R E S E A R C H
Open Access
Coupling constant limits of Schrödinger
operators with critical potentials
Xiaoyao Jia
1*and Yan Zhao
2*Correspondence:
1Mathematics and Statistics School,
Henan University of Science and Technology, No. 263, Luo-Long District, Kai-Yuan Road, Luoyang City, Henan Province 471023, China Full list of author information is available at the end of the article
Abstract
A family of Schrödinger operators,P(
λ
) =P0+λ
V, is studied in this paper. Here P0= –+f(x) withf(x)∼|x1|2 when|x|is large enough andV(x) =O(|x|–2–) for some
> 0. We show that each discrete eigenvalue ofP(
λ
) tends to 0 whenλ
tends to someλ
0. We get asymptotic behavior of the smallest discrete eigenvalue whenλ
tends toλ
0.Keywords: Schrödinger operator; critical potential; asymptotic expansion
1 Introduction
In this paper, we consider a family of Schrödinger operatorsP(λ) which are the perturba-tion ofPin the form
P(λ) =P+λV forλ≥
onL(Rd),d≥. HereP= –+qr(θ). (r,θ) are the polar coordinates onRd, andq(θ) is a
real continuous function.V≤ is a non-zero continuous function satisfying
V(x)≤Cx–ρ for someρ
> . ()
Herex= ( +|x|)/. Let
sdenote the Laplace operator on the sphereSd–. Assume
that
–s+q(θ) > –
(d– )
onLSd–. ()
If () holds, thenP≥ inL(Rd) (see []).
Under the assumption onV, we know thatP(λ) has discrete eigenvalues whenλis large enough, and each discrete eigenvalue tends to zero whenλtends to someλ(see
Sec-tion ). We study the asymptotic behaviors of the discrete eigenvalues ofP(λ) in this paper. The asymptotic behaviors for Schrödinger operators with fast decaying potentials were studied by Klaus and Simon []. In [], they studied the convergence rate of discrete eigen-values ofH(λ) = –+λVwhenλ→λ.λis the value at which some discrete eigenvalue ei(λ) tends to zero. The main method they used in their paper is the Birman-Schwinger
technique.
this paper, we first show that there exists someλsuch that whenλ>λ,P(λ) has
dis-crete eigenvalues. Then, we define the Birman-Schwinger kernelK(α) forP(λ) and find that there is one-to-one correspondence between the discrete eigenvalues ofK(α) and the discrete eigenvalues ofP(λ). Hence, the asymptotic expansion of the discrete eigenvalue ofP(λ) can be got through the asymptotic expansion of the discrete eigenvalue ofK(α). In our main results, we need to use thatK(α) is a bounded operator fromLtoL. To get
that, we add a strong condition onV (i.e.,ρ> in ()). We show thatK(α) is a family of
compact operators converging toK() and obtain the asymptotic expansions of the dis-crete eigenvalues ofK(α) by functional calculus. After that, the convergence rate of the smallest discrete eigenvalue ofP(λ) is obtained.
Here is the plan of our work. In Section , we recall some results ofPand define the
Birman-Schwinger kernelK(α) forP(λ). The relationship between the eigenvalues of these two kinds of operators is studied. In Section , we first study the asymptotic behavior of the discrete eigenvalues ofK(α). Then the convergence rate of the smallest discrete eigenvalue ofP(λ) is obtained. We get the leading term and the estimate of the remainder term of the smallest discrete eigenvalue.
Let us introduce some notations first.
Notation The scalar product onL(R+;rd–dr) andL(Rd) is denoted by·,·and that
onL(Sd–) by (·,·).Hr,s(Rd),r∈Z,s∈R, denotes the weighted Sobolev space of orderr
with volume elementxsdx. The duality betweenH,sandH–,–sis identified with the
Lproduct. DenoteH,s=L,s. NotationL(Hr,s,Hr,s) stands for the space of continuous
linear operators fromHr,stoHr,s. The complex planeCis slit along positive real axis so
thatzν=eνlnzandlnz=ln|z|+iargzwith <argz< πare holomorphic there.
