10.1098/rspa.2001.0851
On the number of negative eigenvalues of Schr¨ odinger operators with an Aharonov–Bohm magnetic field By A. A. Balinsky1 , W. D. Evans1 a n d R. T. L e w i s2 1
School of Mathematics, Cardiff University, 23 Senghennydd Road, Cardiff CF24 4YH, UK (BalinskyA@cardiff.ac.uk; EvansWD@cardiff.ac.uk) 2 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA (
[email protected]) Received 6 March 2001; accepted 11 May 2001
It is proved that for V+ = max(V, 0) in the subspace L1 (R+ ; L∞ (S1 ); r dr) of L1 (R2 ), there is a Cwikel–Lieb–Rosenblum-type inequality for the number of negative eigenvalues of the operator ((1/i)∇ + A)2 − V in L2 (R2 ) when A is an Aharonov–Bohm magnetic potential with non-integer flux. It is shown that the L1 (R+ , L∞ (S1 ), r dr)norm cannot be replaced by the L1 (R2 )-norm in the inequality. Keywords: Schr¨ odinger operator; Aharonov–Bohm magnetic potential; CLR-type inequality; Sobolev-type inequality
1. Introduction The spectral analysis of the Schr¨ odinger operator −∆ − V in L2 (R2 ) is strongly influenced by three related facts. (i) The Sobolev space H 1 (R2 ) is not continuously embedded in L∞ (R2 ). (ii) The Hardy inequality |u(x)|2 dx const. |∇u(x)|2 dx, 2 |x| n n R R
u ∈ C0∞ (Rn \ {0})
(1.1)
does not hold for n = 2. (iii) There is no Cwikel–Lieb–Rosenblum (CLR) inequality of the form V+ (x) dx, V+ = max(V, 0), N (−∆ − V ) const. R2
(1.2)
where N (−∆ − V ) denotes the number of negative eigenvalues of −∆ − V . In 1999, Laptev & Weidl obtained the elegant result that (ii) does hold when n = 2 if the gradient (1/i)∇ is replaced by the ‘magnetic’ gradient (1/i)∇ + A, where curl A is, in particular, a magnetic field of Aharonov–Bohm type. To be specific, they prove, inter alia, the following (cf. their theorem 3). In polar coordinates (r, θ), let A(r, θ) =
Ψ (θ) (sin θ, − cos θ), r
(1.3) c 2001 The Royal Society
Proc. R. Soc. Lond. A (2001) 457, 2481–2489
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A. A. Balinsky, W. D. Evans and R. T. Lewis
where Ψ ∈ L∞ (S1 ), so that B = curl A = 0 for r = 0, and A · x = 0. Then, for all u ∈ C0∞ (R2 \ {0}), 2 1 |u(x)|2 dx C (1.4) ∇ + A u(x) dx, 2 |x| i R2 R2 with the sharp constant 1 Ψ˜ := 2π
C = (min |k − Ψ˜ |)−1 , k∈Z
2π
0
Ψ (θ) dθ.
An important consequence is that the Schr¨ odinger operator SA = ((1/i)∇ + A)2 2 2 does not have zero modes in L (R ), i.e. zero is not an eigenvalue. This encourages the prediction, made in Laptev & Weidl (1999), that, for some appropriate V and norm, there exists a CLR-type inequality for the number of negative eigenvalues of Schr¨ odinger operators TA (V ) that are self-adjoint realizations in L2 (R2 ) of expressions of the form 2 1 ∇ + A − V ≡ SA − V, TA (V ) := i where A is the Aharonov–Bohm magnetic potential given in (1.3). The main objective of this paper is the confirmation of this prediction. Two vital preliminary results, which are of independent interest, are needed. The first is a Sobolev-type embedding 1 defined as the completion of C0∞ (R2 \ {0}) with respect theorem for the space HA to the norm 1/2 2 1 2 1 , u HA = ∇ + A u + u i where · denotes the L2 (R2 )-norm; we prove that if the flux Ψ˜ is not an integer, 1 is continuously embedded in the space HA X = L∞ (R+ ; L2 (S1 ); r dr) ≡ L∞ (R+ ; r dr) ⊗ L2 (S1 ) with norm
u X = ess sup r>0
0
2π
|u(r, θ)|2 dθ
1/2 .
