Φ} denotes the family of ultra-filters whose element satisfies
(4.1).
Remark 4.1. Note that we can use subsequences {αk }∞
k=1 instead of ultrafilters, if H is separable.
Let a family {Tα (t) | − ∞ < t < ∞}α∈A of groups of unitary operators in
H be given.
Aα denotes the generator of {Tα (t)} :
d
Tα (t)φ = Aα Tα (t)φ,
dt
φ ∈ D(Aα ).
Definition 4.2. For an ultra-filter Φ satisfying (4.1), the operators (I − AΦ )−1
and TΦ (t) are defined as follows :
(4.2)
(I − AΦ )−1 f = w- lim (I − Aα )−1 f
(4.3)
TΦ (t)φ = w- lim Tα (t)φ f or
α∈ϕ∈Φ
α∈ϕ∈Φ
f or
∀f ∈ H,
∀φ ∈ H.
In this section we shall show that the existence of a semigroup {TΦ (t)} in
(4.3). As is well known, iAα is self-adjoint : ⟨Aα φ, ψ⟩ = −⟨φ, Aα ψ⟩ for φ, ψ ∈
D(Aα ), and the resolvent (I −Aα )−1 is a contraction : ∥(I −Aα )−1 ∥ ≤ 1. Since
a bounded subset of H is relatively σ(H, H)-compact, ∥(I −Aα )−1 φ∥ ≤ ∥φ∥ and
∥Tα (t)φ∥ = ∥φ∥, there exist w- limα∈ϕ∈Φ (I − Aα )−1 φ and w- limα∈ϕ∈Φ Tα (t)φ.
Hence (I − AΦ )−1 and TΦ (t) are well defined. Note that AΦ may be multivalued. The following condition implies AΦ is single-valued, which will be
verified later (See.Theorem 4.7).
48
KIYOKO FURUYA
Condition
4.3. There exist a dense subspace H0 of H satisfying H0 ⊂
∩
α∈ϕ D(Aα ) and a linear operator A0 : H0 −→ H such that
(4.4)
∀ψ ∈ H0 , ∃α(ψ) ∈ A : A0 ψ = Aα ψ
f or
∀α ≻ α(ψ).
Throughout this paper, we assume Condition 4.3 holds.
Lemma 4.4. For a fixed f ∈ H, put
(4.5)
φα = (I − Aα )−1 f
φΦ = (I − AΦ )−1 f.
and
Under the condition 4.3, we have
(a) Let f = (I − A0 )φ f or φ ∈ H0 then φ = limα∈ϕ∈Φ (I − Aα )−1 f
(b) Let φα = (I − Aα )−1 f and φΦ = w- limα∈ϕ∈Φ φα , then
(4.6)
w - lim Aα φα = AΦ φΦ .
α∈ϕ∈Φ
Moreover if AΦ is single-valued, it follows that
(4.7)
H0 ⊂ D(AΦ )
with
AΦ |H0 = A0
and
(4.8)
⟨AΦ φ, ψ⟩ = −⟨φ, A0 ψ⟩
f or
∀φ ∈ D(AΦ )
and
∀ψ ∈ H0 .
Proposition 4.5. The range of (I − AΦ )−1 : R((I − AΦ )−1 ) = (I − AΦ )−1 H
is dense in H.
We cite Theorem 9 in [9] as Theorem 4.6. Let X be a reflexive Banach space
and {Tα (t)}α∈A be a family of contraction C0 -semigroups in X.
Theorem 4.6 (Kōmura and Furuya [9] Theorem 9). Suppose for some filter
Φ
(4.9)
∀f ∈ X, ∃φΦ = w- lim (I − Aα )−1 f.
α∈ϕ∈Φ
−1
Thus the operator (I −AΦ ) is defined. If the range R((I −AΦ )−1 ) is dense in
X, AΦ is a densely defined closed operator and generates a semigroup {TΦ (t)} :
(4.10)
w- lim Tα (t)x = TΦ (t)x
α∈ϕ∈Φ
for
∀x ∈ X.
Moreover, we have {TΦ (t)} is a contraction C0 -semigroup in X.
Theorem 4.7. Under condition 4.3, AΦ is a closed operator and generates a
contraction C0 -semigroup {TΦ (t)}.