2 Some results forP0
Assume that (r,θ) are the polar coordinates onRd. Then the condition
–s+q(θ) > –
(d– )
onLSd–
implies
–+q(θ)
r ≥ ()
inL(Rd) (see []).
Now, we recall some results on the resolvent and the Schrödinger group for the unper-turbed operatorP. Let
σ∞=
ν;ν=
λ+(d– )
,λ∈σ
–s+q(θ)
.
Denote
Forν∈σ∞, letnνdenote the multiplicity ofλν=ν–(d–)
as the eigenvalue of –s+q(θ).
Letϕν(j),ν∈σ∞, ≤j≤nνdenote an orthogonal basis ofL(Sd–) consisting of
eigenfunc-tions of –s+q(θ):
–s+q(θ)
ϕ(νj)=λνϕ(νj),
ϕν(i),ϕν(j)=δij.
Letπνdenote the orthogonal projection inL(Sd–) onto the subspace spanned by the
eigenfunctions of –s+q(θ) associated with the eigenvalueλν, and letπν(i) denote the
orthogonal projection inL(Sd–) onto the eigenfunctionϕ(i)
ν :
πνf = nν
j=
f,ϕ(νj)⊗ϕν(j), f ∈LSn–,
πν(i)f =f,ϕν(i)⊗ϕν(i), f ∈LSd–, ≤i≤nν.
Denote forν∈σ∞
zν=
zν ifν∈/N,
zlnz ifν∈N.
Hereν=ν– [ν], and [ν] is the largest integer which is not larger thanν. Forν> , let [ν]–
be the largest integer strictly less thanν. Whenν= , set [ν]–= . Defineδνbyδν= , if
ν∈σ∞∩N,δν= , otherwise. One has [ν] = [ν]–+δν.
The following is the asymptotic expansion for the resolventR(z) = (P–z)–.
Theorem .(Theorem . []) The following asymptotic expansion holds for z near zero withz> :
R(z) =δlnzG,π+
N
j= zjFj+
ν∈σN zν
N–
j=[ν]–
zjGν,j+δνπν+R
(N) (z)
inL(–,s; , –s),s> N+ .Here
Gν,j(r,τ) =
bν,j(rτ)j+ν
fj–[ν](r,τ;ν), ν∈/N,
–(irτj!)jfj–[ν](r,τ; ), ν∈N, Fj∈L(–,s; , –s), s> j+ ,
R(N)(z) =O|z|N+∈L(–,s; , –s), s> N+ ,> .
Here
bν,j= –
ije–iνπ/( –ν)
ν(ν+ )· · ·(ν+j)
for≤ν< ,and
fj(r,τ,ν) = (rτ)–
with Pj,ν(ρ)a polynomial inρof degree j:
Pj,ν(ρ) =
ija ν
j!
–
ρ+ θ
j
–θν–dθ, a
ν= –
e–iπ ν/
ν+π/(ν+ /).
First, we show thatP(λ) has discrete eigenvalues whenλis large enough. In fact, we need only to show that there exists a functionψ∈L(Rd) such thatψ,P(λ)ψ< .
From the assumption onV, we know that there exists a pointx∈Rdsuch thatV(x) = infx∈RdV(x). Chooseδ> small enough such that for allx∈B(x,δ),V(x) <
V(x). For
ψ∈C∞(Rd),ψ(x)= ,suppψ⊂B(x
,δ), one has
ψ,P(λ)ψ=ψ,Pψ+λψ,Vψ<ψ,Pψ+
λ
V(x),
whenλis large enough, one hasψ,P(λ)ψ< . This means thatP(λ) has discrete eigen-values whenλis large enough.
