Eigenvalue asymptotics for the Schrödinger operator with a $δ$-interaction on a punctured surface

0.14 MB·English

by  P. Exner

Eigenvalue asymptotics for the Schr¨odinger
operator with a δ-interaction on a punctured
4
0
0 surface
2

n
a P. Exnera,b and K. Yoshitomic
J

5
1

2 a) Department of Theoretical Physics, Nuclear Physics Institute,
v Academy of Sciences, 25068 Rˇeˇz, Czech Republic
2
7 b) Doppler Institute, Czech Technical University, Bˇrehova´ 7,
0 11519 Prague, Czech Republic
3
0 c) Department of Mathematics, Tokyo Metropolitan University,
3 Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan
0
[email protected], [email protected]
/
h
p
-
h
MSC numbers: 35J10, 81V99
t
a
KEYWORDS: Schr¨odinger operators, singular interaction, dis-
m
crete spectrum, manifolds, perturbation
:
v
i
X
Abstract. Given n 2, we put r = min i N; i > n/2 . Let Σ be
r ≥ { ∈ }
a a compact, Cr-smooth surface in Rn which contains the origin. Let
further S be a family of measurable subsets of Σ such that
ǫ 0≤ǫ<η
{ }
sup x = (ǫ) as ǫ 0. We derive an asymptotic expansion for
x∈Sǫ| | O →
the discrete spectrum of the Schro¨dinger operator ∆ βδ( Σ S )
ǫ
− − ·− \
in L2(Rn), where β is a positive constant, as ǫ 0. An analogous
→
result is given also for geometrically induced bound states due to a δ
interaction supported by an infinite planar curve.
1
1 Introduction
Schr¨odinger operators with δ-interactions supported by subsets of a lower
dimension in the configuration space have been studied by numerous authors
– see, e.g., [2]–[5] and references therein. Recently such systems attracted
a new attention as models of “leaky” quantum wires and similar structures;
new results have been derived about a curvature-induced discrete spectrum
[7, 8] and the strong-coupling asymptotics [6, 9, 10, 11, 12].
The purpose of this paper is to discuss another question, namely how
the discrete spectra of such operators behave with respect to a perturbation
of the interaction support. Since the argument we are going to use can
be formulated in any dimension, we consider here generally n-dimensional
Schr¨odinger operators, n 2, with a δ-interaction supported by a punctured
≥
surface. On the other hand, we restrict our attention to the situation when
the surface codimension is one and the Schr¨odinger operator in question is
defined naturally by means of the appropriate quadratic form.
Formally speaking, our result says that up to an error term the eigenvalue
shift resulting from removing an ǫ-neighbourhood of a surface point is the
sameasthatofaddingarepulsiveδ interactionatthispointwiththecoupling
constant proportional to the puncture “area”. We will formulate this claim
precisely in Theorem 1 below for any sufficiently smooth compact surface
in Rn and prove it in Section 3. Furthermore, the compactness requirement
is not essential in the argument; in Section 4 we will derive an analogous
asymptotic formula for an infinite planar curve which is not a straight line
but it is asymptotically straight in a suitable sense.
2 The main result
Put r := min i N; i > n/2 . Let Σ be a compact, Cr-smooth surface in
{ ∈ }
Rn which contains the origin, 0 Σ. Let further S be a family of
ǫ 0≤ǫ<η
∈ { }
subsets of Σ which obeys the following hypotheses:
(H.1) The set S is measurable with respect to the (n 1)-dimensional
ǫ
−
Lebesgue measure on Σ for any ǫ [0,η).
∈
(H.2) sup x = (ǫ) as ǫ 0.
| | O →
x∈Sǫ
2
Next we fix β > 0 and define for 0 ǫ < η the quadratic form q by
ǫ
≤
qǫ[u,v] := ( u, v)L2(Rn) β u(x)v(x)dS, u,v H1(Rn);
∇ ∇ − ∈
ZΣ\Sǫ
it is easily seen to be closed and bounded from below. Let H be the self-
ǫ
adjoint operator associated with q . Since Σ S is bounded, we have
ǫ ǫ
\
σ (H ) = [0, ) and ♯σ (H ) < .
ess ǫ disc 0
∞ ∞
By the min-max principle, there exists a unique β∗ 0 such that σ (H ) is
disc 0
≥
non-empty if β > β∗ while σ (H ) = for β β∗. The critical coupling is
disc 0
∅ ≤
dimension-dependent: a straightforward modification of the usual Birman-
Schwinger argument using [5, Lemma 2.3] shows that β∗ = 0 when n = 2,
while for n 3 we have β∗ > 0 by [5, Thm 4.2(iii)]. Since our aimis to derive
≥
asymptotic properties of the discrete spectrum, we will assume throughout
that
(H.3) β > β∗.