Proof. Since the range R((I − AΦ )−1 ) is dense in H by Proposition 4.5, our
Theorm follows from Theorem 4.6.
FORMALLY SELF-ADJOINT SCHERÖDINGER OPERATORS
49
5. More gener pproximation of U
We consider an approximation of U which is more general than (1.2) :


if
m < U (x),
 m
(5.1) Um,n (x) =
U (x) if
−n ≤ U (x) ≤ m,
for m, n ∈ N.


−n
if
U (x) < −n,
For the approximative equation
(5.2)
d
um,n (t) = Am,n um,n (t), um,n (0) = φ where
dt
Am,n = i(△ − Um,n ).
there exists a unitary group {Tm,n (t)}, where Tm,n (t)φ = um,n (t) is the solution
to (5.2).
Proposition 5.1. We have
(5.3)
(
)
T (t)φ = w- lim Tm,n (t)φ = τ0 - lim Tm,n (t)φ f or ∀φ ∈ L2 (RN ; C).
m,n→∞
m,n→∞
The result is almost the same as the case of approximation {Un }n∈N .
Theorem 5.2 (Furuya[4], Theorem 9). For any U ∈ L∞ (RN \ N ; R), there
exists a closed extension of i(△ − U )|C0∞ (RN \N ;C) in L2 (RN ; C) to L2 (RN ; C)
which generates a contraction C0 -semigroup {T (t) t ≥ 0} such that
(5.4)
T (t)φ = w- lim Tm,n (t)φ
n→∞
f or
∀φ ∈ L2 (RN ; C),
where Tm,n (t)φ is the solution to
d
um,n (t) = Am,n um,n (t), where
dt
and w-lim means the weak convergence.
(5.5)
Am,n = i(△ − Um,n )
+
−
(x) = max{0, −Um,n (x)}.
(x) = max{0, Um,n (x)} and Um,n
Proof. Let Um,n
+
−
Then we obtain that Umn (x) = Um,n (x) − Um,n (x). Note the following:
+
(a) In the case that there exists M ≥ 0 such that Um,n
(x) ≤ M for x ∈ Dm,n
and m, n ∈ N, there exists n0 ∈ N such that Dm,n ⊂ Bn for any n ≥ n0 ≥ M .
−
(b) In the case that there exists M ≥ 0 such that Um,n
(x) ≤ M for x ∈ Dm,n
and m, n ∈ N, there exists m0 ∈ N such that Dm,n ⊂ Bm for any m ≥ m0 ≥ M .
(c) In another case we obtain that
max{B(n), B(m)} ⊃ Dm,n ⊃ min{B(n), B(m)} for m, n ∈ N.
50
KIYOKO FURUYA
Note that
D=
∞
∪
Dm,n.
n,m=1
Therefore from the result of Theorem 2.1 we obtain the consequence.
Note that {T (t)} is independent of the choice of {Dm,n }. Let Φ = {ϕ} be
an ultra-filter of double indexes ϕ = {mk , nj }. Now the proof goes in a same
way as Theorem 4.7 and Theorem 6.5.
6. Weak limit of unitary groups
6.1. Existence in L2 -case.
Theorem 6.1. For each approximation {An } or {Am,n }, the limit TΦ (t) =
limΦ exp(tAn ) or limΦ exp(tAm,n ) exists and {TΦ (t)} is a contraction
C0 -semigroup.
Lemma 6.2. Condition 4.3 is satisfied for H0 = C0∞ (RN \N ; C).
Proof. We have for any ψ ∈ C0∞ (RN \N ), there exists D ∈ D and Bm such that
supp ψ ⊂ D ⊂ Bm , where Bm is defined in (2.4), and if supp ψ ⊂ D ⊂ Bm
then we have Un ψ = Um ψ and An ψ = Am ψ for all n > m. Thus for the case of
{An } Condition 1 is satisfied for α(ψ) = m. For the case of {Am,n }, the proof
is the same.