P(λ) has a continuous spectrum [,∞) forλ≥ becauselim|x|→∞V(x) exists and equals
zero (see []). We know thatσ(P()) =σ(P) = [,∞). Hence, from the continuity of a
dis-crete spectrum ofP(λ), we know that there exists someλsuch that whenλ>λ,P(λ) has
eigenvalues less than zero, and whenλ≤λ,σ(P(λ)) = [,∞). So,P(λ) has an eigenvalue e(λ) < at the bottom of its spectrum forλ>λ. In Section (Proposition .), we prove
thate(λ) is simple and the corresponding eigenfunction can be chosen to be positive
ev-erywhere. (There are many results about the simplicity of the smallest eigenvalue of the Schrödinger operator without singularity, but there is no result which can be used directly, because the potential we use in this paper has singularity at zero. Theorem XIII. [] can treat the Schrödinger operator with the potential which has singularity at zero, but the positivity of potential is needed. Hence, we give this result.) From the discussion above and the continuity of a discrete spectrum, one has thate(λ) tends to zero at someλ. The
asymptotic behavior ofe(λ) is studied in this paper.
To study the eigenvalues ofP(λ), we first define a family of Birman-Schwinger kernel operators. Let
K(z) =|V|/(P–z)–|V|/
forz∈/σ(P), and
K() =|V|/F|V|/.
Then we have the following result.
Proposition . Letα< .Then
(a)Let
A=ψ∈LRd;P(λ) –αψ= ,
B=φ∈LRd;K(α)φ=λ–φ.
Then|V|/is injective from A to B,and(P
(b)The multiplicity ofαas the eigenvalue of P(λ)is exactly the multiplicity ofλ–as the eigenvalue of K(α).
Proof
(a) First, we prove that|V|/is injective fromAtoB. Note that ifψ∈A, then K(α)φ=λ–φ
withφ=|V|/ψ. And ifφ= , then
ψ= –λ(P–α)–Vψ=λ(P–α)–|V|/φ= .
It follows that|V|/is injective fromAtoB.
Next, we show that (P–α)–|V|/is injective fromBtoA. Ifφ∈B, then
P(λ) –αψ= , withψ= (P–α)–|V|/φ.
And ifψ= , then
=|V|/ψ=K(α)φ=λ–φ.
It follows that (P–α)–|V|/is injective fromBtoA.
(b) From (a), one hasdimA=dimB. This means that the multiplicity ofαas the eigen-value ofP(λ) is exactly the multiplicity ofλ–as the eigenvalue ofK(α). From the last proposition, we know that there exists one-to-one correspondence be-tween the discrete eigenvalues of P(λ) and the discrete eigenvalues ofK(α). Hence, we can study the eigenvalues ofK(α) first.
3 Asymptotic expansion of the eigenvalues
IfPandVare defined as above, we show that ifP+Vhas the eigenvalue less than zero,
then the smallest eigenvalue ofP+Vis simple. We use Theorems XIII., XIII. [] to
prove it.
Proposition . Suppose P+V has an eigenvalue at the bottom of its spectrum.Then this eigenvalue is simple and the corresponding eigenfunction can be chosen to be a positive function.
Proof Let ≤χ(t)≤ be a smooth nonincreasing function such thatχ(t) = if|t|< andχ(t) = ift> . Letχn(t) =χ(t/n). SetVn=χn(r)qr(θ)+V–x,H= –+x,Hn=
H+Vn,H=P=P+V. From the proof of Theorem XIII. [], we know thate–tH is
positivity preserving and{e–tH} ∪L∞acts irreducibly onL. Hence, by Theorem XIII.
[], ifHnconverges toHandH–Vnconverges toHin the strong resolvent sense, then e–tHis positivity preserving and{e–tH} ∪L∞acts irreducibly onL. By Theorems XIII.
and XIII. [], we can get the result. SinceC∞(Rd) is the core for allP
nandP, and for
anyψ∈C∞(Rd),V
nψ→(qr(θ)+V)ψ inL, then we have the necessary strong resolvent
Proposition . Assume that /∈σ∞.PFu=u in H–,sfor any u∈H–,s,s> . Proof Ifu∈H–,s, thenF
u∈H,–s. For any test functionφ∈C∞ (Rn), we havePFu,φ=
u,FPφ. If /∈σ∞, then we havelimz→(P–z)–=F inH–,s forz> . It follows
u,FPφ=limz→u, (P–z)–Pφ=limz→u,φ–z(P–z)–φ=u,φbecauseφand Pφbelong toH–,s. Hence,PFu=uinH–,s.
