Inverse Problems For Dirac Operators With a Finite Number of Transmission Conditions

0.15 MB·

by  Yalçın Güldü

INVERSE PROBLEMS FOR DIRAC OPERATORS WITH A
FINITE NUMBER OF TRANSMISSION CONDITIONS
6
1
YALC¸INGU¨LDU¨ ANDMERVEARSLANTAS¸
0
2
n Abstract. Inthispaper,weconsideradiscontinuousDiracoperatordepend-
a ingpolynomiallyonthespectralparameterandafinitenumberoftransmission
J conditions. We get some properties of eigenvalues and eigenfunctions. Then,
we investigate some uniqueness theorems by using Weyl function and some
0
spectral data.
2

]
A
C
.
h 1. Introduction
t
a
Inverse problems of spectral analysis recover operators by their spectral data.
m
Fundamental and vast studies about the classical Sturm-Liouville, Dirac opera-
[ tors,Schr¨odingerequation and hyperbolic equations are well studied (see [1-7]and

1 references therein).
v Studies where eigenvalue dependent appears not only in the differential equa-
0 tion but also in the boundary conditions have increased in recent years (see [8-
2
16] and corresponding bibliography cited therein). Moreover,boundary conditions
3
which depend linearly and nonlinearly on the spectral parameter are considered
5
0 in [8,16-20] and [21-27] respectively Furthermore, boundary value problems with
. transmission conditions are also increasingly studied. These types of studies in-
1
troduce qualitative changes in the exploration. Direct and inverse problems for
0
6 Sturm-Liouville and Dirac operators with transmission conditions are investigated
1 in some papers (see [7, 28–31]and references therein). Then, differential equations
v: with the spectral parameter and transmission conditions arise in heat, mechanics,
i mass transfer problems, in diffraction problems and in various physical transfer
X
problems (see [18, 28, 32-39]and corresponding bibliography).
r Morerecently, someboundary value problemswith eigenparameterinboundary
a
and transmission conditions are spread out to the case of two, more than two or a
finite number of transmission in [40-44]and references therein.
The present paper deals with the discontinuous Dirac operator depending poly-
nomiallyonthe spectralparameteranda finite number oftransmissionconditions.
The aim of the present paper is to obtain the asymptotic formulae of eigenvalues,
eigenfunctions and to prove some uniqueness theorems. Especially, some param-
eters of the considered problem can be determined by Weyl function and some
spectral data.
2000 Mathematics Subject Classification. Primary34A55, Secondary34B24,34L05.
Keywordsandphrases. Diracoperator,Eigenvalues,Eigenfunctions,TransmissionConditions,
WeylFunction.
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2 YALC¸INGU¨LDU¨ ANDMERVEARSLANTAS¸
We consider a discontinuous boundary value problem L with function ρ(x);
(1)
ly :=By′(x)+Ω(x)y(x)=λρ(x)y(x), x∈∪n (ξ ,ξ )=Λ, ξ =a, ξ =b
i=0 i i+1 0 n+1
ρ , a≤x<ξ
0 1
where ρ(x)= ρ , ξ <x<ξ , i=1,n−1 and ρ , i=0,n are given positive
i i i+1 i

ρ , ξ <x≤b
 n n
p(x) q(x)
real numbers;Ω(x)= ,p(x),q(x),r(x)∈L [Λ,R];
q(x) r(x) 2
(cid:18) (cid:19)
0 1 y (x)
B = ,y(x)= 1 ,λ∈Cisarealspectralparameter;boundary
−1 0 y (x)
2
(cid:18) (cid:19) (cid:18) (cid:19)
conditions at the endpoints
(2) l y :=a (λ)y (a)−a (λ)y (a)=0
1 2 2 1 1
(3) l y :=b (λ)y (b)−b (λ)y (b)=0
2 2 2 1 1
with transmission conditions at n points x=ξi, i=1,n
(4) l y :=y (ξi+0)−θ y (ξ −0)=0
3 1 i 1 i
(5) l y :=y (ξ +0)−θ−1y (ξ −0)−γ (λ)y (ξ −0)=0
4 2 i i 2 i i 1 i
where θ , ξi i=1,n are real numbers, a (λ), b (λ),(i=1,2) and γ (λ), are real
i i i i
coefficients polynomials for λ.