Lemma 6.3. Let φΦ = w- limm∈ϕ1 ∈Φ (I − Am )−1 f for f ∈ L2 (RN ; C). Then
on any fixed D ∈ D, the filter {φm ; φm = (I − Am )−1 f, m ∈ ϕ ∈ Φ} strongly
converge to φΦ on D, that is,
(6.1)
∀D ∈ D, ∀ε > 0, ∃ϕ ∈ Φ : (φΦ − φm )D < ε f or all m ∈ ϕ,
and the filter {φΦ |D ; D ∈ D1 } strongly converge to φΦ , that is,
(6.2)
∀ε > 0, ∃D ∈ D such that φΦ − φΦ D < ε.
(2)
Proposition 6.4. Let A = AΦ with domain D(A) = Hloc (RN \N ; C). Then
A is a closed operator from L2loc (RN \N ; C) to L2loc (RN \N ; C).
Proof. Proof follows from Lemma 6.3.
Proof of Theorem 6.1. From Lemma 6.2, Proposition 6.4 and Theorem 4.7 we
obtain Thorem 6.1.
FORMALLY SELF-ADJOINT SCHERÖDINGER OPERATORS
51
6.2. Uniqueness. In this subsection we shall show the uniqueness of TΦ (t) in
(4.3) for the{approximative equation
(2.6).
}
Let Φ = ϕ = {nk } ; nk ∈ N be an ultra-filter of subsequences of natural
numbers.
Theorem 6.5. TΦ (t) does not depend on Φ.
We assume the following Assumption:
Assumption 6.6. AΦ1 ̸= AΦ2 for two ultra-filters Φ1 and Φ2 with Φ1 ̸=Φ2 .
In the following we shall show that Assumption 6.6 implies a contradiction.
We shall begin with several Lemmas.
Lemma 6.7. Suppose TΦ1 (t) ̸= TΦ2 (t). There exists φ0 ∈ C0∞ (RN \N ; C)
satisfies
d
(6.3)
∃t1 > 0, ∃c0 > 0 such that
∥TΦ1 (t)φ0 − TΦ2 (t)φ0 ∥
≥ c0 .
dt
t=t1
Proof. If TΦ1 (t)φ = TΦ2 (t)φ for ∀φ ∈ C0∞ (RN \N ; C), we have TΦ1 (t) =
TΦ2 (t), since C0∞ (RN \N ; C) is dense in L2 (RN ; C). Thus there exisits φ0 ∈
TΦ1 (t)φ0 ̸= TΦ2 (t)φ0 for some t > 0. Since φ0 ∈
C0∞ (RN \N ; C) such that ∩
d
d
N
∞
C0 (R \N ; C) ⊂ D(AΦ1 ) D(AΦ2 ), dt
TΦ1 (t)φ0 and dt
TΦ2 (t)φ0 are continud
ous in t, and hence dt ∥TΦ1 (t)φ0 − TΦ2 (t)φ0 ∥ is continuous in t. Thus there
exists t0 > 0 such that
∫ t0
d
(6.4)
0 < ∥TΦ1 (t0 )φ0 − TΦ2 (t0 )φ0 ∥ =
∥TΦ1 (t)φ0 − TΦ2 (t)φ0 ∥dt.
dt
0
d
If dt
∥TΦ1 (t)φ0 − TΦ2 (t)φ0 ∥ ≤ 0 for 0 ≤ ∀t ≤ t0 , we have ∥TΦ1 (t0 )φ0 −
TΦ2 (t0 )φ0 ∥ ≤ 0. This is cotradiction to (6.4) and (6.3) is verified.
Put φ1 = TΦ1 (t1 )φ0 and φ2 = TΦ2 (t1 )φ0 . (6.3) means
d
(6.5)
∥TΦ1 (t)φ1 − TΦ2 (t)φ2 ∥t=0 ≥ c0 > 0.
dt
Note that φ1 ∈ D(AΦ1 ) and φ2 ∈ D(AΦ2 )
Case 1. In the case that φ2 ∈ D(AΦ1 ).