Let N be the number of negative eigenvalues of H . Since
0
0 q [u,u] q [u,u] 0 as ǫ 0 for u H1(Rn),
ǫ 0
≤ − → → ∈
there exists η′ (0,η) such that for ǫ (0,η′) the operator H has exactly N
ǫ
∈ ∈
negative eigenvalues denoted by λ (ǫ) < λ (ǫ) λ (ǫ), and moreover
1 2 N
≤ ··· ≤
λ (ǫ) λ (0) as ǫ 0 for 1 j N
j j
→ → ≤ ≤
(see [14, Chap. VIII, Thm 3.15]). Let ϕ (x) N be an orthonormal system
{ j }j=1
of eigenfunctions of H such that H ϕ = λ (0)ϕ for 1 j N. Pick a
0 0 j j j
≤ ≤
sufficiently small a > 0 so that the set x Rn : x < a Σ consists of
{ ∈ | | } \
two connected components, which we denote by B . We have ϕ Hr(B )
± j ±
∈
by the elliptic regularity theorem (see [1, Sec. 10]), because the form domain
H1(Rn) of q is locally invariant under tangential translations along the sur-
0
face Σ. Since r > n/2 by assumption, the Sobolev trace theorem implies
that the function ϕ is continuous on a Σ-neighbourhood of the origin. We
j
also note that one can suppose without loss of generality that ϕ (x) > 0 in
1
Rn. For a given µ σ (H ) we define
disc 0
∈
m(µ) := min 1 j N; µ = λ (0) ,
j
{ ≤ ≤ }
n(µ) := max 1 j N; µ = λ (0) ,
j
{ ≤ ≤ }
C(µ) := ϕ (0)ϕ (0) .
i j
m(µ)≤i,j≤n(µ)
(cid:16) (cid:17)
3
Let s s s be the eigenvalues of the matrix C(µ). In
m(µ) m(µ)+1 n(µ)
≤ ≤ ··· ≤
particular,ifµ = λ (0)isasimpleeigenvalueofH ,wehavem(µ) = n(µ) = j
j 0
and s = ϕ (0) 2. Our main result can be then stated as follows.
j j
| |
Theorem 1 Adopt the assumptions (H.1)–(H.3). Let µ σ (H ), then
disc 0
∈
the asymptotic formula
λ (ǫ) = µ+βmeas (S )s +o(ǫn−1) as ǫ 0
j Σ ǫ j
→
holds for m(µ) j n(µ), where meas ( ) stands for the (n 1)-dimensional
Σ
≤ ≤ · −
Lebesgue measure on Σ.
It should be stressed that our problem involves a singular perturbation
and thus it cannot be reduced to the general asymptotic perturbation theory
of quadratic forms described in [14, Sec. VIII.4]. Indeed, we have
q [u,u] = q [u,u]+βmeas (S ) u(0) 2+ (ǫn) as ǫ 0
ǫ 0 Σ ǫ
| | O →
for u C∞(Rn) and the quadratic form C∞(Rn) u u(0) 2 R does not
∈ 0 0 ∋ 7→ | | ∈
extend to a bounded form on H1(Rn), because the set
u C∞(Rn); u = 0 in a neighbourhoodof the origin
{ ∈ 0 }
is dense in H1(Rn). We eliminate this difficulty by using the compactness
of the map H1(Rn) f f L2(Σ), which will enable us to prove
Σ
∋ 7→ | ∈
Theorem 1 along the lines of the asymptotic-perturbation theorem proof.
Let us remark that our functional-analytic argument has a distinctive
advantage over another technique employed in such situations, usually called
the matching of asymptotic expansions – see [13] for a thorough review –
since the latter typically requires a sort of self-similarity for the perturbation
domains. Our technique needs no assumption of this type.