(cid:0) (cid:1)
2. Operator Formulation and Properties of Spectrum
In this section, we present the space
n
H :=L2(Λ)⊕Cma⊕Cmb⊕ Cri wherema =max{degai(λ)},mb =max{degbi(λ)},
i=1
X
r =max{degγ (λ)}. We define the norm on space H by
i i
b ma
(6) kYk2 := ρ(x) |y (x)|2+|y (x)|2 dx+ Y1Y1
1 2 i i
Za n o Xi=1
mb n n
+ Y2Y2+ Y3jY3j
j j i i
j=1 j=1i=1
X XX
for
Y ∈H,Y = f(x),Y1,Y2,Y3 ,Y1 = Y1,Y1,...,Y1 ,Y2 = Y2,Y2,...,Y2 ,
1 2 ma 1 2 mb
Y3 = Y3i,Y3i,...,Y3i i=1,n .
1 2 (cid:0) ri (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Consider the operator T defined by the domain
(cid:0) (cid:1) (cid:0) (cid:1)
y (x)
i) y(x)= 1 ∈AC(Λ), lf ∈L (Λ)
y (x) 2
2
 (cid:18) (cid:19) 
D(T)=Y ∈H : iiiivi)i))yYY11(21ξ=i=+ba0m)a−2yyθ2(i(bya)1)−(−ξiba−m0ay)1y=(1b(0)a) 
1 mb2 2 mb1 1
 v) Y13i =γriiy1(ξi−0) 
INVERSE PROBLEMS FOR DIRAC OPERATORS 3
such that
TY :=W =(ly,W1,W2,W3I˙), W1 = W11,W21,...,Wm1a ,
W2 = W12,W22,...,Wm2b , W3I˙ = (cid:0)W13i,W23i,...,Wr3ii(cid:1),Wr3n+1 i=1,n
(cid:0) (cid:1) (cid:16) (cid:17) (cid:0) (cid:1)
Wi1 =Yi1+1−ama−i,2y2(a)+ama−i,1y1(a), i=1,ma−1
W1 =−a y (a)+a y (a)
ma 02 2 01 1
Wj2 =Yj1+1−bmb−j,2y2(b)+bmb−j,1y1(b), j =1,mb−1
W2 =−b y (b)+b y (b)
mb 02 2 01 1
W13 =YK31+1−γr1−k,1y1(ξ1−0), k =1,r1−1
W23 =YK32+1−γr2−k,2y1(ξ2−0), k =1,r2−1
.
.
.
Wr3n =YK3n+1−γrn−k,ny1(ξn−0), k=1,rn−1
W3 =−γ y (ξ −0)+y (ξ +0)−θ−1y (ξ −0), t=1,n
rn+1 0t 1 t 2 t i 2 t
Thus, we can rewrite the considered problem (1)-(5) i(cid:0)n the op(cid:1)erator form as
TY =λY.
We define the solutions
ϕ(x,λ)=ϕ (x,λ), x∈(ξ ,ξ ); ψ(x,λ)=ψ (x,λ), x∈(ξ ,ξ ) i=0,n
i i i+1 i i i+1
(cid:0) (cid:1)
ϕ (x,λ)=(ϕ (x,λ),ϕ (x,λ))Tandψ (x,λ)=(ψ (x,λ),ψ (x,λ))T , i=0,n
1 i1 i2 1 i1 i2
of equation (1) by the initial conditions
(cid:0) (cid:1)
(7) ϕ (a,λ)=a (λ), ϕ (a,λ)=a (λ)
01 2 02 1
and similarly;
(8) ψ (b,λ)=b (λ), ψ (b,λ)=b (λ)
n1 2 n2 1
respectively.
