SAITO, YOSHIMI

Osaka City University Repository

Saitδ, Y.
Osaka J. Math.
11 (1974), 295-306
THE PRINCIPLE OF LIMITING ABSORPTION
FOR THE NON-SELFADJOINT SCHRODINGER
OPERATOR IN R2
YOSHIMI SAITO
(Received November 13, 1973)
Introduction
The present paper is a continuation of [3] and is devoted to extending the
results obtained in [3] to the non-selfadjoint Schrϋdinger operator in R2.
In the paper [3] we considered the non-selfadjoint Schrϋdinger operator
(0.1) L=-Σhp-
in RN, where N is a positive integer such that iV#=2, and the complex-valued
function ζ)(#) and the real-valued functions bj(x) (j=ly 2, •••, N) are assumed to
satisfy some asymptotic conditions at infinity. Among others we have shown
the following: Let us define a Hubert space L2β=L2β(RN) (β^R) by
(0.2) Ufi = {/(*)/(l+ \x\yf(X)<EL2(R»)}
with its inner product
(0.3) (f,g)
β
=
and norm
(0.4)
If /ceC+={/cGC//c=|=0 and Im # ^ 0 } does not belong to an exeptional set which
is called the set of the singular points of L, then the operator (L—tc2)'1 is well-
defined as a bounded linear operator from L2tCl+o/2 into L2,-ci+ε)/2
the estimate
(0.5)
Here u=(L—κ2)~1f^L2..(:i+zy2 (/GJL 2 ) ( 1 + ? ) / 2 ) is a unique solution of the equation
(0.6) {L-κ2)u=f
with a sort of "radiation condition", and \\{L—κ,?)~λ\\ means the operator norm
296 Y. SAITO
of (L-K2)-1 from L
 2>Cl+ε)/2 into L2_Cl+ε)/2
l:>
.
In this paper, modifying the method of [3], we shall show that the estimate
(0.5) holds good for L defined in R2 with bj(x)=0yj=l, 2. In our case L takes
the form
(0.7) L = -A+Q(x).
At the same time it will be shown that the other results obtained in [3] also
hold for L in R2. Throughout this paper we shall use the same notations as
in [3]2). For example djU=^-, £)jU=£}fκ>u=djU+(%jl(2r))u-iκZjU, r=\x\9
OX 3
)2 etc.
1. A priori estimates
Let us define a differential operator L in R2 by (0.7), where Q(x) is a
complex-valued function on R2 and L is regarded as an operator from H2loc into
L2yioc* We decompose Q(x) as Q(x)=V0(x)-{-V(x). Throughout this paper the
following is assumed3-*: V0(x) is a real-valued, measurable function such that the
radial derivative exists and
(l.i) IFo^i^qi+kDΛ^-^qi+i^i)-1-8 (XEΞR2).
d\x\
V(x) is a complex-valued, measurable function which satisfies
(1.2) I V(x)\ ^ C ( l + I*!)- 1 " 8 (XEΞR2) .
Here C and δ are positive constants.
Now let us note that with no loss of generality V0(x) can be assumed to
satisfy
(1.3) V0(x) = 0 ( |* | ^ R )
by replacing V
o
 and V with aV0 and (1—a)V0+V, respectively, a(x) being a
real-valued, C°°-function such that
n
. , , , JO(1 4) ( H
Henceforth we assume (1.3) with R=Ί as well as (1.1) and (1.2).
1) In this regard we note that Ikebe-Saίto [1] has shown the boundedness of | |(L —yc2)"1!! for /c
moving in any compact set contained in C+, where L is a self-adjoint Schrδdinger operator in
R-& and N is an arbitrary positive integer.
2) The list of the notation is given in the end of Introduction of [3].
3) This aptίssumon is the same as the one imposed on Q(x) in [3].