3 Proof of Theorem 1
We denote R(ζ,ǫ) = (H ζ)−1 for ζ ρ(H ) and R(ζ) = (H ζ)−1 for
ǫ ǫ 0
− ∈ −
ζ ρ(H ). Put κ := 1 dist( µ ,σ(H ) µ ). Since λ ( ) is continuous at
∈ 0 2 { } 0 \ { } j ·
the origin for 1 j N, there is an η (0,η′) such that
0
≤ ≤ ∈
σ(H ) [µ κ,µ+κ] = σ(H ) (µ κ/2,µ+κ/2)
ǫ ǫ
∩ − ∩ −
= λ (ǫ),λ (ǫ),...,λ (ǫ)
m(µ) m(µ)+1 n(µ)
{ }
4
holds if 0 < ǫ η . Choosing the circle C := z C; z µ = 3κ we put
≤ 0 { ∈ | − | 4 }
w (ζ,ǫ) := R(ζ,ǫ)ϕ R(ζ)ϕ for 0 < ǫ η , ζ C.
j j j 0
− ≤ ∈
Our first aim is to check that
wj(ζ,ǫ) H1(Rn) = (ǫ(n−1)/2) as ǫ 0 (1)
k k O →
holds uniformly with respect to ζ C for m(µ) j n(µ). Notice that
∈ ≤ ≤
there exists a K > 0 such that
0
u 2 2 (q ζ)[u,u] +K u 2
k kH1(Rn) ≤ | ǫ − | 0k kL2(Rn)
for ζ C, u H1(Rn), and 0 < ǫ η . This implies that there exists a
0
∈ ∈ ≤
K > 0 such that
1
R(ζ,ǫ)u H1(Rn) K1 u L2(Rn)
k k ≤ k k
for ζ C, u H1(Rn), and 0 < ǫ η . Moreover, by the Sobolev trace
0
∈ ∈ ≤
theorem, there exists a constant K > 0 such that
2
u L2(Σ) K2 u H1(Rn) for u H1(Rn).
k k ≤ k k ∈
Combining these three estimates we get
w (ζ,ǫ) 2
k j kH1(Rn)
2 (qǫ ζ)[wj(ζ,ǫ),wj(ζ,ǫ)] +K0(wj(ζ,ǫ),wj(ζ,ǫ))L2(Rn)
≤ | − |
= 2 β R(ζ)ϕ w (ζ,ǫ)dS +K (q q )[R(ζ)ϕ ,R(ζ,ǫ)w (ζ,ǫ)]
j j 0 0 ǫ j j
− −
(cid:12) ZSǫ (cid:12)
(cid:12) (cid:12)
≤ βk(cid:12)(cid:12)R(ζ)ϕjkL2(Sǫ)(2kwj(ζ,ǫ)k(cid:12)(cid:12)L2(Sǫ) +K0kR(ζ,ǫ)wj(ζ,ǫ)kL2(Sǫ))
4β
= ϕ (2 w (ζ,ǫ) +K R(ζ,ǫ)w (ζ,ǫ) )
3κk jkL2(Sǫ) k j kL2(Sǫ) 0k j kL2(Sǫ)
4β
≤ 3κK2(2+K0K1)kϕjkL2(Sǫ)kwj(ζ,ǫ)kH1(Rn). (2)
Since ϕ = (ǫ(n−1)/2) as ǫ 0 by the assumptions (H.1), (H.2) and
k jkL2(Sǫ) O →
the continuity of ϕ at the origin, we arrive at the relation (1).