Proposition . Assume that /∈σ∞.K(α)is a compact operator forα≤.And K(α)
converges to K()in operator norm sense.
Proof Forα< ,K(α) =|V|/(P–α)–|V|/. Since (P–α)–is a bounded operator from L(Rd) toH(Rd), andV is a compact operator fromH(Rd) toL(Rd), thenV(P
–α)–
is a compact operator onL(Rd). Using a similar method to that in Proposition ., we
can show thatV(P–α)–andK(α) have the same non-zero eigenvalues, and for the same
eigenvaluee(α), the multiplicity ofe(α) as the eigenvalue ofV(P–α)–and the multiplicity
ofe(α) as the eigenvalue ofK(α) are the same. Hence,K(α) is a compact operator. Because
K(α) –K() =|V|/(P–α)––F
|V|/=|V|/R() |V|/
and ifρ> , then|V|/R() |V|/=o(|α|) inL(Rd). Hence,K(α)→K() in operator
norm sense asα→. This means thatK() is a compact operator.
Lemma . Suppose A,Aare two bounded self-adjoint operators on a Hilbert space H. Set
μn(Ai) = inf φ,...,φn
sup ψ=,ψ∈[φ,...,φn]⊥
(ψ,Aiψ),
then|μn(A) –μn(A)| ≤ A–A. Proof By the definition ofμn(Ai), one has
μn(A) –μn(A)
= inf
φ,...,φn
sup ψ=,ψ∈[φ,...,φn]⊥
ψ,Aψ– inf
φ,...,φn
sup ψ=,ψ∈[φ,...,φn]⊥
ψ,Aψ
=– sup
φ,...,φn
– sup ψ=,ψ∈[φ,...,φn]⊥
ψ,Aψ
+ sup
φ,...,φn
– sup ψ=,ψ∈[φ,...,φn]⊥
ψ,Aψ
≤ sup
φ,...,φn
– sup ψ=,ψ∈[φ,...,φn]⊥
ψ,Aψ+ sup
ψ=,ψ∈[φ,...,φn]⊥
ψ,Aψ
= sup
φ,...,φn
sup
ψ=,ψ∈[φ,...,φn]⊥
ψ,Aψ– sup
ψ=,ψ∈[φ,...,φn]⊥
ψ,Aψ
≤ sup
φ,...,φn
sup ψ=,[φ,...,φn]⊥
ψ, –Aψ–ψ, –Aψ
≤ A–A.
Lemma . Suppose T(α)is a family of compact self-adjoint operators on a separable Hilbert space H,and T(α) =T+o(|α|)forαnear zero.Set
μk(α) = inf φ,...,φk
sup ψ=,ψ∈[φ,...,φk]⊥
ψ,T(α)ψ.
Then:
(a)μk(α)is an eigenvalue of T(α),andμk(α)converges whenα→.Moreover,ifμk(α)→
μk,thenμkis an eigenvalue of T.
(b)Suppose that E= is an eigenvalue of Tof the multiplicity of m.Then there are m eigenvalues(counting multiplicity),Ek(α) (≤k≤m),of T(α)near E.Moreover,we can choose{φk(α); ≤k≤m}such that(φk(α),φj(α)) =δkj(≤k,j≤m),φk(α)is the
eigenvec-tor of T(α)corresponding to Ek(α) (Ek(α)→E),andφk(α)converges asα→.Ifφk(α)
converges toφk,thenφkis the eigenvector of Tcorresponding to E. Proof
(a) By the min-max principle, we know that μk(α) is an eigenvalue of T(α). By
Lemma ., one has
μk(α) –μk()≤T(α) –T=O
|α|.
It follows thatμk(α) converges to the eigenvalue ofT.
