The Dirac Operator with Mass m_{0} ≥ 0 :

Non-Existence of Zero Modes

and of Threshold Eigenvalues

Hubert Kalf, Takashi Okaji, and Osanobu Yamada

Received: November 21, 2014

Communicated by Heinz Siedentop

Abstract. A simple global condition on the potential is given which excludes zero modes of the massless Dirac operator. As far as local conditions at infinity are concerned, it is shown that at energy zero the Dirac equation without mass term has no non-trivial L^{2}-solutions at infinity for potentials which are either very slowly varying or decaying at most like r^{−s} with s ∈ (0, 1). When a mass term is present, it is proved that at the thresholds there are again no such solutions when the potential decays at most like r^{−s} with s ∈ (0, 2). In both situations the decay rate is optimal.

2010 Mathematics Subject Classification: Primary 35P15; Secondary 81Q10. Keywords and Phrases: Dirac operators, virial theorem, threshold eigenvalue, zero mode.

1 Introduction

In their 1986 study of the stability of matter in the relativistic setting of the Pauli operator J. Fröhlich, E.H. Lieb and M. Loss recognised that there was a restriction on the nuclear charge if and only if the three-dimensional Dirac operator with mass zero has a non-trivial kernel (see [LS], Chapters 8, 9 and the references there). An example of such a zero mode was first given by M. Loss and H.T. Yau [LY]; for many more examples see [LS], p.167. Later, the Loss-Yau example was used in a completely different setting, viz. to show the necessity of certain restrictions in analogues of Hardy and Sobolev inequalities [BEU].

An observation with remarkable technological consequences is that in certain situations of non-relativistic quantum mechanics the dynamics of wave packets in crystals can be modelled by the two-dimensional massless Dirac operator (see [FW] and the references there). When the potential is spherically symmetric, a detailed spectral analysis of the Dirac operator with mass zero was given in two and three dimensions by K.M. Schmidt [S]. In particular he showed that a variety of potentials with compact support can give rise to zero modes. In Theorem 2.1 of the present paper we give a simple global condition on the potential which rules out zero modes of the massless Dirac operator in any dimension. Theorem 2.7 deals with a fairly large class of massless Dirac operators under conditions on the potential solely at infinity. It is shown, for example, that for energy zero there is no non-trivial solution of integrable square at infinity if the potential is very slowly varying or decaying like r^{−s} with s ∈ (0, 1). This decay rate is in a certain sense the best possible one (see Appendix B). To rule out non-trivial L^{2}-solutions at infinity for the threshold energies ±m_{0} turns out to be more complicated. Here the asymptotic analysis of Appendix B suggests 1/r^{2}-behaviour as the borderline case and Theorem 2.10 indeed permits potentials which tend to zero with a rate at most like r^{−s} with s ∈ (0, 2). Theorem 2.10 relies on a transformation of the solutions which is intimately connected with the block-structure that can be given to the Dirac matrices. In connection with this theorem we should like to draw attention to the very interesting paper [BG] where global conditions are used. In broad outline the proof of our Theorems 2.7 and 2.10 follows from the virial technique which was developed by Vogelsang [V] for the Dirac operator and later extended in [KOY], but basic differences in the assumptions on the potential are required in the situations considered here. At the beginning of §4 the general strategy of proof is outlined and comparison with [KOY] made, but the present paper is self-contained.

Acknowledgements. H.K. and O.Y. are indebted to Malcolm Brown, Maria Esteban, Karl Michael Schmidt and Heinz Siedentop for the invitation to the programme ’Spectral Theory of Relativistic Operators” at the Isaac Newton Institute in the summer of 2012. H.K. also gratefully acknowledges the hospitality given to him at Kyoto University and Ritsumeikan University in the autumn of 2013. T.Ō and O.Y. are partially supported by JSPS Grant-in-Aid No.22540185 and No.230540226, respectively.

2 Results and Examples

Given n ≥ 2 and N := 2^{[(n+1)/2]}, there exist n + 1 anti-commuting Hermitian N × N matrices α_{1}, α_{2}, ..., α_{n}, β := α_{n+1} with square one. No specific representation of these matrices will be needed except in Appendix B. Using the notation

α · p = ∑_{j=1}^{n} α_{j}(−i∂_{j}),

we start by considering the differential expression

τ := α · p + Q

in L^{2}(R^{n})^{N}. Here Q : R^{n} → C^{N×N} is a matrix-valued function with measurable entries. Let ϕ ∈ C^{N}. Then

u(x) := \frac{1 + iα · x}{(1 + |x|^{2})^{n/2}} ϕ

satisfies

τu = 0 with Q(x) = −\frac{n}{1 + r^{2}} .  (2.1)

This is the example of Loss and Yau [LY] mentioned in the Introduction. (Note that u is in L^{2} iff n ≥ 3.) For n = 3 the potential in (2.1) is in fact the first in a hierarchy of potentials with constants larger than 3, all giving rise to zero modes; see [SU]. In contrast, we have the following result.

Theorem 2.1. Let Q : R^{n} → C^{N×N} be measurable with

sup_{x∈R^{n}} |x||Q(x)| ≤ C for some 0 < C < \frac{n − 1}{2}.

Then any solution u ∈ H_{loc}^{1}(R^{n})^{N} ∩ L^{2}(R^{n}, r^{−1}dx)^{N} of τu = 0 is identically zero.

Remark 2.2. a) In case Q is Hermitian, Theorem 2.1 can be rephrased as follows: The self-adjoint realisation H of τ with

∫ \frac{|u|^{2}}{r} dx < ∞ (u ∈ D(H))  (2.2)

does not have the eigenvalue zero. (For the existence of such a self-adjoint realisation see [Ar] and the references therein.)

b) Suppose that V is a real-valued scalar potential with sup_{x∈R^{n}} |x||V (x)| < \frac{n−1}{2}. Then it needs but a small additional assumption to show by means of the virial theorem that the self-adjoint realisation H of α · p + V + m_{0}β (m_{0} ≠ 0) with property (2.2) does not have the eigenvalues ±m_{0} (see, e.g., [L], Theorem 2.4 in conjunction with [We], p.335).

c) It seems to be difficult to compare our C with the size of the compactly supported potentials in [S] which produce zero modes for n ≥ 2.