+
This means ddt TΦ1 (t)φ2 t=0 exists, where
d+
dt
denotes the right derivative:
(f (t + h) − f (t))
d+
f (t) = lim
.
h↓0
dt
h
52
KIYOKO FURUYA
We have
d+
∥TΦ1 (t)φ1 − TΦ1 (t)φ2 ∥
≤ 0,
dt
t=0
(6.6)
or equivalently,
d+
⟨TΦ1 (t)φ2 − TΦ1 (t)φ1 , φ2 − φ1 ⟩
dt
t=0
1 d+
=
∥TΦ1 (t)φ1 − TΦ1 (t)φ2 ∥2 ≤ 0,
2 dt
t=0
since TΦ1 (t) is a contraction. We have by (6.5) and (6.6)
d+
∥TΦ1 (t)φ2 − TΦ2 (t)φ2 ∥t=0
dt
d+
d+
≥
∥TΦ1 (t)φ1 − TΦ2 (t)φ2 ∥t=0 −
∥TΦ1 (t)φ1 − TΦ1 (t)φ2 ∥t=0
dt
dt
> 0.
(6.7)
Re
Hence
d+
⟨(TΦ2 (t) − TΦ1 (t))φ2 , φ2 ⟩
dt
t=0
1 d+
2
=
∥(TΦ2 (t) − TΦ1 (t))φ2 ∥ > 0,
2 dt
t=0
Thus we have AΦ2 φ2 ̸= AΦ1 φ2 . Since C0∞ (RN \N ; C) is dense in L2 (RN ; C),
there exists ψ ∈ C0∞ (RN \N ; C) such that
(6.8)
Re⟨(AΦ2 − AΦ1 )φ2 , φ2 ⟩ = Re
⟨(AΦ2 − AΦ1 )φ2 , ψ⟩ ̸= 0.
Nevertheless, from (6.9) of lemma 6.8 we obtain that
⟨(AΦ2 − AΦ1 )φ2 , ψ⟩ = ⟨φ2 , (t AΦ2 − t AΦ1 )ψ⟩ = −⟨φ2 , (A0 − A0 )ψ⟩ = 0.
This is a contradiction.
Lemma 6.8. Let A0 = A|C0∞ (RN \N ;C) (see Condition 4.3). We have
(6.9)
A0 = −t AΦ1 |C0∞ (RN \N ;C) = −t AΦ2 |C0∞ (RN \N ;C) .
Proof. Proof follows from (4.8) by Lemma 6.2.
C0∞ (RN \N ; C).
Corollary 6.9. Let f ∈ L (R ; C) and ψ ∈
Then ⟨TΦ2 (t)f, ψ⟩
is differentiable in t ≥ 0 :
d
⟨TΦ2 (t)f, ψ⟩ = ⟨TΦ2 (t)AΦ2 f, ψ⟩ = ⟨f, t (TΦ2 (t)t AΦ2 )ψ⟩ = −⟨f, t TΦ2 (t)A0 ψ⟩.
dt
2
N
FORMALLY SELF-ADJOINT SCHERÖDINGER OPERATORS
53
Case 2. In the case that φ2 ∈
/ D(AΦ1 ).
This means
∫
2
(6.10)
∥AΦ1 φ2 ∥ = lim
|AΦ1 φ2 |2 dx = ∞,
D∈δ∈D1
since φ2 ∈ D(AΦ2 ) ⊂
by Proposition 6.4.
(2)
Hloc (RN \N ; C)
D
and AΦ1 φ2 = A0 φ2 ∈ L2loc (RN \N ; C)
Lemma 6.10. There exists δ > 0 such that
d
(6.11)
0 < Re ⟨TΦ2 (t)φ2 − TΦ1 (t)φ1 , ψ⟩t=0 ,
dt
if ∥φ2 − φ1 − ψ∥ < δ and ψ ∈ C0∞ (RN \N ; C).
Proof. Let δ satisfy
0<δ<
c0 ∥φ2 − φ1 ∥
.
2∥AΦ2 φ2 − AΦ1 φ1 ∥
From Lemma 6.7 we have
1 d
c0 ∥φ2 − φ1 ∥ ≤
∥TΦ1 (t)φ1 − TΦ2 (t)φ2 ∥2 t=0
2 dt
d
= Re ⟨TΦ2 (t)φ2 − TΦ1 (t)φ1 , φ2 − φ1 ⟩t=0
dt
≤ Re⟨AΦ2 φ2 − AΦ1 φ1 , ψ⟩ + |Re⟨AΦ2 φ2 − AΦ1 φ1 , φ2 − φ1 − ψ⟩|
≤ Re⟨AΦ2 φ2 − AΦ1 φ1 , ψ⟩ + ∥AΦ2 φ2 − AΦ1 φ1 ∥∥φ2 − φ1 − ψ∥
1
< Re⟨AΦ2 φ2 − AΦ1 φ1 , ψ⟩ + c0 ∥φ2 − φ1 ∥, if ∥φ2 − φ1 − ψ∥ < δ.