(b) BecauseTis a compact operator andE= is an eigenvalue ofT, thenEis a
dis-crete spectrum ofT. Then there exists a constantδ> small enough such thatThas only
one eigenvalueEinB(E,δ) (={z∈C;|z–E|<δ}). Forαsmall enough,T(α) has exactly meigenvalues (counting multiplicity) inB(E,δ) because the eigenvalues ofT(α) converge
to the eigenvalues ofTby part (a) of lemma. Suppose themeigenvalues, nearE, ofT(α)
areE(α),E(α), . . . ,Em(α), and the corresponding eigenvectors areψ(α),ψ(α), . . . ,ψm(α)
such thatψk(α),ψj(α)=δkj. Let
Pα= –
πi
|E–E|=δ
T(α) –E–dE.
ThenPα=
m
k=·,ψk(α)ψk(α). LetPα(k)=·,ψk(α)ψk(α), thenPα=
m
k=P(αk). Forαnear
zero, one has
Pα–P=
–
πi
|E–E|=δ
T(α) –E–– (T–E)–dE
=– πi
|E–E|=δ
T(α) –E–(T–Tα)(T–E)–dE
=O|α|.
LetA={φ;φ= ,φ∈RanP}. Letφkbe an element inAsuch thatφ–ψk(α)acquires
the minimum value. Then we have
Pα(k)φk–φk≤P(αk)φk–ψk(α)+ψk(α) –φk
and
ψk(α) –φk≤
Pψk(α) Pψk(α)
–ψk(α)
≤ Pψk(α) Pψk(α)
–Pψk(α)
+Pψk(α) –Pαψk(α)=O
|α|.
In the last equality, we use the fact
Pψk(α)=Pψk(α) –ψk(α)
+ψk(α)= +O
|α|,
and
Pψk(α) Pψk(α)
–Pψk(α)
=Pψk(α)
Pψk(α)
– =O|α|.
It follows thatP(k)
α φk–φk=O(|α|). Letφk(α) = P
(k)
α φk Pα(k)φk
. Thenφk(α),φj(α)= fork=
jbecauseP(k)
α P
(j)
α = ifk=j, andφk(α) –φk ≤ Pα(k)φk–φk+( – P(αk)φk
)P(k)
α φk=
O(|α|). This ends the proof.
Let <α<α<· · ·<αi<· · ·<αmand
T(β) =T+
m
i=
βαi(lnβ)δiT
i+Tr(β).
Here, δi = or , T≥,Ti (≤i ≤m), Tr(β) are compact operators, and Tr(β) =
O(|β|αm+) forβnear zero. Set
es= inf φ,...,φs
sup ψ=,ψ∈[φ,...,φs]⊥
ψ,Tψ.
Then, by the min-max principle,esis an eigenvalue ofT. Moreover, ifes= , thenesis a
discrete eigenvalue ofTbecauseTis a compact operator. Ifes= is an eigenvalue ofT
of multiplicitym, without loss, we can suppose thates=es+=· · ·=es+m–. Then there exist
exactlymeigenvalues (counting multiplicity),es(β),es+(β), . . . ,es+m–(β), ofT(β) neares.
By Lemma ., we know that there exists a family of normalized eigenvectors{φj(β);j=
s,s+ , . . . ,s+m– }ofT(β) such thatT(β)φj(β) =ej(β)φj(β),φj(β),φk(β)=δjk (j,k=
s,s+ , . . . ,s+m– ), andφj(β) (j=s,s+ , . . . ,s+m– ) converge asβ→. Suppose that
φj(β) converge toφjfor alljsuch thatej= . Thenφs,φj=δsj.{φs}can be extended to a
standard orthogonal basis. Set
T(β) =
m
s=
βαs(lnβ)δiT
s+Tr(β), Tsj(β) =
φs,T(β)φj
.
Lemma . T(β),esare given as before.Then the eigenvalue of T(β),ej(β) (j=s,s+, . . . ,s+
m– )has the following form:
ej(β) =es+
∞
n=a (j)
n(β)
∞
n=b (j)
n(β)
.