Now we replace the whole space with the exterior domain

E_{R} := {x ∈ R^{n} | |x| > R},

where R > 0 can be arbitrarily large. On E_{R} we consider the Dirac equation

(α · D + Q)u = 0 with D := p − b.  (2.3)

We assume for simplicity that the vector potential b is in C^{1}(E_{R}, R^{n}). Further conditions will not be imposed on b but on the magnetic field

B := (∂_{i}b_{k} − ∂_{k}b_{i}).

Note that B can be identified with the scalar function ∂_{1}b_{2} − ∂_{2}b_{1} if n = 2 and the vector-valued function curl b if n = 3. Solutions of (2.3) will be functions in H_{loc}^{1}(E_{R})^{N} which satisfy (2.3), if Q is locally bounded in E_{R}. The following result and its remark are essentially contained in [KOY], Example 6.1 and final remark on p.40.

Theorem 2.3. Let m_{0} ≥ 0, λ ∈ R and Q := V + m_{0}β − λ + W, where

(I) V = V^{∗} ∈ C^{1}(E_{R}, C^{N×N}), (II) W is a measurable and bounded matrix function (not necessarily Hermitian).

Suppose V, W and the magnetic field B satisfy the following conditions:

a) r^{1/2}V = o(1) = r∂_{r}V uniformly with respect to directions;

b) there exist numbers K ∈ (0, 1/2) and M > 0 such that

r|W | ≤ K, |Bx| ≤ M on E_{R}.

Assertion: If u ∈ L^{2}(E_{R})^{N} is a solution of (2.3) for

|λ| > \sqrt{m_{0}^{2} + M^{2}}/(1 − 2K),

then u = 0 on E_{R_{1}} for some R_{1} > R. If r|W | = o(1) or |Bx| = o(1) uniformly w.r.t. directions, then K = 0 or M = 0 is permitted.

Remark 2.4. If V ∈ C^{2}(R, ∞) is a real-valued (scalar) function, condition a) can be replaced by

V (r) = o(1) = rV′(r), rV′′(r) = o(1) as r → ∞.

Remark 2.5. Using a unique continuation result, e.g., the simple one [HP] or the more sophisticated one in [DO], one can conclude that u = 0 on R^{n}.

Remark 2.6. It follows immediately from Theorem 2.3 and Remark 2.5 that the potential in (2.1) cannot create a non-zero eigenvalue.

Theorem 2.3 will now be supplemented by Theorems 2.7 and 2.10.

Theorem 2.7. Let m_{0} = 0, λ ≤ 0 and V, W as in (I), (II) of Theorem 2.3. Let q ∈ C^{2}(R, ∞) be a positive bounded function with the properties

(i) [r(q − λ)]′ ≥ δ_{0}(q − λ) for some δ_{0} ∈ (0, 1),

(ii) \frac{q′}{q^{2}} = o(1) = r \frac{q′′}{q^{2}} .

Suppose V, W and B satisfy the following conditions:

(H.1) r(V − q) = O(1), ∂_{r}V − q′ = o (\frac{q}{r} );

(H.2) r|W | ≤ K for some K ∈ [0, δ_{0}/2);

(H.3) there is a function a(r) with |Bx| ≤ a(r) and \frac{a}{q} = o(1).

Assertion: Any solution u ∈ L^{2}(E_{R})^{N} of (2.3) with Q = V + W − λ vanishes identically on E_{R_{1}} for some R_{1} > R.

Remark 2.8. a) To prove Theorem 2.7, it will be important to observe that condition (i) implies

r(q − λ) ≥ const. r^{δ_{0}} (r > R).

b) If q is a negative bounded function with property (ii) and which satisfies [r(λ − q)]′ ≥ δ_{0}(λ − q) for some δ_{0} ∈ (0, 1), then Theorem 2.7 holds for λ ≥ 0.

c) In case V decays at infinity, hypothesis (H.3) demands a corresponding stronger decay of B to prevent the existence of eigenvalues. (The contrasting situation that V and B become large at infinity is considered in [MS].)

\underline{Examples}. For simplicity we assume V = q and W = 0. Let q_{0} be a positive number. Then the functions

q = q_{0} [2 + sin (log log r)],  (2.4)
q = q_{0} (log r)^{−s} (s > 0),  (2.5)
q = q_{0} r^{−s} (0 < s < 1),  (2.6)

have the required properties (i), (ii). In addition, a magnetic field with the decay property (H.3) is allowed. As far as (2.5) and (2.6) are concerned, Remarks 2.4–2.5 already rule out any eigenvalue λ ≠ 0. In case there is no vector potential, it follows from [S], Corollary 1 that the self-adjoint operator associated with τ = α · p + q in L^{2}(R^{n})^{N} has purely absolutely continuous spectrum outside [q_{0}, 3q_{0}].

Remark 2.9. More realistic potentials than (2.4) will have the property

lim_{r→∞} q(r) =: q_{∞} ≠ 0.

In such situations, however, it may be possible to use Theorem 2.3 or Remark 2.4 to show that

(α · D + Q − q_{∞})u = λu

has no non-trivial solution of integrable square at infinity if λ = −q_{∞}. A case in point is the potential

q = \frac{r}{1 + r} ,

which does not obey condition (i). Assuming |Bx| = o(1) uniformly w.r.t. directions, it follows from Theorem 2.3 that

(α · D + q − 1)u = λu

has no solution u ≠ 0 in L^{2} at infinity if λ ≠ 0. In particular, α · D + q has no zero mode.

For the equation

(α · D + V + m_{0}β)u = −m_{0}u, D := p − b  (2.7)

our result is as follows.