j Σ
|
In the next step we are going to demonstrate that the convergence is in
fact slightly faster, namely
sup wj(ζ,ǫ) H1(Rn) = o(ǫ(n−1)/2) as ǫ 0. (3)
k k →
ζ∈C
5
We will proceed by contradiction. Suppose that (3) does not hold; then there
wouldexist aconstantδ > 0, asequence ǫ ∞ (0,η )which tendstozero,
{ i}i=1 ⊂ 0
and ζ ∞ C such that
{ i}i=1 ⊂
ǫ−i (n−1)/2kwj(ζi,ǫi)kH1(Rn) ≥ δ for all i ∈ N. (4)
Notice that the map H1(Rn) f f L2(Σ) is compact due to the
Σ
∋ 7→ | ∈
boundedness of the map H1(Rn) g g H1/2(Σ) and the compactness
Σ
∋ 7→ | ∈
of the imbedding H1/2(Σ) h h L2(Σ) – cf. [15, Chap. 1, Thms 8.3
∋ 7→ ∈
and 16.1]. Since the two sequences
∞ ∞
−(n−1)/2 −(n−1)/2
ǫ w (ζ ,ǫ ) and ǫ R(ζ ,ǫ )w (ζ ,ǫ )
i j i i i i i j i i
i=1 i=1
n o n o
are bounded in H1(Rn), there is a subsequence i(k) ∞ of i ∞ such that
{ }k=1 { }i=1
∞ ∞
−(n−1)/2 −(n−1)/2
ǫ w (ζ ,ǫ ) and ǫ R(ζ ,ǫ )w (ζ ,ǫ )
i(k) j i(k) i(k) i(k) i(k) i(k) j i(k) i(k)
k=1 k=1
n o n o
converge in L2(Σ). Let us denote
g := lim ǫ−(n−1)/2w (ζ ,ǫ ) L2(Σ);
k→∞ i(k) j i(k) i(k) ∈
then we have
−(n−1)/2
ǫ w (ζ ,ǫ )
i(k) j i(k) i(k)
(cid:13) (cid:13)L2(Sǫi(k))
(cid:13) (cid:13) 1/2
(cid:13) (cid:13)
ǫ−(n−1)/2w (ζ ,ǫ ) g + g(x) 2dS 0
≤ (cid:13) i(k) j i(k) i(k) − (cid:13)L2(Σ) ZSǫi(k) | | ! →
(cid:13) (cid:13)
(cid:13) (cid:13)
as k . Similarly we obtain
→ ∞
−(n−1)/2
ǫ R(ζ ,ǫ )w (ζ ,ǫ ) 0 as k .
(cid:13) i(k) i(k) i(k) j i(k) i(k) (cid:13)L2(Sǫi(k)) → → ∞
(cid:13) (cid:13)
Combi(cid:13)ning these result with the inequalit(cid:13)ies (2) we infer that
−(n−1)/2
ǫi(k) kwj(ζi(k),ǫi(k))kH1(Rn) → 0 as k → ∞,
which violates the relation (4); in this way we have proved (3).
6
Now we denote by P the spectral projection of H associated with the
ǫ ǫ
interval (µ 3κ/4,µ+3κ/4). It follows from (3) that
−
√ 1
P ϕ ϕ = − w (ζ,ǫ)dζ
ǫ j j j
− 2π
I|ζ−µ|=3κ/4
= o(ǫ(n−1)/2) in H1(Rn) as ǫ 0
→
holds for m(µ) j n(µ). Consequently, we have
≤ ≤
(HǫPǫϕi,Pǫϕj)L2(Rn) µδi,j βϕi(0)ϕj(0)measΣ(Sǫ)
− −
= q [P ϕ ,P ϕ ] q [ϕ ,ϕ ] βϕ (0)ϕ (0)meas (S )
ǫ ǫ i ǫ j 0 i j i j Σ ǫ
− −
= q [ϕ ,ϕ ] q [ϕ ,ϕ ] q [(I P )ϕ ,(I P )ϕ ]
ǫ i j 0 i j ǫ ǫ i ǫ j
− − − −
βϕ (0)ϕ (0)meas (S )
i j Σ ǫ
−
= q [(I P )ϕ ,(I P )ϕ ]+β ϕ (x)ϕ (x)dS
ǫ ǫ i ǫ j i j
− − −
ZSǫ
βϕ (0)ϕ (0)meas (S )
i j Σ ǫ
−
= o(ǫn−1) (5)
and
(Pǫϕi,Pǫϕj)L2(Rn) = δi,j +o(ǫn−1) (6)
as ǫ 0 for m(µ) i,j n(µ), where we have used, in the last step of (5),
→ ≤ ≤
the assumptions (H.1), (H.2), the continuity of the restrictions ϕ and ϕ
i Σ j Σ
| |
at the origin, and the uniform boundedness of q on H1(Rn) with respect to
ǫ
0 < ǫ η . Let us now introduce the matrices
0
≤
L(ǫ) := ((HǫPǫϕi,Pǫϕj)L2(Rn))m(µ)≤i,j≤n(µ),
M(ǫ) := ((Pǫϕi,Pǫϕj)L2(Rn))m(µ)≤i,j≤n(µ).