Here
a(j)(β) =Tjj(β),
a(j)(β) = –
{k;ek=es}
(ek–es)–Tjk(β)Tkj(β),
a(j)(β) =
k=j=l
(ek–es)–(el–es)–Tjk(β)Tkl(β)Tlj(β)
–
{k;ek=es}
(ek–es)–Tjk(β)Tkj(β)Tjj(β),
a(nj)(β) = –(–)
n
πi
|E–es|=δ
(es–E)–
i,i,...,in
(ei–E) –· · ·(e
in–E) –
×TjiTii· · ·TinjdE for n> , b(j)(β) = ,
b(j)(β) = ,
b(j)(β) =
{k;ek=es}
(es–ek)–Tjk(β)Tkj(β),
b(nj)(β) = –(–)
n
πi
|E–es|=δ
(es–E)–
i,i,...,in–
(ei–E) –· · ·(e
in––E) –
×TjiTii· · ·Tin–jdE for n> .
Proof Ifes= , thenesis the discrete eigenvalue ofT. Suppose that the multiplicity ofes
ism, and suppose thates=es+=· · ·=es+m–as before. Hence, we can chooseδ> small
enough such that there is only one eigenvalueesinB(es,δ) ={z∈C;|z–es|<δ}. We know
thatej(β) (j=s,s+ , . . . ,s+m– ) converge toes. It follows that ifδis small enough, there
are exactlymeigenvalues (counting multiplicity) ofT(β) inB(es,δ) forβsmall. Set
Pβ–
πi
|E–es|=δ
T(β) –E–dE.
Then
ej(β) =
φj,T(β)Pβφj φj,Pβφj
=φj,TPβφj
φj,Pβφj
+φj,T(β)Pβφj
φj,Pβφj
=es+
φj,T(β)Pβφj φj,Pβφj
Since
T(β) –E–= (T–E)–
I+T(β)(T–E)–
–
= (T–E)– ∞
n=
(–)nT(β)(T–E)–
n
,
then
φj,Pβφj= –
πi
|E–es|=δ
φj, (T–E)– ∞
n=
(–)nT(β)
(T–E)–
n
φj
dE.
Then
b(nj)(β) = –(–)
n
πi
|E–es|=δ
φj, (T–E)–
T(β)
(T–E)–
n
φj
dE.
In particular,
b(j)(β) = – πi
|E–es|=δ
φj, (T–E)–φj
dE= ,
b(j)(β) = –
– πi
|E–es|=δ
φj, (T–E)–
T(β)
(T–E)–
φj
dE
= –– πi
|E–es|=δ
(es–E)–
φj,T(β)φj
dE= ,
b(j)(β) = –(–)
πi
|E–es|=δ
φj, (T–E)–
T(β)
(T–E)–
φj
dE
= – πi
|E–es|=δ
(es–E)–
φj,
T(β)
(T–E)–
T(β)
φj
dE
= – πi
|E–es|=δ
(es–E)–
k
φj,
T(β)
φk
(ek–E)–
φj,
T(β)
φk
dE
= – πi
|E–es|=δ
(es–E)–
{k;ek=es}
φj,
T(β)
φk
(ek–E)–
φj,
T(β)
φk
dE
– πi
|E–es|=δ
(es–E)–
{k;ek=es}
φj,
T(β)
φk
(ek–E)–
φj,
T(β)
φk
dE
=
{k;ek=es}
(es–ek)–Tjk(β)Tkj(β).