Theorem 2.10. Let m_{0} > 0 and μ := \sqrt{q(q + 2m_{0})}. Let q ∈ C^{2}(R, ∞) be a positive function with q = o(1) and the following properties:

(i) \frac{(rμ)′}{μ} ≥ δ_{0} for some δ_{0} ∈ (0, 1);

(ii) r \frac{q′}{q} = O(1), r \frac{q′′}{q^{3/2}} = o(1).

Suppose V ∈ C^{1}(E_{R}, R) and B satisfy the following conditions:

(H.1) r^{2}(V − q) = O(1), ∂_{r}V − q′ = o(\frac{q}{r} );

(H.2) there is a function a(r) with |Bx| ≤ a(r) and \frac{a}{\sqrt{q}} = o(1).

Assertion: Any solution u ∈ L^{2}(E_{R})^{N} of (2.7) vanishes identically on E_{R_{1}} for some R_{1} > R.

Remark 2.11. a) The function μ originates from a transformation in Appendix A (see (A.13)–(A.18)). Theorem 2.10 holds good for solutions of

(α · D + V + m_{0}β)u = m_{0}u,

if q = o(1) is a negative function and μ := \sqrt{q(q − 2m_{0})}.

b) Since

\frac{(rμ)′}{μ} = 1 + \frac{rq′}{2q} + o(1),

q may decay like r^{−s} with 0 < s < 2.

3 Proof of Theorem 2.1

Since

∫ r|α · pu|^{2} = ∫ r|Qu|^{2} ≤ C^{2} ∫ \frac{|u|^{2}}{r} < ∞,

we can find a sequence of functions {u_{j}} in C_{0}^{∞}(R^{n})^{N} with

r^{−1/2}u_{j} → r^{−1/2}u, \sqrt{r} (α · p)u_{j} → \sqrt{r} (α · p)u in L^{2}(R^{n})^{N}. Let U_{j} = r^{(n−1)/2}u_{j}. We write ∥U_{j}∥ rather than ∥U_{j}(r·)∥ for the norm in L^{2}(S^{n−1})^{N} and similarly for the scalar product. We use the decomposition in (A.3) of Appendix A and note that the symmetric operator S with b = 0 has a purely discrete spectrum with

− (N_{0} + \frac{n − 1}{2}) ∪ (N_{0} + \frac{n − 1}{2})

as eigenvalues. Hence

∫ r|α · pu_{j}|^{2} = ∫_{0}^{∞} r ∥α_{r} (−i∂_{r}U_{j} + \frac{i}{r} SU_{j})∥^{2}

= ∫_{0}^{∞} r ⟨−i∂_{r}U_{j} + \frac{i}{r} SU_{j}, −i∂_{r}U_{j} + \frac{i}{r} SU_{j}⟩

= ∫_{0}^{∞} r (∥∂_{r}U_{j}∥^{2} + \frac{1}{r^{2}} ∥SU_{j}∥^{2}) + ∫_{0}^{∞} ∂_{r}⟨−U_{j}, SU_{j}⟩

= ∫_{0}^{∞} r (∥∂_{r}U_{j}∥^{2} + \frac{1}{r^{2}} ∥SU_{j}∥^{2})

≥ (\frac{n − 1}{2})^{2} ∫_{0}^{∞} \frac{∥U_{j}∥^{2}}{r} ,

and the assertion follows in the limit j → ∞. □

4 Preliminaries to the proof of Theorem 2.7

To explain the general strategy of the proof of Theorem 2.7, let u be a solution of (2.3). We multiply U := r^{(n−1)/2}u by functions e^{φ}, φ = φ(| · |) real-valued, and χ = χ(| · |) with support in E_{R} and

0 ≤ χ ≤ 1, χ = 1 on [s, t_{k}], χ = 0 outside [s − 1, t_{k+1}],

where {t_{k}} is a sequence tending to infinity as k → ∞. Then ξ := χe^{φ}U satisfies

[−iα_{r}D_{r} + iα_{r} (\frac{S}{r} + φ′) + Q] ξ = g := −iα_{r}χ′e^{φ}U,  (4.1)

where D_{r} = ∂_{r} − i(x/r) · b. As in proofs of unique continuation results by means of Carleman inequalities, the idea is (as it was in the earlier papers ([V], [KOY]) to prove the existence of a constant C > 0 such that for large s and in the limit t_{k} → ∞ the inequality

∫_{s}^{∞} e^{2φ}∥U∥^{2} ≤ C ∫_{s−1}^{s} e^{2φ}∥U∥^{2}  (4.2)

holds. (For the precise inequality see (5.3) below.) If, for example φ = (ℓ/2)r^{b} is permitted for some b > 0, (4.2) implies

e^{ℓ(s+1)^{b}} ∫_{s+1}^{∞} ∥U∥^{2} ≤ Ce^{ℓs^{b}} ∫_{s−1}^{s} ∥U∥^{2},  (4.3)

and in the limit ℓ → ∞ the desired conclusion U = 0 on E_{s+1} follows. The present paper differs from [KOY] in three important respects. Firstly, the function q in Theorem 2.7 and 2.10 is allowed to tend to zero at infinity, while it was absolutely necessary to require ±q ≥ const. > 0 in [KOY]. Secondly, in contrast to [KOY], Proposition 3.1, the virial relation (A.8) from which we set out here

∫ ⟨[∂_{r}r(V − λ)]ξ, ξ⟩

= \underbrace{− ∫ ⟨(α · Bx)ξ, ξ⟩}_{I_{1}} + \underbrace{∫ 2Re⟨r W ξ, D_{r}ξ⟩}_{I_{2}} + \underbrace{∫ 2rφ′Re⟨−iα_{r}D_{r}ξ, ξ⟩}_{I_{3}}