1 This proves, in particular, that the set of radial functions in HA is continuously ∞ 2 embedded in L (R ). The second crucial result is that for V in the space
Y = L1 (R+ ; L∞ (S1 ); r dr) ≡ L1 (R+ ; r dr) ⊗ L∞ (S1 ), the operator TA (V ) = SA − V is defined as a form sum and has the same essential spectrum as SA , which is [0, ∞). Therefore, the negative spectrum of TA (V ) consists only of eigenvalues. Our results are the following. Theorem 1.1. Suppose that the magnetic flux Ψ˜ is not an integer. Then, for all 1 and A defined by (1.3), u ∈ HA 2 1 ∇ + A u (1.5) u 2X C , i where C = (mink∈Z |k − Ψ˜ |)−1 and X = L∞ (R+ ; L2 (S1 ); r dr). Proc. R. Soc. Lond. A (2001)
On the number of negative eigenvalues
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Theorem 1.2. Suppose that Ψ˜ is not an integer, and let V ∈ Y = L1 (R+ ; L∞ (S1 ); r dr). Then V is form bounded relative to SA with SA -bound zero, and hence SA − V 1 . Moreover, the essential is defined as a form sum with form domain Q(SA ) = HA spectra of SA − V and SA coincide. Theorem 1.3. Let A be given by (1.3), V ∈ L1loc (R2 \ {0}) and V+ ∈ Y = L1 (R+ , L∞ (S1 ), r dr), and suppose that the magnetic flux Ψ˜ is not an integer. Then N (TA (V )), the number of negative eigenvalues of TA (V ), is finite and 1 N (TA (V )) (1.6) V , ˜| + Y 2|m + Ψ m∈Z indicates that all summands less than 1 are omitted. where Note that, in (1.6),
V+ Y =
0
∞
{ess sup |V+ (r, θ)|}r dr. θ∈S1
Theorem 1.4. Suppose that the hypothesis of theorem 1.3 is satisfied. Then N (TA (V )) c(Ψ˜ ) V+ Y ,
(1.7)
where c(Ψ˜ ) is a constant depending only on Ψ˜ . To prove theorem 1.3, we make use of a result of Bargmann (1952). For theorem 1.4, we apply theorem 1.2 in Laptev & Netrusov (1999) concerning an operator of the form b −∆ + − V, b > 0, |x|2 in L2 (R2 ). This and our operator TA (V ) have similar characteristics, and indeed both satisfy (1.7), but are not in fact comparable. As noted earlier, Hardy’s inequality is not valid in R2 and there is no CLR inequality in terms of the L1 (R2 )-norm of V . A Hardy-type inequality is introduced in Laptev & Netrusov (1999) by the presence of the term b/|x|2 , while in our case, equation (1.3) fulfils the role. The resulting CLR inequalities require V+ to belong to a subspace of L1 (R2 ), namely L1 (R+ , L∞ (S1 ), r dr) in our case. In § 5 we show that the L1 (R+ , L∞ (S1 ), r dr)-norm of V+ cannot be replaced by the L1 (R2 )-norm on the right-hand side of (1.7). Remark 1.5. If the flux Ψ˜ is an integer, the operator TA (V ) is unitarily equivalent to −∆−V and hence there is no CLR inequality. Therefore, the arithmetic properties of the flux have a vital role and this implies that theorems 1.3 and 1.4 cannot be consequences of general results in which the singular and local behaviour of the vector and scalar potentials are the main feature. Remark 1.6. Our results confirm the known connection between the Sobolev and CLR inequalities: if u ∈ X, then u2 ∈ Z = L∞ (R+ ; L1 (S1 ); r dr) and V+ must lie in the dual of Z. Proc. R. Soc. Lond. A (2001)
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2. A Sobolev-type inequality The magnetic Schr¨ odinger operator SA is defined to be the non-negative self-adjoint operator in L2 (R2 ) associated with the closable non-negative sesquilinear form 2 1 dx, ϕ ∈ C0∞ (R2 \ {0}). sA [ϕ] = (2.1) ∇ + A ϕ i R2 1/2
1/2
1 1 defined in § 1. Note that HA is the domain D(SA ) of SA It has form domain HA with the graph norm.