PRINCIPLE OF LIMITING ABSORPTION 297
Let £ be a positive number such that 0 < £ ^ l and 0<£^δ/2. As in
Definition 1.2 of [3] we define by Σ = Σ ( L ) = Σ (£> £) the set of the singular
points of L. i .e , ,«GΣ if and only if KeC+= {/ceC\κ4= 0, Im #^>0} and there
exists a non-trivial solution w of the equation
( ί 5) ί (L—tc2)u = 0 ,
For fc^C+ with Im K>0 and the above £ we put
(1.6) D
κ2 = DK= {u<=H2locΠL2I-(1+8)/2/(I-
As is easily seen, Lemma 2.1 and Proposition 2.3, (i), (ii) of [3] are true in R2>
too, and hence we have
Proposition 1.1. Let u^D
κ
 with Λ G C + and Im/c>0. Then u, 3^, 9 2 M£
estimate
(1.7)
a constant C=C(κy L, £)5 ). As a function of /c, C is bounded when K
moves in a compact set contained in \jc e C/Im K. > 0}.
The purpose of this section is to prove the following estimates for u^D
κ
.
Theorem 1.2. Let M be an open set such that MdM
a
={/c^CI\κ\ >a,
Imκ>0} with some a>Q. andMf)*Σ]=φj M being the: closure of M in C. Let
K^M and let u^D
κ
. Then there exists a constant C=C(M, L, 6) such that we
have the estimates
(1.8) ||5to||c-i+β>/2,
(1.9) IMI-Ci+eV^p
where f=(L—κ2)u and Έ?={x^R2\\x\ ^ p } .
REMARK 1.3. Cf. Theorem 2.7 of [3]. In R2 the relation
for M G 5 K is not necessarily true, because u\\x\ (u^H2j0C) is not always square
4) As in [3] we put
2 f| | ^M| | 2 C_I+ S ) /2= Σ \ (1 + \x\)~1+2\&)ju\2dx .y=* i J
5) Here and inlthe sequel we mean by C=C(A, B, •••) that C is a positive constant depend-
ing only on A, B, •••.
298 Y. SAITO
integrable on a neighborhood of the origin x=0. But, of course, for
In order to prove Theorem 1.2 we prepare several propositions. Let us
first show that the above estimates (1.8) and (1.9) can be easily obtained for
if Im K is sufficiently large. Set
(1.10) β0 = max [{2(sup | V0(x)+ V&) I +1)}1/2, sup | V£x) I ]
(V^x) = Re V(x), V2(x) = Im V(x)).
Then we have
Proposition 1.4. Let ueDk with lmκ^β0. Then the estimates
(1.11) ^
and
(1.12)
hold with a constant C0=C0(β0), where f=(L—κ2)u and \\\\ means the usual
L2-norm.
Proof. Take the real and imaginary part of ((L—κ2)u> u)=(fy u) to obtain
(1.13) ± (djU, drf+dKl-Kl+Vo+VJu, u) = Re (/, u),
2
y=»i
(1.14) {{V2-2Klκ2)u, u) = Im (/, u),
where Λ,=Re«, κ2=ϊmκ, and (,) is the L2-inner product. It follows from
(1.10) that
(1.15) «l
and
(1.16) \
for all x^R2. By the use of the relations (1.13)~(1.16) we can show
(1.17) |M%
In fact, if \κ1\ < 1 and κ2^βωwe have from (1.13) and (1.15)
(1.18) IMI^(2/(/9
Λ
)) | |/ | |^(4/(/3
o
k|)) | |/ | | ,
where we should note that | K \ ^  | κ
x
\ -{-fc2^2/c2. If | κ11 ^ 1 and κ2^βQ> we can
PRINCIPLE OF LIMITING ABSORPTION
see from (1.14) and (1.16) that
(1.19) I^IJCIIH
whence we obtain
(1.20) | |«| |έ(|/
299
(1.11) is a direct consequence of (1.17).
Next let us prove (1.12). Since
(1.21)
2|*| +ll««lli
El
and (1.17) has been established, we have only to show
(1.22)
This follows from (1.1), (1.2), (1.13) and (1.17). Q.E.D.
In the rest of this section it is enough to consider u^D
κ
 with 0<Im κ<β0.
Proposition 1.5. Let a be a positive number and let u^D
κ
 with \κ\>a and
0 < Im K < β^where β0 is as above. Then the estimate
(1.23) | |^«| |c-i+,v..β l^C {|M|.C l + f 3 Λ+||/ | |C l + e 5 Λ} {f={L-κ*)u)
holds with a positive constant C=C(a, β
oy L, £).