\underbrace{− ∫ 2rRe⟨g, D_{r}ξ⟩}_{T_{1}},  (4.4)

does not contain a term involving q′/q. Such a term arose in [KOY] as it was necessary to divide ξ by (±q)^{1/2} in order to cope with the case that the potential (and possibly a variable mass) became large at infinity. Thirdly, we use a more refined cutoff function. Given t_{k} := 2^{k} and s < t_{k}, there exists a function χ ∈ C^{∞}(0, ∞) with

χ(r) = 1 for s ≤ r ≤ t_{k} and χ(r) = 0 for r ≥ t_{k+1}

such that

0 ≤ −rχ′(r) ≤ const.\frac{r}{t_{k+1} − t_{k}} ≤ const.\frac{2^{k+1}}{2^{k}}

for r ∈ [t_{k}, t_{k+1}] and all k ∈ N. Moreover,

r^{ℓ}|χ^{(ℓ)}(r)| ≤ const. (r ∈ [t_{k}, t_{k+1}], k ∈ N)

for ℓ = 2 and ℓ = 3. Estimates of the five terms in (4.4) will lead us to inequality (5.1) below, from which an inequality of type (4.3) will eventually emerge with the help of a bootstrap argument. We start with the left-hand side of (4.4) and write

∂_{r}[r(V − λ)] = V − q + r(∂_{r}V − q′) + [r(q − λ)]′
≥ [\frac{O(1/r)}{q − λ} + o(1)\frac{q}{q − λ} + δ_{0}] (q − λ)

by means of (i) and (H.1) in Theorem 2.7. Since 0 < q/(q − λ) ≤ 1 and (q − λ)^{−1} ≤ O(r^{1−δ_{0}}) at infinity in view of Remark 2.8, we have

∫ ⟨[∂_{r}r(V − λ)]ξ, ξ⟩ ≥ ∫ [δ_{0} + o(1)](q − λ)∥ξ∥^{2}.  (4.5)

The four terms I_{1}, I_{2}, I_{3} and T_{1} on the right-hand side of (4.4) are estimated as follows.

The Dirac Operator 45

Lemma 4.1.

a) I_{1} := − ∫ ⟨(α · Bx)ξ, ξ⟩ ≤ ∫ o(1)(q − λ)∥ξ∥^{2},

b) I_{2} := ∫ 2Re⟨r W ξ, D_{r}ξ⟩

≤ ∫ [2K − \frac{K}{(q − λ)^{2}} (\frac{φ′}{r} + φ′′) + o(1)] (q − λ)∥ξ∥^{2} + T_{2},

where

T_{2} := K ∫ \{\frac{(χ′)^{2}}{q − λ} + |χ′| [\frac{const.}{r(q − λ)} + o(1)]\} ∥e^{φ}U∥^{2}  (4.6)
≤ const. {∫_{s−1}^{s} r∥e^{φ}U∥^{2} + ∫_{t_{k}}^{t_{k+1}} r^{−δ_{0}}(q − λ)∥e^{φ}U∥^{2}} . (4.7)

\underline{Proof}. a) follows immediately from assumption (H.3) since α · Bx is an Hermitian matrix with square |Bx|^{2}. To prove b), we observe

I_{2} = ∫ 2Re⟨r W ξ, D_{r}ξ⟩ ≤ K (∫ (q − λ)∥ξ∥^{2} + ∫ \frac{1}{q − λ}∥D_{r}ξ∥^{2})

and use relations (A.9), (A.11) with

h = \frac{1}{q − λ}, j = r (\frac{h}{r})′ = h′ − \frac{h}{r}.

Then

∫ \frac{1}{q − λ}∥D_{r}ξ∥^{2} ≤ ∫ \frac{1}{q − λ}∥g − Qξ∥^{2} − ∫ \frac{1}{r(q − λ)}⟨Aξ, ξ⟩

+ ∫ j Im⟨Qξ, α_{r}ξ⟩ + ∫ [\frac{j′}{2} − h (\frac{φ′}{r} + φ′′)] ∥ξ∥^{2}

+ ∫ jχχ′ ∥e^{φ}U∥^{2},  (4.8)

where Q = V + W − λ, A = −α_{r}(α · Bx). Before turning to the first term on the right-hand side of (4.8) we note

Im⟨Qξ, α_{r}ξ⟩ = Im ⟨(V − q)ξ, α_{r}ξ⟩ + Im ⟨W ξ, α_{r}ξ⟩

and

|Im ⟨(V − q)ξ, α_{r}ξ⟩| ≤ \frac{const.}{r(q − λ)}(q − λ)∥ξ∥^{2} ≤ \frac{const.}{r^{δ_{0}}}(q − λ)∥ξ∥^{2}  (4.9)

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as well as

|Im ⟨W ξ, α_{r}ξ⟩| ≤ \frac{const.}{r(q − λ)}(q − λ)∥ξ∥^{2} ≤ \frac{const.}{r^{δ_{0}}}(q − λ)∥ξ∥^{2}  (4.10)

(see hypotheses (H.1), (H.2) and Remark 2.8 a). Similarly, the term

−2rRe⟨g, Qξ⟩ = 2rRe ⟨iα_{r}χ′e^{φ}U, Qχe^{φ}U⟩
= 2rRe ⟨iα_{r}χ′e^{φ}U, (V − q + W)χe^{φ}U⟩

can be estimated by

| − 2rRe⟨g, Qξ⟩| ≤ const.|χ′|∥e^{φ}U∥^{2}.

Next,

∥g − Qξ∥^{2} = ∥(q − λ + V − q + W)ξ∥^{2} + ∥g∥^{2} − 2Re ⟨g, Qξ⟩

= (q − λ)^{2}∥ξ∥^{2} + ∥(V − q + W)ξ∥^{2} + 2(q − λ)Re ⟨ξ, (V − q + W)ξ⟩

+∥g∥^{2} − 2Re ⟨g, Qξ⟩
≤ [1 + o(1)](q − λ)^{2}∥ξ∥^{2} + [(χ′)^{2} + \frac{const.}{r}|χ′||] ∥e^{φ}U∥^{2}.  (4.11)

Using Remark 2.8 a) again, the second term in (4.8) can be majorised by

const. ∫ \frac{a}{r^{δ_{0}}} ∥ξ∥^{2}.