Since P ϕ isa basis ofthe spectralsubspace RanP , we see that
ǫ j m(µ)≤j≤n(µ) ǫ
{ }
λ (ǫ),λ (ǫ),...,λ (ǫ)aretheeigenvaluesofthematrixL(ǫ)M(ǫ)−1,
m(µ) m(µ)+1 n(µ)
which by (5), (6) is equal to
L(ǫ)M(ǫ)−1 = µI +βmeas (S )C(µ)+o(ǫn−1),
Σ ǫ
where I stands for the identity matrix. This concludes the argument.
7
4 Perturbation of an infinite curve
As we have mentioned, the compactness of Σ did not play an essential role in
the above argument, and we can use the same technique for punctured non-
compact manifolds of unit codimension as well, as long as the corresponding
Hamiltonian has a discrete spectrum. At present this is known to be true in
the case n = 2 without restriction to the coupling constant β, see [7], and
for n = 3 and β large enough [9].
Weshallthusconsider“puncture”perturbationsofinfiniteasymptotically
straight curves. Let Λ : R R2 be a C2-smooth curve parameterized by its
→
arc length. Fix β > 0 and assume that Λ(0) = 0. Given ǫ 0, we define
≥
tǫ[u,v] := ( u, v)L2(R2) β u(x)v(x)dS, u,v H1(R2).
∇ ∇ − ∈
ZΛ(R\(−ǫ,ǫ))
Let T be the self-adjoint operator associated with the quadratic form t . We
ǫ ǫ
adopt the following assumptions about the curve Λ.
(H.4) The curve Λ is not a straight line.
(H.5) There exists c (0,1) such that Λ(s) Λ(t) c t s for s,t R.
∈ | − | ≥ | − | ∈
(H.6) There exist d > 0, ρ > 1/2, and w (0,1) such that the inequality
∈
Λ(s) Λ(s′)
1 | − | d 1+ s+s′ 2ρ −1/2
− s s′ ≤ | |
| − |
(cid:2) (cid:3)
holds in the sector (s,s′) R2; w < s < w−1 .
∈ s′
From [7, Prop 5.1 and Thm 5.2] we know that under these conditions
(cid:8) (cid:9)
σ (T ) = [ β2/4, ) and 1 ♯σ (T ) .
ess 0 disc 0
− ∞ ≤ ≤ ∞
Let K := j N; j ♯σ (T ) . For j K, we denote by κ (ǫ) the
disc 0 j
{ ∈ ≤ } ∈
j-th eigenvalue of T counted with multiplicity. The function κ ( ) is mono-
ǫ j
·
tone non-decreasing, continuous function in a neighbourhood of the origin.
Let ψ (x) be an orthonormal system of eigenfunctions of T such that
j j∈K 0
{ }
T ψ = κ (0)ψ for j K. Each function ψ is continuous on Λ. For
0 j j j j
∈
µ σ (T ), we define
disc 0
∈
p(µ) := min j K; µ = κ (0) ,
j
{ ∈ }
r(µ) := max j K; µ = κ (0) ,
j
{ ∈ }
D(µ) := ψ (0)ψ (0) .
i j
p(µ)≤i,j≤r(µ)
(cid:16) (cid:17)
8
Let e e e be the eigenvalues of the matrix D(µ).
p(µ) p(µ)+1 r(µ)
≤ ≤ ··· ≤
As in the compact case, if µ = κ (0) is a simple eigenvalue of H , we have
j 0
p(µ) = r(µ) = j and e = ψ (0) 2. The asymptotic behaviour now looks as
j j
| |
follows.
Theorem 2 Assume that (H.4)–(H.6) and take µ σ (T ). Then
disc 0
∈
κ (ǫ) = µ+2βe ǫ+o(ǫ) as ǫ 0
j j
→
holds for p(µ) j r(µ).
≤ ≤
Proof is analogous to that of Theorem 1.
Let us mention in conclusion that the results derived here raise some
interesting questions, for instance, what is the following term in the expan-
sion, what the asymptotic behaviour looks like for non-smooth surfaces, and
whether similar formulae are valid in the case of codimΣ = 2,3 when the
corresponding generalized Schr¨odinger operator has to be defined by means
of appropriate boundary conditions.
Acknowledments
The authors are grateful for the hospitality extended to them, P.E. in the
Department ofMathematics, Tokyo MetropolitanUniversity, andK.Y.inthe
Nuclear Physics Institute, AS CR; during these visits a part of this work was
done. The research has been partially supported by GAAS and the Czech
Ministry of Education within the projects A1048101 and ME482. Useful
comments by the referees are also appreciated.
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