Proof. It follows from the formula (2.21) given in Lemma 2.5 of [3], which
is true in the case N=2, too, that
<' 2 4 >
r l
=
 \ i-
jBtr 4
l a
2 3r
2 T
d χ
+ Re ( φj{β
r
u)dx
where 0 < ί < 1 < Γ < o o , Bps={x<=R2lp^ \x\ <>s}, r= \x\, φ=Φ(r) is a real-
valued, piecewise continuously differentiable function on [0, oo], and we put in
300 Y. SAITO
(2.21) of [3] cN=c2= —1/4 and B)k(x)=0. Set
(1.25) φ(r) = I _l_ ( 1 + r ) i ( f > 1 )
in (1.24). Then we estimate the both sides of (1.24) as follows:
(1.26) the left-hand side of (1.24)
2=( r\3)
r
u\*dx+\ ^ (
J Bn J B\τ Z
and
(1.27) the right-hand side of (1.24)
Jβ<r 2 \ or or y
\ φ\V\\u\\Wru\dχ+\
Btτ
Let us estimate each / fe. First we obtain
( 1 . 2 8 ) Jk£Ck\\u\\*-<1+t* (A = 1 , 2 ) ,
where C1=C1(β0, 6), C1=C2(L, ε) and we used (1.1). It follows from (1.1) and
(1.3) with i?=7 that
(1.29)
L-o,2,B2)\MU+ty2 (C3'=C3'(L, €)).
On the other hand in quite a similar way to the one used to prove Proposition
2.3, (iii) of [3] we can show
(1.30) K J N a - ^
(C3"=C3"(a, L, 6)) ,
which, together with (1.29), yields
PRINCIPLE OF LIMITING ABSORPTION 301
(1.31) /,ίgC8{||«||icι+8>/2
+ll«ll-ci+«v2ll 0«llc-i+«>/i..Bi
+ H«ll-c1-o/kll/llα+ώΛ} (C3=C3(a, L, 6)).
As t o / 4 and/ 5 we have, using (1.2),
where Ck=Ck{L, S), k=4, 5. Here we should note that rx'χS)rυ)<=LiR2)loc
because u&H2(R*)ιoc is a continuous function on R2 by the Sobolev lemma.
l im/ 6 =0 and l im/ 7 =0 follow from the fact that r(\£)u\2+ | Vo\ \u\") and
r*{2\£)
r
u\2+(\V0\+l/(4 r2))\u\2} are integrable on B1 and Ely respectively.
Summing up these estimates and letting £->0 and T->ooy we arrive at
(1.33) lk 1/ 2(^W) |
.^)} {C'=c\ay βQy L, ε)).
(1.23) is a direct consequence of (1.33). Q.E.D.
REMARK 1.6. In the present paper we assume that the magnetic potentials
bj(x)=0 (j=ly 2). This assumption is used only to prove the above Proposi-
tion 1.5. Technically, it is possible to adopt a weaker assumption. For
example it is enough to assume that B12(x)=d1b2(x)—d2b1(x)=0 in a neighborhood
of the origin x=0.
Proposition 1.7. Let u<=D
κ
 with \ K \ >a {a>0) and 0 < I m K<β0. Then
there exists a constant
C=C(a, β0> Ly S) such that we have
(1.34)
where f=(L—κ2)u.
Proof. The proof will be divided into two steps. In Step I the estimate
(1.34) with p2^1 will be proved, where Proposition 1.5 will be useful. In Step
302 Y. SAITO
II we shall show (1.34) with p=0. From these results we can easily obtain
(1.34) for all p^O.
Step I. We can find positive numbers b and c such that {κ=fc1-\-iκ2l \κ\>a,
2, where
1 1 / | , 2 0 , 4
(1.35) l 2
[
 > oo2</?0, 4-4^
e.g., we may put b=a\A and c=aj2. Consider the case κ^K^ Then, proceed-
ing as in the first half of the proof of Proposition 2.6 of [3], we arrive at (1.34)
with p ^ l . Next consider the case κ^K2. Then we can proceed as in the
second half of the proof of Proposition 2.6 of [3] to obtain (1.34) with p ^ l .
Step II. Set * 0 =(2, 0)e/22 and set for M E D ,
(1.36) u(x) = u(x-x0).