With hypothesis (H.2) we see that it is

∫ o(1)(q − λ)∥ξ∥^{2}.

The same is true of the third term in (4.8), since (4.9) and (4.10) hold and j = o(1) by Remark 2.8 a) and the first part of assumption (ii). This leaves us with

\frac{j′}{q − λ} = \frac{1}{r^{2}(q − λ)^{2}} + \frac{q′}{r(q − λ)^{3}} − \frac{q′′}{(q − λ)^{3}} + \frac{2(q′)^{2}}{(q − λ)^{4}} ,

which, by assumptions (ii) and Remark 2.8 a), is again o(1). Collecting terms, we finally obtain (4.6) from which (4.7) follows, employing the properties of our cutoff function and Remark 2.8 a). □

Lemma 4.2. Let φ′ ≥ 0 and

k_{φ} := −rφ′φ′′ − (φ′)^{2} + \frac{1}{2} (rφ′′)′,  (4.12)
c := \frac{a}{q − λ} + \frac{const.}{r(q − λ)} + \frac{1}{2} \frac{q′}{(q − λ)^{2}} + \frac{1}{2} \frac{rq′′}{(q − λ)^{2}}  (4.13)

The Dirac Operator 47

Then,

I_{3} := ∫ 2rφ′Re⟨−iα_{r}D_{r}ξ, ξ⟩
≤ ∫ [\frac{k_{φ}}{(q − λ)^{2}} + \frac{φ′}{q − λ}c + const.\frac{|φ′′|}{(q − λ)^{2}} ] (q − λ)∥ξ∥^{2} (4.14)
+T_{3},

where

T_{3} := ∫ \frac{r}{q − λ}[ φ′(χ′)^{2} + |φ′′||χ′′|] ∥e^{φ}U∥^{2}
≤ const. {∫_{s−1}^{s} r^{2}(φ′ + |φ′′|) ∥e^{φ}U∥^{2}  (4.15)

+ ∫_{t_{k}}^{t_{k+1}} r^{2−2δ_{0}} (\frac{φ′}{r} + |φ′′|) (q − λ)∥e^{φ}U∥^{2}\} .

\underline{Proof}. By setting ξ = \sqrt{q − λ} w, I_{3} can be written as

I_{3} = ∫ 2rφ′Re⟨−iα_{r}D_{r}ξ, ξ⟩ = ∫ 2rφ′Re⟨−iα_{r}D_{r}w, (q − λ)w⟩
= ∫ 2rφ′Re⟨−iα_{r}D_{r}w, Qw⟩ − ∫ 2rφ′Re⟨−iα_{r}D_{r}w, (V − q + W)w⟩.

Using (A.10), (A.12) with h = rφ′, j = rφ′′, we obtain I_{3} = ∫ rφ′(∥f∥^{2} − ∥D_{r}w∥^{2} − ∥f + iα_{r}D_{r}w∥^{2}) − ∫ φ′⟨Aw, w⟩

+ ∫ [rφ′φ′′ + \frac{1}{2}(rφ′′)′ − [r(φ′)^{2}]′ + \frac{1}{2} (\frac{rφ′}{q − λ}q′)′ − \frac{rφ′′}{2(q − λ)}q′] ∥w∥^{2}

+ ∫ rφ′′Im⟨Qw, α_{r}w⟩ + ∫ \frac{rφ′′}{q − λ}χχ′∥e^{φ}U∥^{2}

− ∫ 2rφ′Re⟨−iα_{r}D_{r}w, (V − q + W)w⟩.  (4.16)

The last term can simply be estimated by

∫ rφ′ (∥D_{r}w∥^{2} + \frac{const.}{r^{2}}∥w∥^{2}) .

Replacing ξ by w in (4.9) and (4.10), we have

I_{3} ≤ ∫ \{k_{φ} + φ′ [a + \frac{const.}{r} + \frac{1}{2} (\frac{rq′}{q − λ})′] + const.|φ′′|\} ∥w∥^{2}

+ ∫ \frac{r}{q − λ} [φ′(χ′)^{2} + |φ′′||χ′|] ∥e^{φ}U∥^{2},

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which leads to (4.12)–(4.14). The estimate (4.15) is again a consequence of the properties of χ and of Remark 2.8 a). □

Lemma 4.3. Let φ′ ≥ 0. Then

T_{1} := − ∫ 2rRe⟨g, D_{r}ξ⟩
≤ const. {∫_{s−1}^{s} r [(1 + φ′)∥e^{φ}U∥^{2} + e^{2φ}∥D_{r}U∥^{2}]  (4.17)
+ ∫_{t_{k}}^{t_{k+1}} r^{1−δ_{0}} (1 + |φ′′|) (q − λ)∥e^{φ}U∥^{2}} .

\underline{Proof}. Let η := e^{φ}U. Since

Re⟨−iα_{r}χ′η, ∂_{r}(χη)⟩ = χχ′ Re⟨−iα_{r}η, ∂_{r}η⟩,

we have

T_{1} = − ∫ 2rRe⟨g, D_{r}ξ⟩ = ∫ 2rRe⟨iα_{r}χ′e^{φ}U, D_{r}ξ⟩
= ∫ 2rχχ′Re⟨iα_{r}η, D_{r}η⟩.

On the interval [s − 1, s] the integral can simply be estimated by

∫_{s−1}^{s} 2rχ′∥η∥∥D_{r}η∥ ≤ ∫_{s−1}^{s} re^{2φ}∥U∥∥D_{r}U + φ′U∥.