In the last step, we use that
{k;ek=es}
(ek–ej)–Tjk(β)Tkj(β)≤C
{k;ek=es}
Tjk(β)
+Tkj(β)
≤CT*(β)φj
+T(β)φj
Similarly, we can get
b(nj)(β) = –(–)
n
πi
|E–es|=δ
(es–E)–
i,i,...,in–
(ei–E) –· · ·(e
in––E) –
×TjiTii· · ·Tin–jdE;
and
a(j)(β) =Tjj(β),
a(j)(β) = –
{k;ek=es}
(ek–es)–Tjk(β)Tkj(β),
a(j)(β) =
{l,k;ek=ej=el}
(ek–es)–(el–es)–Tjk(β)Tkl(β)Tlj(β)
–
{k;ek=es}
(ek–es)–Tjk(β)Tkj(β)Tjj(β),
a(nj)(β) = –(–)
n
πi
|E–es|=δ
φj,
T(β)
(T–E)–
n+
φj
dE
= –(–)
n
πi
|E–es|=δ
(es–E)–
φj,
T(β)
(T–E)–
n
T(β)φj
dE
= –(–)
n
πi
|E–es|=δ
(es–E)–
i
φj,
T(β)
(T–E)–
n–
×T(β)φi
(es–E)–Tij(β)dE
=· · · = –(–)
n
πi
|E–es|=δ
(es–E)–
i,i,...,in
(ei–E) –· · ·(e
in–E) –
×TjiTii· · ·TinjdE.
First, we study the asymptotic expansion of the smallest eigenvaluee(λ) of P(λ). By
Proposition ., we know thate(λ) is a simple eigenvalue ofP(λ), and the corresponding
eigenfunction can be chosen to be positive. We supposeu(λ) is a positive eigenfunction corresponding toe(λ). Then u˜(λ) =|V|/u(λ)∈L(Rd). Without loss of generality, we
can suppose that˜u(λ)L(Rd)= . Then we can get the following result.
Lemma . Assume that /∈σ∞. Setν=min{ν;ν∈σ∞}.u is defined as above˜ .Then
˜
u(λ)converges in L(Rd)whenλ→λ.Ifν< andu˜(λ)converges toφ,thenφ is the eigenfunction of K(),andφ,|V|/Gν,πν|V|
/φ = .
Proof By the assumption ofu˜(λ), one hasK(e(λ))u˜(λ) =λ–u˜(λ). One can check thatu˜(λ)
converges inL(Rd) asλ→λ
by Lemma .. And also, by Lemma ., we know thatφis
the normalized eigenfunction ofK() corresponding toE.φis a positive function since
˜
because|V|/u=|V|/F
|V|/φ=K()φ=λ–φ. Then
φ,|V|/Gν,πν|V| /φ
=λ|V|/u,|V|/Gν,πν|V|
/|V|/u
=λVu,Gν,πνVu
=λCνVu,|y| –n– +νϕ
ν
= .
In the last equality, we use the fact that
Gν,= (rτ)
νb
ν,f=dνbν,(rτ)
–d– +ν=C
ν(rτ) –d– +ν
withdν= –
e– iπ ν
ν+(ν+), andCν=dνbν,. This ends the proof.
Theorem . Assume that /∈σ∞.φis defined in Lemma..Ifρ> ,one of three ex-clusive situations holds:
(a)Ifσ=∅,then
e(λ) = –c(λ–λ) +o(λ–λ) with c= (λF|V|
φ)–= .
(b)Ifν= ,then
e(λ) = –c
λ–λ ln(λ–λ)
+o
λ–λ ln(λ–λ)
with c=λ– φ,|V|/G
,π|V|/φ–= .
(c)Ifν< ,then e(λ) =c
(λ–λ)
ν+o(λ–λ)
ν
with c=λ– φ,|V|/G
ν,πν|V|
/φ–= . Proof
(a) By Theorem ., one has
R(α) =F+αF+R() (α), inL(–,s; , –s),s> .
Then ifρ> , we can getK(α) =K() +|V|/(αF+R() (α))|V|/inL(, ; , ). Because e(λ) is the simple eigenvalue ofP(λ), thenλ–is the simple eigenvalue ofK(e(λ)). Since V ≤, one has thatP(λ) is monotonous with respect toλ and so is thee(λ). Hence, K(e(λ)) and the eigenvalues ofK(e(λ)) are monotonous with respect toλ. Therefore,
we have that λ–is the biggest eigenvalue ofK(e(λ)). If not, suppose thata>λ–is an
eigenvalue ofK(e(λ)), then by the continuity and monotony of the eigenvalue ofK(e(λ))
It follows thate(λ) <e(λ) is an eigenvalue ofP+λV. This is contradictory to thate(λ)
is the smallest eigenvalue. By Lemma ., we know the normalized eigenfunctionu˜(λ) of
K(e(λ)) converges toφ. It followsu˜(λ) =PPλλφφ withPλ= –
πi
|E–E|=δ(K(e(λ)) –E)
–dE.