Proof of theorem 1.1. Following § 1 of Laptev & Weidl (1999), we observe that SA is the operator 2 1 ∂ ∂2 1 ∂ + + Ψ (θ) , i SA = − 2 − ∂r r ∂r r2 ∂θ where the operator Kθ = i(∂/∂θ) + Ψ (θ) with domain H 1 (S1 ) in L2 (S1 ) has eigenvalues λk = k + Ψ˜ , k ∈ Z, and eigenfunctions
θ 1 ˜ Ψ (η) dη . ϕk (θ) = √ exp −i θ(k + Ψ ) − 2π 0 The sequence {ϕk (θ)} is an orthonormal basis of L2 (S1 ). Any u ∈ L2 (S1 ) can therefore be written uk (r)ϕk (θ), u(r, θ) = k∈Z
where
uk (r) =
1 , and, for any u ∈ HA
sA [u] =
0
k∈Z
∞
0
2π
u(r, θ)ϕk (θ) dθ,
|uk (r)|2
λ2k 2 + 2 |uk (r)| r dr . r
(2.2)
For any t ∈ (0, ∞), |uk (t)|2 = 2 Re
0
2
t
t
0
u ¯k (r)uk (r) dr
|uk (r)|2 r dr
t
1/2
t
0 1/2
2 dr
|uk (r)|
t
1/2
r
dr 2 λ2k = |uk (r)|2 r dr |uk (r)|2 |λk | 0 r 0 ∞ 2 1 λ |uk (r)|2 + 2k |uk (r)|2 r dr . r |λk | 0 Proc. R. Soc. Lond. A (2001)
1/2
On the number of negative eigenvalues Hence, on using (2.2), 2π 0
|u(t, θ)|2 dθ =
2485
|uk (t)|2
k∈Z
2 1 ∇ + A u , mink∈Z |λk | i 1
whence (1.5).
3. Relative compactness 1/2
It follows from (1.5) that SA has no zero mode, and hence SA is injective and has 1/2 1/2 1 dense domain D(SA ) and range R(SA ) in L2 (R2 ). Let DA denote the completion 1/2 of D(SA ) with respect to 1 1/2 ϕ DA1 := SA ϕ = ∇ + A ϕ (3.1) . i This is not a subspace of L2 (R2 ), but in view of the Laptev–Weidl inequality (1.4), it is a function space. In the next theorem, the space Y = L1 (R+ ; L∞ (S1 ); r dr) = L1 (R+ , r dr) ⊗ L∞ (S1 ) has norm
u Y =
∞
0
ess sup |u(r, θ)|r dr. −1/2
−1/2
P ≡ SA and hence that
(3.3)
θ∈(0,2π)
Proof of theorem 1.2. We shall prove that SA is sufficient to prove that −1/2
|V |SA
−1/2
T = |V |1/2 SA
−1/2
V SA
(3.2)
is compact in L2 (R2 ). It
is compact, is compact,
since P = T ∗ T . Given ε > 0, choose W ∈ C0∞ (R+ ; L∞ (S1 )) with support in Ωε = B(0, kε ) \ B(0, 1/kε ) and such that W L∞ (R2 ) kε for some constant kε > 0 and V − W Y < ε. Let −1/2 1 ϕn ' 0 in L2 (R2 ). Then, with ψn = SA ϕn , ψn ' 0 in DA and T ϕn 2 = |V |1/2 ψn 2 |W |1/2 ψn 2 + |V − W |1/2 ψn 2 kε |ψn |2 dx + V − W Y ψn 2X Ωε 1/2 kε |ψn |2 dx + Cε SA ψn 2 Ωε
Proc. R. Soc. Lond. A (2001)
(3.4)
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A. A. Balinsky, W. D. Evans and R. T. Lewis
by (1.5). For any ψ ∈ C0∞ (R2 \ {0}), we have that there exists a constant C(ε), depending on ε, such that 2 1 2 2 ψ L2 (Ωε ) C(ε) ψ X C(ε) ∇ + A ψ i by (1.5) and ∇ψ L2 (Ωε )
2 2 1 1 2 ∇ + A ψ + C(ε) ψ L2 (Ωε ) C(ε) ∇ + A ψ . i i
1 Hence DA is continuously embedded in the standard Sobolev space H 1 (Ωε ). Since 1 H (Ωε ) is compactly embedded in L2 (Ωε ) by Rellich’s theorem, it follows that T ϕn → 0 by (3.4) and the theorem is proved.