Lemma 1 The following asymptotic behaviours hold for |λ|→∞:
am22λm2cosλρ0(x−a)+o(λm2exp|Imλ|(x−a)ρ0, dega2(λ)>dega1(λ)
 −am11λm1sinλρ0(x−a)+o(λm1exp|Imλ|(x−a)ρ0, dega1(λ)>dega2(λ)
ϕ (x,λ)=
01  λm(am22cosλρ0(x−a)−am11sinλρ0(x−a))
+o(λmexp|Imλ|(x−a)ρ , dega (λ)=dega (λ)
0 1 2
 am22λm2sinλρ0(x−a)+o(λm2exp|Imλ|(x−a)ρ0, dega2(λ)>dega1(λ)
 am11λm1cosλρ0(x−a)+o(λm1exp|Imλ|(x−a)ρ0, dega1(λ)>dega2(λ)
ϕ (x,λ)=
02  λm(am11cosλρ0(x−a)+am22sinλρ0(x−a))
+o(λmexp|Imλ|(x−a)ρ , dega (λ)=dega (λ)
0 1 2

4 YALC¸INGU¨LDU¨ ANDMERVEARSLANTAS¸
−γr11am22λr1+m2cosλρ0(ξ1−a)sinλρ1(x−ξ1)
 +o(λr1+m2exp|Imλ|((ξ1−a)ρ0+(x−ξ1)ρ1), dega2(λ)>dega1(λ)
ϕ11(x,λ)= +γro1(1λλrr11++mm1(e−xγaprm1|1I2am2mcλ1o1|sλ(λ(rξρ11+0(−mξ11as−)inρa0λ)+ρs0in((xξλ1−ρ−1ξ(a1x))−sρi1nξ)1,λ)ρd1e(gxa−1(ξλ1))>dega2(λ)
 −++γoar(m1λ11ra11m+si2mn2λλerxρ1p0+(|ξmI1m2−cλo|as()λ(sξρi1n0(−λξρ1a1−)(ρxa0)−+siξn(1xλ)−ρ1ξ(1x)−ρ1ξ)1,)dega1(λ)=dega2(λ)
 +o(λr1+m2exp|Imλ|((ξ1−a)ρ0+(x−ξ1)ρ1),dega2(λ)>dega1(λ)
ϕ11(x,λ)= +γro1(1λarm11+1mλ1r1e+xmp1|Isminλλ|ρ(0((ξξ11−−aa))ρs0in+λ(ρx1−(xξ−1)ξρ11)),dega1(λ)>dega2(λ)
γr11λr1+m(−am22cosλρ0(ξ1−a)sinλρ1(x−ξ1)+am11sinλρ0(ξ1−a)sinλρ1(x−ξ1)
 +γro1(1λarm12+2mλre1x+pm|2Imcoλs|λ(ρ(ξ01(ξ−1−a)aρ)0c+os(λxρ−1(ξx1−)ρξ11)),dega1(λ)=dega2(λ)
 +o(λr1+m2exp|Imλ|((ξ1−a)ρ0+(x−ξ1)ρ1), dega2(λ)>dega1(λ)
ϕ12(x,λ)= +−oγ(rλ11ra1+mm111λerx1p+m|I1msiλn|λ((ρξ01(−ξ1a−)ρa0)c+os(xλρ−1(ξx1)−ρ1ξ)1,) dega1(λ)>dega2(λ)
γr11λr1+m(am22cosλρ0(ξ1−a)cosλρ1(x−ξ1)am11sinλρ0(ξ1−a)cosλρ1(x−ξ1)
 +γro1(1λγrr12+2amme2x2pλr|1Im+rλ2+|m((2ξ1co−sλaρ)0ρ(0ξ+1−(xa−)siξn1)λρρ11)(,ξ2d−egξa11)(sλin)λ=ρ2d(exg−a2ξ(2λ))
 +o(λr1+r2+m2exp|Imλ|((ξ1−a)ρ0+(ξ2−ξ1)ρ1+(x−ξ2)ρ2), dega2(λ)>dega1(λ)
ϕ21(x,λ)= +−γroγ1(1rλ1γ1rrγ12+2r2λr22ra+1m+mr1121+eλxmr1p(+a|rIm2m+22λmc|1o(s(siξnλ1ρλ−0ρ(0aξ(1)ξρ1−0−a+)a(s)ξisn2inλ−ρλ1ξρ1(1)ξ(ρ2ξ12−+−ξ1(ξ)x1s)−isniξnλ2ρλ)2ρρ(22x)(,x−−dξe2ξg)2a)1(λ)>dega2(λ)