It follows from the relation (L—κ2)u=f that
(1.37) ( L - « 2 ) « = / ,
where
ίL=-A+ΫJix)+Ϋ(x),
(1.38) V0(x) = VJix-xt), Ϋ(x) = V(x-x0),
[f(*)=A*-*o)
Obviously V(x) satisfies (1.2) with the same δ and some positive C. It can be
also shown that V0(x) satisfies (1.1). In fact we have
(I 39) dVo(x)
 =
 ^o(^—^o) d\x—x
o
\
κ
 "
 ;
 d\x\ d\x-x
o
\ d\x\
_ 9^o („ ^\ \x\ — ko|cos<9
— — - X - Xo) j j ,
d\x\ \x—x
o
\
θ being the angle between x and x0. By (1.3) with R=7 -(x—xo)=O for
ό I x I
\x—x
o
\ ^ 5 , and for \x—#0| >5 it follows that
(\x0\/\x\)cosθ 1+2/3 _ g
l*-*ol " i-( l*ol/l*l) 1-2/3
where we have used the fact that | x0 \ /1 x \ <2/3 if | x—x01 >5. Thus we obtain,
together with (1.1),
PRINCIPLE OF LIMITING ABSORPTION 303
d]x\
Hence the result obtained in Step I can be applied to L to show
(1.42) p||L
α +o/2.B l=S<?{yiyPlliα+ 8>/2
^ ^ } (C=C(a, β
n
 L, £)) .
Since the unit disc B
λ
 of R2 is contained in the set {ίcei?2/1 * + Λ O | S: 1}, we have
(1.43) ||«||i
α+βV2 Bl = \ ( l + l ^ i r ' l φ - ^ P ώ
(1+1 *—«b)-χ-1 «(*) I *Λc
Therefore it folloss from (1.42) and the boundedness of ( 1 + | # | ) / ( 1 + \x—x
o
\)
and (1+1 x—x01 )/(l + | x | ) on the whole space R2 that
(1.44) ||iί| |ϊ.α+.)Λ.Λl^C/
(1.34) with p = 0 can be easily obtained from (1.44) and (1.34) with ρ = l .
Q.E.D.
Proof of Theorem 1.2. Set
fΛfi = {A: = fc1+iκ2lκ(ΞM, κ2^β0
[M2 ={fc = κ1
We have M = M 1 U M 2 . For M G D K with Λ E M J we have (1.8) and (1.9) from
(1.12) and (1.11), respectively. Next suppose that u^D
κ
 with « E M 2 . Then,
since the estimates (1.23) and (1.34) have been shown, we can proceed as in the
proof of Theorem 2.7 of [3] to obtain
(1.46) INI-α+oΛ^^rll/llα+o/k (C = C(a, β0, L, €)).
\κ\
(1.8) and (1.9) for « e Z )
κ
 with « G M 2 follow from (1.23), (1.34) and (1.46),
which completes the proof. Q.E.D.
304 Y. SAITO
2. The limiting absorption principle for L
Now that a priori estimates for L have been established (Theorem 1.2), it
can be shown by arguements quite similar to those used in §3 and §4 of [3] that the
main results of [3] hold in our case, too. We sum up these in the following
three theorems whose proof will be omitted.
Theorem 2.1 (the properties of the set Σ of the singular points of L). Let
(1.1) and (1.2) be satisfied and let 0 < £ ^ m a x (1, δ/2).
(i) Then the set Σ — Σ (L, £) of the singular points of L is a bounded set of
C+={«GC//cφ0, Im/c^0}. Σ Λ = Σ Γ I > B is a bounded set with the Lebesgue
measure 0.
(ii) For any α > 0 Σ n { / ί G C + / | / ί | ^a} is a compact set of C+y Further,
Σ ~ ~ Σ Λ is an isolated, bounded set having no limit point in {tc e C+/Im K > 0}.
(iii) Let /C<ΞC+ and Im κ>0. Then ι > e Σ *7 and only if κ2 belongs to the
point spectrum of H, where H is a densely defined, closed linear operator in L2 given
by
(2.1)
V J
For / c e C + — Σ with Im/c>0 belongs to the resolvent set ofH.