On [t_{k}, t_{k+1}] we use the estimate

∫_{s−1}^{s} r(−χ′)(∥η∥^{2} + ∥D_{r}η∥^{2})

and observe that (A.9) and (A.11) hold with χ = 1 (i.e., g = 0) and ξ = η, provided h and j have compact support. Hence, with h = r(−χ′), j = r(h/r)′ = −rχ′′ we have

∫ r(−χ′) [∥D_{r}η∥^{2} + ∥ (\frac{S}{r} + φ′) η∥^{2}]

= ∫ r(−χ′)∥Qη∥^{2} + ∫ χ′⟨Aη, η⟩ − ∫ rχ′′Im⟨(V − q + W)η, α_{r}η⟩

+ ∫ [−\frac{1}{2}(χ′′ + rχ′′′) + χ′φ′ + rχ′φ′′] ∥η∥^{2}.

The integration extends over [t_{k}, t_{k+1}] only where χ′φ′ ≤ 0. Using Remark 2.8 a) and the estimates of the derivatives of χ, the assertion follows. □

Remark 4.4. The constant in (4.6), (4.13), and (4.14) is the sum of the constants which occur in assumptions (H.1) and (H.2). In view of the hypotheses of Theorem 2.7 the function c in (4.13) is o(1) at infinity. It is important that the constants in (4.7), (4.15) and (4.17) are independent of φ and t_{k} := 2^{k}.

The Dirac Operator 49

5 Proof of Theorem 2.7 Before proving Theorem 2.7 we prepare the following Proposition 5.1. Suppose f > 0, g ≥ 0 are functions on (0, ∞) with

∫ ^{∞} \frac{1}{f} = ∞, ∫ ^{∞} g < ∞,

and f is continuous and non-decreasing. Let t_{k} := 2^{k} (k ∈ N). Then we have

lim inf_{k→∞} ∫_{t_{k}}^{t_{k+1}} fg = 0

\underline{Proof}. Assume to the contrary that there are numbers ε_{0} > 0 and N ∈ N such that

∫_{t_{k}}^{t_{k+1}} fg ≥ ε_{0}

for N ≤ k ∈ N. Then

∫_{t_{N}}^{∞} g = ∑_{k=N+1}^{∞} ∫_{t_{k−1}}^{t_{k}} (fg) \frac{1}{f} ≥ ε_{0} ∑_{k=N+1}^{∞} \frac{1}{f(t_{k})} ≥ ε_{0} ∫_{t_{N+1}}^{∞} \frac{1}{f} = ∞

gives the desired contradiction. Proof of Theorem 2.7 From (4.4) and (4.5) and Lemma 4.1–4.3 we see

∫ [δ_{0} − 2K + o(1)](q − λ)e^{2φ}∥χU∥^{2}  (5.1)
≤ ∫ [\frac{k_{φ}}{(q − λ)^{2}} + \frac{φ′}{q − λ}c + const.\frac{|φ′′|}{(q − λ)^{2}}

−\frac{K}{(q − λ)^{2}} (\frac{φ′}{r} + φ′′)] (q − λ)e^{2φ}∥χU∥^{2}

+const. ∫_{s−1}^{s} r^{2}e^{2φ}[(1 + φ′ + |φ′′|)∥U∥^{2} + ∥D_{r}U∥^{2}] +const. ∫_{t_{k}}^{t_{k+1}} r^{1−δ_{0}}e^{2φ}(1 + φ′ + r|φ′′|)|(q − λ)∥U∥^{2},

where t_{k} := 2^{k}.

a) We claim

∫_{s}^{∞} r^{ℓ}(q − λ)∥U∥^{2} < ∞

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for all s > R and ℓ > 0. Let j ∈ N. We choose φ = (j/2) log log r in (5.1) and note

φ′ = \frac{j}{2r log r} , φ′′ = −\frac{j}{4r^{2} log r} (1 + \frac{1}{log r} ) ,

k_{φ} = \frac{j}{4r^{2} log r} [1 + \frac{2}{log r} + \frac{j + 2}{(log r)^{2}} ] ≤ \frac{j(j − 1)}{4r^{2}(log r)^{3}} + \frac{2j}{4r^{2} log r}

for r > R_{0} if R_{0} is sufficiently large. Using Remark 2.8 a), the first integral on the right-hand side of (5.1) can be majorised by

const. ∫ \{\frac{1}{r^{2δ_{0}}} [\frac{j(j − 1)}{(log r)^{3}} + \frac{j}{(log r)^{2}} + \frac{j}{(log r)}]

+\frac{1}{r^{δ_{0}}} \frac{j}{(log r)}o(1)\} (log r)^{j}(q − λ)∥χU∥^{2}.

The last integral on the right-hand side of (5.1) can be estimated by

∫_{t_{k}}^{t_{k+1}} r^{1−δ_{0}}(1 + const. j)(log r)^{j}(q − λ)∥U∥^{2}.  (5.2)

With (q − λ) being bounded, (q − λ)∥U∥^{2} is in L^{1}(R_{0}, ∞). Thus there is a sub-sequence {t_{k_{ℓ}}}_{ℓ=1}^{∞} on which (5.2) tends to zero in view of Proposition 5.1. This proves

∫_{R_{0}}^{∞}(log r)^{j}(q − λ)∥U∥^{2} < ∞.

Moreover, for some c_{0} > 0 the inequality (5.1) implies

c_{0} ∫_{R_{0}}^{∞} ∑_{j=0}^{L} \frac{(ℓ log r)^{j}}{j!} (q − λ)∥U∥^{2}

≤ const. ∫_{s−1}^{∞} [\frac{ℓ^{2}}{r^{2δ_{0}} log r} ∑_{j=2}^{L} \frac{(ℓ log r)^{j−2}}{(j − 2)!} + \frac{ℓ}{r^{δ_{0}}} ∑_{j=1}^{L} \frac{(ℓ log r)^{j−1}}{(j − 1)!} ]

·(q − λ)∥U∥^{2} + const. ∫_{s−1}^{s} r^{2} ∑_{j=0}^{L} \frac{(ℓ log r)^{j}}{j!} [(1 + j)∥U∥^{2} + ∥D_{r}U∥^{2}].