 −+−γoa(rmλ111r1γ1+rs2irn22a+λmmρ202e(λξx1rp1−|+Imra2)+λsm|in(2(λcξo1ρs1−λ(ξρa20)(−ρξ01ξ+−1)(asξi)2ns−λinρξλ21ρ()1xρ(−1ξ2+ξ−2()x)ξ1−)cξo2s)λρ2ρ)2,(xde−gξa21)(λ)=dega2(λ)
 +o(λr1+r2+m2exp|Imλ|((ξ1−a)ρ0+(ξ2−ξ1)ρ1+(x−ξ2)ρ2), dega2(λ)>dega1(λ)
ϕ22(x,λ)= +γγrro11(11λγγrrr122+22aλr2mr+11+m1rλ12r+e1xm+pr(2−|+Imamm1λ2s|2i(nc(oξλ1sρλ−0ρ(ξa01()ξ−ρ10−a+)as()iξns2iλn−ρλ1ξρ1(1ξ)2(ρξ1−2+−ξ1()ξx1c)−ocsoξλ2sρ)λ2ρρ(22x)(,−xd−ξe2g)ξ2a)1(λ)>dega2(λ)
 ++oa(mλ1r11+sirn2+λmρ0e(ξx1p−|Ima)λs|in((λξ1ρ1−(ξa2)−ρ0ξ+1)(cξo2s−λρξ21()xρ1−+ξ2()x−ξ2)ρ2), dega1(λ)=dega2(λ)
INVERSE PROBLEMS FOR DIRAC OPERATORS 5
,...,
n
(−1)nam22λA+m2 × γrii ×cosλρ0(ξ1−a)sinλρn(x−ξn)×
 !
i=1
ϕn1(x,λ)= +×(+×−oo 1((λλ)iiYY==nnnAA+22++1ssmmiiann12mλλee1ρρxx1iippλ−−A||11II+mm((ξξmiiλλ1−−||×((((ξξYbb ii−−−−iY11=n))ξξ1!!nnγ))riρρinn!++sinXi=nnλ1ρ((0ξξ(iiξ−1−−ξξiia−−)11s))iρρnii−−λρ11))n,,(xddee−ggξaan12)((×λλ))>>ddeeggaa21((λλ))
i=1
 +λ×Ao+(λmA+×+((m−− e11Yix=))nnnp1+γa|Ir1mmiai2!m2λc|1×1o((ssbiλn−ρλ0ξρ(nξ01)(ξρ−1n−a+)asXi)i=nnsX1iλn(ρξλniρ−(nxξ(−xi−−ξ1n)ξ)ρni×−)1× ),Yi =nd2iY=nes2ignsaiλn1ρ(λiλ−ρ)1i−=(1ξid(ξ−eig−ξai−2ξ1(i−λ)!1))! 
6 YALC¸INGU¨LDU¨ ANDMERVEARSLANTAS¸
n
(−1)n+1am22λA+m2 × γrii cosλρ0(ξ1−a)cosλρn(x−ξn)×
 i=1 !
ϕn2(x,λ)= +×(+×−oo 1((λλ)iiYY==nnnAA22a++ssmmmiinn1121λλeeλρρxxAiipp−−+||11mIImm((1ξξ×iiλλ−−|| ((((ξξiYbbii=n−−−−1Y11γ))ξξr!!nini))!ρρsnnin++λXiρ=nn01(ξ((1ξξii−−−aξξ)ii−−co11s))λρρρii−−n11())x,, −ddeeξggn)aa12×((λλ))>>ddeeggaa21((λλ))
i=1
X
 +λ×Ao+(λmA(+−×+(m1− )en1iYx=+n)pn11γ|aaIrmmmii21!λ21|c×s(ion(sbλλ−ρρ00ξ((nξξ)11ρ−−naa+))cXci=oonss1λλ(ξρρinn−((xxξ−i−−1ξξ)nnρ))i××−1 ),YiiY==nnd22essgiinnaλλ1ρρ(iλi−−)11=((ξξdiie−−gξaξii2−−(11λ))!)! 
where A=r +r +···+r .