Corollary 3.1. Let V ∈ Y . Then the essential spectrum of SA and SA − V is [0, ∞).
4. Proof of theorems 1.3 and 1.4 We need some preliminary results and remarks. Note that we may assume, without loss of generality, that Ψ ≡ Ψ˜ in (1.3) since the corresponding magnetic potentials A generate the same magnetic field and hence the corresponding operators are gauge equivalent. The operator TA (V ) is defined as the (Friedrichs) operator associated with the quadratic form 2 1 ∇ + A u − V |u|2 dx, u ∈ C0∞ (R2 \ {0}), Q[u] = (4.1) i R2 with
V ∈ L1loc (R2 ). 1
+
∞
(4.2)
1
If V+ ∈ L (R , L (S ), r dr), it follows from theorem 1.2 that Q is bounded below in L2 (R2 ). It is also closable and TA (V ) is defined as the self-adjoint operator in L2 (R2 ) whose form domain is the domain of the closure of Q, which we still denote by Q. Let W (r) := V+ (r, ·) L∞ (S1 ) , (4.3) so that W L1 (R+ ,r dr) = W L1 (R+ ;L∞ (S1 );r dr) = V+ L1 (R+ ;L∞ (S1 );r dr) < ∞.
(4.4)
Then, from theorem 1.2 again, TA (W ) is a lower semi-bounded self-adjoint operator with essential spectrum [0, ∞). Since TA (V ) TA (W ), we have N (TA (V )) N (TA (W )),
(4.5)
and so the theorems will follow if we prove them for TA (W ). Hereafter in the proof we take V to be replaced by W in (4.1). Proc. R. Soc. Lond. A (2001)
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We need the following facts from Adami & Teta (1998) and Laptev & Weidl (1999): L2 (R2 ) = L2 (R+ ; r dr) ⊗ L2 (S1 )
imφ e = L2 (R+ ; r dr) ⊗ √ , 2π m∈Z where [·] denotes the linear span; SA =
(4.6)
{Dm ⊗ 1m },
(4.7)
m∈Z
where Dm is the Friedrichs operator in L2 (R+ ; r dr) associated with the quadratic form ∞ (m + α)2 2 2 hm [u] = |u (r)| + (4.8) |u(r)| r dr, α = Ψ˜ ; 2 r 0 and, for any u in the form domain of TA (W ), we have 1 imθ u(r, θ) = um (r) √ e 2π m∈Z and Q[u] =
hm [um ] −
m∈Z
0
∞
(4.9)
2
W (r)|um (r)| r dr ,
where Q is given by (4.1), but with V replaced by W . It follows that TA (W ) = {(Dm − W ) ⊗ 1m },
(4.10)
(4.11)
m∈Z
and the negative spectrum of TA (W ) is the aggregate of the negative eigenvalues of the Dm − W . To prove theorem 1.3, we use the estimate ∞ 1 N (Dm − W ) W (r)r dr, 2|m + α| 0 which follows from Bargmann’s (1952) result for the operator d (m + α)2 1 d r + Dm = − . r dr dr r2 In view of (4.11), this gives N (TA (W ))
1 2|m + α|
0
∞
W (r)r dr,