Since we can let L → ∞, this establishes the claim.

b) Next we assert

∫_{s}^{∞} e^{ℓr^{b}}(q − λ)∥U∥^{2} < ∞

for all s > R, ℓ > 0 and b ∈ (0, δ_{0}).

We insert φ = (jb/2) log r into (5.1) and observe

φ′ = \frac{jb}{2r} , φ′′ = −\frac{jb}{2r^{2}} , k_{φ} = \frac{jb}{4r^{2}} .

In view of Part a) there is a sub-sequence {t_{k_{ℓ}}} with

lim_{ℓ→∞} ∫_{t_{k_{ℓ}}}^{t_{k_{ℓ}+1}} r^{1−δ_{0}}(1 + jb)r^{jb}(q − λ)∥U∥^{2} = 0.

Using Remark 2.8 a) again, we see that (5.1) implies

c_{0} ∫_{s}^{∞} r^{jb}(q − λ)∥U∥^{2} ≤ const. ∫_{s−1}^{∞} (\frac{1}{r^{2δ_{0}}} + \frac{1}{r^{δ_{0}}} ) jbr^{jb}(q − λ)∥U∥^{2}

+ const. ∫_{s−1}^{s} r^{2+jb}[(1 + jb)∥U∥^{2} + ∥D_{r}U∥^{2}]

or

c_{0} ∫_{s}^{∞} ∑_{j=0}^{L} \frac{(ℓr^{b})^{j}}{j!} (q − λ)∥U∥^{2}

≤ const. ∫_{s−1}^{∞} \frac{ℓb}{r^{δ_{0}−b}} ∑_{j=1}^{L} \frac{(ℓr^{b})^{j−1}}{(j − 1)!} (q − λ)∥U∥^{2}

+ const. ∫_{s−1}^{s} r^{2} ∑_{j=0}^{L} \frac{(ℓr^{b})^{j}}{j!} [(1 + jb)∥U∥^{2} + ∥D_{r}U∥^{2}].

For b ∈ (0, δ_{0}) we can again move the first term of the right-hand side to the left and let L → ∞.

c) In order to show that U vanishes a.e. on E_{R_{1}} for some R_{1} > R, we choose φ = (ℓ/2)r^{b} where ℓ > 0 and b ∈ (0, δ_{0}). From

φ′ = \frac{ℓb}{2}r^{b−1}, φ′′ = −\frac{ℓb}{2}(1 − b)r^{b−2}

we observe (φ′/r) + φ′′ > 0, so that the last part of the first integral on the right-hand side of (5.1) can be discarded. On account of Part b) there is a sequence {t_{k_{ℓ}}} on which the last integral vanishes. Finally we note

k_{φ} = −\frac{ℓb}{4} [ℓb^{2} − \frac{(1 − b)^{2}}{r^{b}} ] r^{2(b−1)}.

With

X := \frac{ℓb}{2(q − λ)}r^{b−1}

52 Kalf, Okaji, and Yamada we therefore have

−\frac{k_{φ}}{(q − λ)^{2}} − \frac{φ′}{q − λ}c(r) − const.\frac{|φ′′|}{(q − λ)^{2}} = bX^{2} − d(r)X,

where

d(r) := \frac{1 − b}{r(q − λ)} (\frac{1 − b}{2} + const.) + c(r) = o(1).

Hence

∫_{s}^{∞} [δ_{0} − 2K + o(1) − \frac{d(r)^{2}}{4b} ] (q − λ)e^{ℓr^{b}}∥U∥^{2}

≤ const. ∫_{s−1}^{s} r^{2}e^{ℓr^{b}}[(1 + ℓbr^{b})∥U∥^{2} + ∥D_{r}U∥^{2}].  (5.3)

Now there is an R_{1} > R with the property that the left-hand side of (5.3) can be estimated from below by

const. e^{ℓ(s+1)^{b}} ∫_{s+1}^{∞} ∥U∥^{2}

for s > R_{1}. The assertion therefore follows in the limit ℓ → ∞. 6 Proof of Theorem 2.10 From (A.16)-(A.17) with λ = −m_{0} we see

μ := [(q + m_{0})^{2} − m_{0}^{2}]^{1/2} = \sqrt{q}\sqrt{q + 2m_{0}}, F := (\frac{q + 2m_{0}}{q} )^{1/4} ,  (6.1)

and

\sqrt{μ}F = \sqrt{q + 2m_{0}}, \frac{\sqrt{μ}}{F} = \sqrt{q}.

As a consequence

ξ := χe^{φ} (\begin{matrix}F U_{1} \\ (1/F)U_{2} \end{matrix}) = μ^{−1/2}χe^{φ}ζ with ζ := (\begin{matrix}\sqrt{q + 2m_{0}} U_{1} \\ \sqrt{q} U_{2} \end{matrix})

solves

[−iα_{r}D_{r} + iα_{r} (\frac{S}{r} + φ′) + Q] ξ = g := −μ^{−1/2}(iα_{r})χ′e^{φ}ζ,

where

Q = μI_{N} + (V − q) (\begin{matrix}F^{−2}I_{N/2} & 0 \\ 0 & F^{2}I_{N/2} \end{matrix}) − \frac{m_{0}q′}{2q(q + 2m_{0})}(iα_{r}β).