Theorem 2.2. (the limiting absorption principle for L). Let (1.1) and
(1.2) be satisfied and let 6 and Σ be as above. Assume that M is an open set of C
such that Mf] Σ = Φ and MczM
a
 with some a>0, M being the closure of M and
M
a
 being given as in Theorem 1.2.
(i) Then for any pair ( Λ , / ) G M X L 2 I ( 1 + 5 ) / 2 there exists a unique solution
u=u(κ, f) of the equation
I
[Hu = Lu.
(2 2) 0C
ΠL2t _
(ii) The solution u=u(κ, f), (/c, f)^M xL2,\1+1!y2, satisfies the estimates
11 ^ * ^-^
(2.3)
with a positive constant C=C(M, L> S).
6) D(T) is the domain of T.
PRINCIPLE OF LIMITING ABSORPTION 305
(iii) If we define an operator (L—K2)'1 by
(2.4) (L-KTV = «(«. /) C/eL,,
 t l + ΰ Λ)
for κ^My then (L—A:2)"1 is a B(L2Cl+2)/2y L2 ^Cl^o/2)-valuedy continuous function on
MΌy and we have
(2.5) ||(L-<rΊI^ SL («εί , C = C(M, L, 6)),
where \\(L—K2)'^] ifiαzro ίAβ operator norm of B(L2tCl+t)/29 L2 _Cl+βy2).
(iv) (L—K 2 ) ' 1 ^C(L2fci+β)/2> ^ 2,-ci+ε)/2)8:>. Moreover we have the following',
let {f
n
} be any bounded sequence of L2til+ty2 and let {κn} be any sequence contained
in M. Then the sequence {(L—/c^)"1/^} ώ relatively compact in L2 -Cl+(!y2.
(v) (L—K2)'1 is a B(L2fCl+t)/29 L2Cl+ti)/2)-valued, analytic function on M.
Finally let us show some properties of the spectrum σ{H) of H defined by
(2.1). Its point spectrum, continuous spectrum and residual spectrum are
denoted by σp(H), σc(H) and <rr(H), respectively. We define the essential
spectrum <r
e
(H) of H as in [3]9).
Theorem 2.3 (the properties of σ(H)). Let (1.1) and (1.2) be satisfied and
let H be as defined in (2.1). Then we have the following (i)^(iv):
( i )
(ii)
(iii) <r(H)Γ\(C-[0, oo))(Zσ,(H) and σp(H)Π(0, oo)=φ, and hence
σ
e
(H)=)(0, oo).
(iv) The eigenvalues in C— [0, oo), if they exist
 y are of finite multiplicity and
they form an isolated, bounded set having no limit point in C— [0, oo).
OSAKA CITY UNIVERSITY
References
[1] T. Ikebe and Y. Saito: Limiting absorption method and absolute continuity for the
Schrδdinger operator, J. Math. Kyoto Univ. 12 (1972), 513-542.
7) B(X, Y) denotes the set of all bounded linear operators on X into Y, X and Y being Banach
spaces.
8) C(X, Y) is all compact operators on X into F, where X and Y are Banach spaces.
9) See (1.20) of [3].
10) (iv) can be obtained from the fact that in our case the relation Hu = 0 (uEίD(H)) is equi-
valent to H*U=0, where H* is the adjoint of H and U{x) is the conjugate of u(x). See
Mochizuki [2], Remark 1.1 (p. 425). See also Remark 1.8 of [3].
306 Y. SAITO
[2] K. Mochizuki: Eigenfunction expansions associated with the Schrδdinger operator
with a complex potential and the scattering theory, Publ. RIMS Kyoto Univ. Ser.
A 4 (1968), 419-466.
[3] Y. Saitδ: The principle of limiting absorption for the non-self adjoint Schrδdinger
operator in RN(Nφ2), Publ. Res. Inst. Math. Sci. Kyoto Univ. 9 (1974), 397-428.


THE PRINCIPLE OF LIMITING ABSORPTION FOR THE NON-SELFADJOINT SCHRODINGER OPERATOR IN R^2

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THE PRINCIPLE OF LIMITING ABSORPTION FOR THE NON-SELFADJOINT SCHRODINGER OPERATOR IN R^2

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