1 2 n
Lemma 2 The following asymptotic behaviours hold for |λ|→∞:
bm42λm4cosλρn(x−b)+o(λm4exp|Imλ|(x−b)ρn, degb2(λ)>degb1(λ)
 −bm31λm3sinλρn(x−b)+o(λm3exp|Imλ|(x−b)ρn, degb1(λ)>degb2(λ)
ψ (x,λ)=
n1  λmb(bm42cosλρn(x−b)−bm31sinλρn(x−b))
+o(λmbexp|Imλ|(x−b)ρn), degb1(λ)=degb2(λ)

bm42λm4sinλρn(x−b)+o(λm4exp|Imλ|(x−b)ρn, degb2(λ)>degb1(λ)
 bm31λm3cosλρn(x−b)+o(λm3exp|Imλ|(x−b)ρn, degb1(λ)>degb2(λ)
ψ (x,λ)=
n2  λmb(bm42sinλρn(x−b)+bm31cosλρn(x−b))
+o(λmbexp|Imλ|(x−b)ρn), degb1(λ)=degb2(λ)

INVERSE PROBLEMS FOR DIRAC OPERATORS 7
bm42γrnnλrn+m4cosλρn(ξn−b)sinλρn−1(x−ξn)
 +o(λrn+m4exp|Imλ|((ξn−b)ρn+(x−ξn)ρn−1),degb2(λ)>degb1(λ)
ψn−1,1(x,λ)= +−obm(λ3r1nγ+rnmn3λerxnp+|mIm3sλin|(λ(ρξnn(−ξnb−)ρbn)+sin(xλρ−n−ξn1)(xρn−−1ξ)n,) degb1(λ)>degb2(λ)
γrnnλrn+mb(−bm31sinλρn(ξn−b)+bm42cosλρn(ξn−b))sinλρn−1(x−ξn)
 +o(λrn+mbexp|Imλ|((ξn−b)ρn+(x−ξn)ρn−1), degb1(λ)=degb2(λ)
−bm42γrnnλrn+m4cosλρn(ξn−b)cosλρn−1(x−ξn)
 +o(λrn+m4exp|Imλ|((ξn−b)ρn+(x−ξn)ρn−1), degb2(λ)>degb1(λ)
ψn−1,2(x,λ)= b+mo3(1λγrrnn+nmλ3rne+xmp3|Isminλλ|ρ(n(ξ(nξn−−b)b)ρnco+sλ(ρxn−−1ξ(nx)ρ−n−ξn1)), degb1(λ)>degb2(λ)
γrnnλrn+mb(bm31sinλρn(ξn−b)−bm42cosλρn(ξn−b))cosλρn−1(x−ξn)
 +o(λrn+mbexp|Imλ|((ξn−b)ρn+(x−ξn)ρn−1), degb1(λ)=degb2(λ)
−bm42γrnnγrn−1n−1λrn+rn−1+m4×
 ×cosλρn(ξn−b)sinλρn−1(ξn−ξn−1)sinλρn−2(x−ξn−1)
ψn−2,1(x,λ)= ,+b×+moods3((ie1nλλgγrrλrnnbnρ2++nn(rrγ(λnnξr−−)nn11−>++−1mmndb−43e)1geesλxxibnrpp1nλ||(+IIλρmmrn)n−−λλ11||+(((((mξξξn3nn×−−−ξbbn))−ρρ1nn)++sin((ξξλnnρ−−n−ξξ2nn(−−x11−))ρρξnnn−−−111++) ((xx−−ξξnn−−11))ρρnn−−22))
, degb (λ)>degb (λ)
1 2
 ×+γ,ron(d(bneλmγgrr3nbn11+−s(r1iλnnn−)−λ11=ρ+λnmrd(nbξe+negxrb−np2−|(b1Iλ)+m)−mλbb|sm(i(n4ξ2λncρ−ons−bλ2)ρρ(nnx(+ξ−n(ξ−ξnn−b−1)))ξsnin−λ1)ρnρ−n−11(ξ+n−(xξ−n−ξ1n)−×1)ρn−2)
8 YALC¸INGU¨LDU¨ ANDMERVEARSLANTAS¸
bm42γrnnγrn−1n−1λrn+rn−1+m4×
 ×cosλρn(ξn−b)sinλρn−1(ξn−ξn−1)cosλρn−2(x−ξn−1)
ψn−2,2(x,λ)= ,+−×+obodsm((ienλλ3grr1λnnbγρ2++rnn(rr(λnnnξ−−)γn11r>++n−−mmd1b43en)gee−sxxib1npp1λλ||r(IInλρmm+n)−rλλn1||−(((1((ξ+ξξnmnn−3−−×ξbbn))−ρρ1nn)++sin((ξξλnnρ−−n−ξξ2nn(−−x11−))ρρξnnn−−−111++) ((xx−−ξξnn−−11))ρρnn−−22))
, degb (λ)>degb (λ)
1 2
 ×γ,+ron(d(−neλγgbrrmnbn1+−31(r1λnns−)i−n11=+λλmρrdnnbe+(egξxrbnnp2−−|(1Iλ+mb)m)λb+|c(ob(msξnλ42ρ−cno−bs)2ρλ(nρxn+−(ξ(ξnξnn−−−1b))ξs)nin−λ1)ρρnn−−11(ξ+n(−xξ−n−ξn1)−×1)ρ2)
,...,
n
(−1)nbm42λr2+...+rn+m4 × γrii ×
 i=2 !