2
Thus we obtain that 0 < 21 c0 ∥φ2 − φ1 ∥ < Re⟨AΦ2 φ2 − AΦ1 φ1 , ψ⟩.
Lemma 6.11. Let δ be in Lemma 6.10. Then there exists ψ1 ∈ D(AΦ1 ) with
∥φ2 − φ1 − ψ1 ∥ < δ, such that
d
(6.12) Re ⟨TΦ1 (t)φ2 − TΦ1 (t)φ1 , ψ1 ⟩t=0 = Re⟨AΦ1 φ2 − AΦ1 φ1 , ψ1 ⟩ ≤ 0,
dt
where
d
d
⟨TΦ1 (t)φ2 − TΦ1 (t)φ1 , ψ1 ⟩ = ⟨φ2 − φ1 , t TΦ1 (t)ψ1 ⟩
dt
dt
(
)
d
= ⟨φ2 − φ1 , t TΦ1 (t)ψ1 ⟩ .
dt
Proof. We recall ∥AΦ1 φ2 ∥ = ∞ (see (6.10)). That is, for any L > 0 and
δ in Lemma 6.10, there exists ψε ∈ C0∞ (RN \N ; C) such that |⟨AΦ1 φ2 −
AΦ1 φ1 , ψε ⟩| > L and ∥ψε ∥ < δ/2. Therfore ⟨AΦ1 φ2 − AΦ1 φ1 , eiθ ψε ⟩ < −L
54
KIYOKO FURUYA
for some real θ.
For ψ0 ∈ C0∞ (RN \N ; C) satisfying ∥φ2 − φ1 − ψ0 ∥ < δ/2, put
L := |⟨AΦ1 (φ2 − φ1 ), ψ0 ⟩|.
For ψ1 = ψ0 + e ψε we have ∥φ2 − φ1 − ψ1 ∥ < δ. Hence by Lemma 6.10
iθ
Re⟨AΦ1 φ2 − AΦ1 φ1 , ψ1 ⟩ = Re⟨AΦ1 φ2 − AΦ1 φ1 , ψ0 + eiθ ψε ⟩
≤ |⟨AΦ1 φ2 − AΦ1 φ1 , ψ0 ⟩| + ⟨AΦ1 φ2 − AΦ1 φ1 , eiθ ψε ⟩
≤ L − L = 0.
Lemma 6.12. For ψ1 ∈ C0∞ (RN \N ; C) in Lemma 6.11 we have
d
(6.13)
Re ⟨(TΦ2 (t) − TΦ1 (t))φ2 , ψ1 ⟩t=0 > 0,
dt
d
d
where dt ⟨(TΦ2 (t) − TΦ1 (t)) φ2 , ψ1 ⟩ = dt
⟨φ2 , (t TΦ2 (t) − t TΦ1 (t)) ψ1 ⟩.
Proof. By using (6.11) and (6.12) we get
d
d
0 < Re ⟨TΦ2 (t)φ2 − TΦ1 (t)φ1 , ψ1 ⟩t=0 + Re ⟨TΦ1 (t)φ1 − TΦ1 (t)φ2 , ψ1 ⟩t=0
dt
dt
d
= Re ⟨TΦ2 (t)φ2 − TΦ1 (t)φ2 , ψ1 ⟩t=0 .
dt
On the other hand, from Lemma 6.8 and Corollary 6.9 it follows that
d
Re ⟨TΦ2 (t)φ2 − TΦ1 (t)φ2 , ψ1 ⟩t=0 = Re⟨φ2 , (t AΦ2 − t AΦ1 )ψ1 ⟩
dt
= Re⟨φ2 , (A0 − A0 )ψ1 ⟩ = 0.
This is a contradiction to (6.13).
Thus in both cases we get a contradiction and the proof of Theorem 6.5 is
complete.
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Kiyoko Furuya
Graduate School of Humanities and Science Ochanomizu University
2-1-1 Ōtsuka,bunkyou-ku, Tokyo Japan
E-mail address: [email protected]