whence (1.6). For the proof of theorem 1.4 we apply theorem 1.2 in Laptev & Netrusov (1999). We may assume, without loss of generality, that α ∈ (0, 1), since the gauge transformation f (r, θ) → einθ f (r, θ), for n ∈ Z, takes Ψ˜ into Ψ˜ + n and gives rise to unitary equivalent operators. For m 0, we have d α2 m2 1 d r + 2 + 2 (4.12) Dm − r dr dr r r Proc. R. Soc. Lond. A (2001)
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A. A. Balinsky, W. D. Evans and R. T. Lewis
and, for m < 0, Dm
d (1 − α)2 (m + 1)2 1 d r + − + . r dr dr r2 r2
(4.13)
The two-dimensional case of the operator considered in Laptev & Netrusov (1999) has the form b − W = {(Lm (b) − W ) ⊗ 1m }, b > 0, L(b; W ) ≡ −∆ + |x|2 m∈Z
in (4.6), where
d b 1 d m2 r + 2+ 2 Lm (b) = − r dr dr r r
and Lm (b)−W is defined by the associated quadratic form in L2 (R+ ; r dr). It follows from (4.12) and (4.13) that N ( ⊕ {(Dm − W ) ⊗ 1m }) N ( ⊕ {(Lm (α2 ) − W ) ⊗ 1m }) m0
m0
N (L(α2 ; W ))
(4.14)
and N ( ⊕ {(Dm − W ) ⊗ 1m }) N ( ⊕ {(Lm ((1 − α)2 ) − W ) ⊗ 1m }) m 0, there exists a compactly supported V ∈ L1 (R2 ) such that V L1 (R2 ) < ε and N (T0 (V )) 1. Hence there exists f ∈ C0∞ (R2 ) such that, for some δ = δ(ε) > 0, (T0 (V )f, f ) < −δ. We may suppose that V and f are supported in B(0, R), the ball with centre at the origin and radius R = R(ε). Define the translation τ : x → x + (2R, 0) Proc. R. Soc. Lond. A (2001)
On the number of negative eigenvalues
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and set V˜ = V ◦τ , f˜ = f ◦τ , so that both V˜ and f˜ are supported in B(−2R, R), which lies in the half-plane P = {(x, y) : x < −R}. Since the magnetic field B = curl A is zero in the simply connected set P , we have, by Poincar´e’s theorem, that there exists a gauge transformation φ(x) → eik(x) φ(x),
k : P → R,
which is such that −∆ and SA are equivalent in P . Thus, with g(x) = eik(x) f˜(x), we have (TA (V˜ )g, g) = (T0 (V˜ )f˜, f˜) = (T0 (V )f, f ) < −δ. Our claim is therefore established. The first two authors are grateful to the European Union for support under the TMR grant FMRX-CT96-0001.
References Adami, R. & Teta, A. 1998 On the Aharonov–Bohm Hamiltonian. Lett. Math. Phys. 43, 43–53. Bargmann, V. 1952 On the number of bound states in a central field of force. Proc. Natl Acad. Sci. USA 38, 961. Laptev, A. & Netrusov, Yu. 1999 On the negative eigenvalues of a class of Schr¨ odinger operators. In Differential operators and spectral theory. Am. Math. Soc. Transl. 2 189, 173–186. Laptev, A. & Weidl, T. 1999 Hardy inequalities for magnetic Dirichlet forms. In Mathematical results in quantum mechanics, pp. 299–305. Basel: Birkh¨ auser.
Proc. R. Soc. Lond. A (2001)