ψ11(x,λ)= ××+(−ocs1(ionλ)snrλλ−2ρ+ρ1nn.b.((.m+ξξnnr3n1−−λ+rmbb2)+)4ss.e.iix.nn+pλrλ|nρρI+1m1m((xλx3|−×−((Yξξ ξ222Yi))−=n××2ξγ 1 r)iiYiiYρ=!=nn133×s+siinnXi=λnλρ3ρii(−−ξ1i1−((ξξiiξ−i−−1ξξ)ii−−ρ11i−))!!1), degb2(λ)>degb1(λ)
n
 ×+λ+roo 2((+λλiY.=n.rr.223+++sri..n..n..+++λmrrρnnbi++−×mm1 3b(ξeeiYixx=n−pp2γ||ξIIrimmi−i!1λλ)||×!(((((sξξi22n−−+(λ−ρξξ(11−11)))(n1xρρ−)11−n1++bbξmm2XXii4)==3n21×33cs((oiξξnsiiλλ−−ρρnnξξi(i(−−ξξnn11))−−ρρiibb−−))11))),, ddeeggbb11((λλ))=>ddeeggbb22((λλ))
INVERSE PROBLEMS FOR DIRAC OPERATORS 9
n
(−1)n−1bm42λr2+...+rn+m4 × γrii ×
 !
i=2
ψ12(x,λ)= +××(−ocs1(ionλ)snrλλ2bρ+ρmnn..3(.(1+ξξλnnrnr−−2++mbb..)4).+cceoroxnssp+λλ|mρIρm131(×(λxx| −−((iYξ=ξξn2222Y))γ−×r×iξi1 ! )iYiYρ×==nn133+ssiinnXi=nnλλ3ρρi(i−−ξi11−((ξξiξii−−−1ξξ)ii−−ρ1i1−))!1!), degb2(λ)>degb1(λ)
 +×+λroo 2((+λλYi.=n.rr.223+++sri..n..n..+++λmrrρnnbi++−×mm1 3b(ξeeiYixx=n−pp2γ||ξIIrimmi−i!1λλ)||×!((((cξξo22s−−+λρξξ((111−−))(11xρρ))11n−n++−bξ1mX2Xiib==)n3m133×4s((2iξξnciioλ−−sρλnξξiiρ(−−ξn11n())ξ−ρρniib−−−)11b))),, ddeeggbb11((λλ))=>ddeeggbb22((λλ))
10 YALC¸INGU¨LDU¨ ANDMERVEARSLANTAS¸
n
(−1)n−1bm42λA+m4 × γrii ×
 !
i=1
ψ01(x,λ)= ×+(×−ocs1(ionλ)snAλλb+ρρmnmn3((41ξξλennxA−−p+|mbbI)m)3ss×iiλnn| λλ((ρρiYξ=0n011(Y(γ−xxr−i−ai!)ξξρ11×0))+××X i =nnYii2Y==nn(22ξssiiin−nλλξρρi−ii−−11)1ρ((ξξi−ii−1−),ξξii−d−1e1)g)!!b2(λ)>degb1(λ)
 +×λ+Aoo +((λλYim=nAAb2++×smmin 3bλeeYiρxx=nipp1−γ||1IIrmmi(iξ!iλλ−||×((((ξsξξii−11n1−−λ)ρ!aa0))(ρρx00−+++(ξ(XXii−1−==n)2121×))((nnξξ−iib1−−mb3ξξm1ii−−4s2i11nc))oλρρsρii−−λn11ρ())ξn,,n(ξ−ddneeb−gg)bbb11)((λλ))=>ddeeggbb22((λλ))