Then
μe(λ)
=u˜(λ),Ke(λ)
˜
u(λ)=φ,K(e(λ))Pλφ
φ,Pλφ
.
Hereμ(e(λ)) is the eigenvalue ofK(e(λ)) corresponding to the eigenfunctionu˜(λ). By
Lemma ., we should computeφ,|V|/F
|V|/φ. Letψ=F|V|/φ. From the
defini-tion ofφ, one hasK()φ=λ–φ. Hence,
(P+λV)ψ=PF|V|/φ+λVF|V|/φ= .
In the last equality, we use the factPF|V|/φ=|V|/φ, which can be obtained by
Propo-sition .. Sinceν> , we haveψ∈L(Rd) by Theorem . []. So,ψis the ground state
ofP(λ). We also have
|V|/ψ=K()φ=λ– φ, (P–α)–|V|ψ=λ–
ψ+αR(α)ψ
. Hence,
|V|/F|V|/φ=λ|V|/F|V|/|V|/ψ
=λα–|V|/
R(α) –F
|V|ψ+O|α|
=λα–|V|/λ–
ψ+αR(α)ψ–ψ
+O|α|
=|V|/R(α)ψ+O
|α|.
It follows
φ,|V|/F|V|/φ
=lim
α→
|V|/φ,R(α)ψ
=F|V|/φ,ψ
=ψ= .
So, μ(α) =λ– +cα+o(|α|+) with c=F|V|
φ. By the Proposition ., one has
μ(e(λ)) =λ–. It follows
λ–=λ– +ce(λ) +O
|e(λ)|+
.
Sinceλ–=λ–
–λ– (λ–λ) +O(|λ–λ|), we can get the leading term ofe(λ) is –c(λ–λ)
withc= (λF|V| φ)–.
(b) Ifν= , then
By Lemma ., one has
φ,|V|/G,π|V|/φ
= . Then we have
μ(α) =λ– +cαlnα+o(α)
withc=φ,|V|/G,π|V|/φ. As in (a), usingμ(e(λ)) =λ–and
λ–=λ– –λ– (λ–λ) +O
|λ–λ|
,
one has –λ–(λ–λ) +O(|λ–λ|) =ce(λ)lne(λ) +o(e(λ)). To get the leading term of e(λ), we can suppose thate(λ) = (λ–λ)f(λ). Then, by comparing the leading term, we
can getf(λ) = /ln(λ–λ). It follows
e(λ) = –c
λ–λ ln(λ–λ)
+o
λ–λ ln(λ–λ)
withc=λ– φ,|V|/G
,π|V|/φ–.
(c) Ifν< , one has K(α) =K() +
<ν≤
αν|V|/Gν,πν|V|/+O
|α|.
By Lemma ., we know thatφ,|V|/G
ν,πν|V|/φ–= . Using the same argument as
before, we can conclude
μ(α) =λ– +cαν+o
|α|ν
withc=φ,|V|/Gν,πν|V|/φ. As above, we can get that
e(λ) =c(λ–λ)
ν +o(λ–λ)
ν
withc=λ–
φ,|V|/Gν,πν|V|/φ–.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the manuscript and read and approved the final manuscript.
Author details
1Mathematics and Statistics School, Henan University of Science and Technology, No. 263, Luo-Long District, Kai-Yuan
Road, Luoyang City, Henan Province 471023, China. 2College of Civil Engineering and Architecture, Zhejiang University,
B505 Anzhong Building, 866 Yuhangtang Road, Hangzhou, Zhejiang Province 310058, China.
Acknowledgements
This research is supported by the Natural Science Foundation of China (11101127,11271110) and the Natural Science Foundation of Educational Department of Henan Province (2011B110014).
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doi:10.1186/1687-2770-2013-62