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The Dirac Operator 53

So our virial relation (A.8) now reads

∫ {⟨[∂_{r}(rQ)]ξ, ξ⟩ + ⟨(α · Bx)ξ, ξ⟩}
= ∫ 2rφ′Re⟨−iα_{r}D_{r}ξ, ξ⟩ − ∫ 2rRe⟨g, D_{r}ξ⟩.  (6.2)

Before beginning our estimates we observe that the assumption (i) implies

rμ ≥ const. r^{δ_{0}} or q^{−1/2} ≤ const. r^{1−δ_{0}}.  (6.3)

In view of (6.1) and our assumptions (ii), (H.1) we therefore have

\frac{1}{μ}(rF^{−2})′(V − q) = \frac{1}{μF^{2}} [1 − \frac{m_{0}rq′}{q(q + 2m_{0})}] (V − q)
= o(1)\frac{V − q}{q} = o(1),
\frac{1}{μ}(rF^{2})′(V − q) = \frac{F^{2}}{μ} [1 + \frac{m_{0}rq′}{q(q + 2m_{0})}] (V − q)
= O(1)\frac{V − q}{q} = o(1),

and from

[\frac{rq′}{q(q + 2m_{0})}]′ = \frac{1}{q + 2m_{0}} [ \frac{q′}{q} + \frac{rq′′}{q} − r (\frac{q′}{q} )^{2} − \frac{r(q′)^{2}}{q(q + 2m_{0})}]

we find

\frac{1}{μ} [\frac{rq′}{q(q + 2m_{0})}]′ = o(1),

taking advantage of (6.3) again. Since

∫ ⟨(α · Bx)ξ, ξ⟩ ≥ − ∫ \frac{a}{μ} μ ∥ξ∥^{2} = − ∫ o(1) μ ∥ξ∥^{2},

the left-hand side of (6.2) can be estimated from below by

∫ [δ_{0} + o(1)]μ∥ξ∥^{2}.

Lemma 6.1. Let φ′ ≥ 0 and

k_{φ} := −rφ′φ′′ − (φ′)^{2} + \frac{1}{2}(rφ′′)′,
c := \frac{a}{μ} + \frac{r(V − q)^{2}}{μq} + \frac{1}{2μ} [(r \frac{μ′}{μ} )′ + m_{0} |(r \frac{q′}{μ^{2}} )′|] .

Documenta Mathematica 20 (2015) 37–64

Then

I := ∫ 2rφ′Re⟨−iα_{r}D_{r}ξ, ξ⟩
≤ ∫ (\frac{k_{φ}}{μ^{2}} + \frac{φ′}{μ} c + const. \frac{|φ′′|}{μ^{2}}) μ∥ξ∥^{2} + J,

where

J := ∫ \frac{r}{μ^{2}} [φ′(χ′)^{2} + |φ′′||χ′′|] ∥e^{φ}ζ∥^{2}
≤ const. {∫_{s−1}^{s} r^{3}(φ′ + |φ′′|) ∥e^{φ}ζ∥^{2}

+ ∫_{t_{k}}^{t_{k+1}} r^{2−2δ_{0}} (\frac{φ′}{r} + |φ′′|) ∥e^{φ}ζ∥^{2}\}

with a constant which is independent of φ and t_{k} := 2^{k}. (Note that the assumptions of Theorem 2.10 imply c = o(1).)

\underline{Proof}. Since w := μ^{−1/2}ξ satisfies

[−iα_{r}D_{r} + iα_{r} (\frac{S}{r} + φ′) + Q − i\frac{μ′}{2μ}α_{r}] w = f := μ^{−1/2}g,

we have

∫ j ⟨(\frac{S}{r} + φ′) w, w⟩ + \frac{1}{2} ∫ (j′ − j \frac{μ′}{μ} ) ∥w∥^{2}

+ ∫ j Im⟨Qw, α_{r}w⟩ + ∫ \frac{j}{μ^{2}} χχ′∥e^{φ}ζ∥^{2} = 0

as a substitute for identity (A.12). Let

I_{1} := ∫ 2rφ′ Re ⟨−iα_{r}D_{r}w, \frac{m_{0}q′}{2q(q + 2m_{0})}(iα_{r}β)w⟩ ,
I_{2} := ∫ 2rφ′ Re ⟨−iα_{r}D_{r}w, (V − q) (\begin{matrix}F^{−2}w_{1} \\ F^{2}w_{2} \end{matrix})⟩ .

Replacing q by μ in (A.10), we can write

I = ∫ 2rφ′ Re ⟨−iα_{r}D_{r}w, μw⟩ = ∫ 2rφ′ Re ⟨−iα_{r}D_{r}w, Qw⟩ + I_{1} + I_{2}

= ∫ rφ′(∥f∥^{2} − ∥D_{r}w∥^{2} − ∥f + iα_{r}D_{r}w∥^{2}) − ∫ φ′⟨Aw, w⟩

+ ∫ [k_{φ} + \frac{1}{2} (rφ′ \frac{μ′}{μ} )′ − \frac{1}{2}rφ′′ \frac{μ′}{μ} ] ∥w∥^{2} + ∫ \frac{rφ′′}{μ^{2}} χχ′∥e^{φ}ζ∥^{2}

+ ∫ rφ′′ Im⟨Qw, α_{r}w⟩ + I_{1} + I_{2}.

The Dirac Operator 55

Integrating by parts in I_{1}, we obtain

I_{1} = − ∫ 2rφ′ Re ⟨∂_{r}w, \frac{q′}{2q(q + 2m_{0})}(m_{0}β)w⟩  (6.4)

= \frac{1}{2} ∫ ⟨w, \{rφ′′ \frac{q′}{q(q + 2m_{0})}(m_{0}β) + φ′ [\frac{rq′}{q(q + 2m_{0})}]′ (m_{0}β)\} w⟩

and note that the first term in (6.4) cancels

∫ rφ′′ Im⟨Qw, α_{r}w⟩ = −\frac{1}{2} ∫ rφ′′ ⟨\frac{q′}{q(q + 2m_{0})}(m_{0}β)w, w⟩ .

Furthermore, since

F^{−4}|w_{1}|^{2} + F^{4}|w_{2}|^{2} = \frac{q}{q + 2m_{0}} |w_{1}|^{2} + \frac{q + 2m_{0}}{q} |w_{2}|^{2},

we have

I_{2} ≤ ∫ rφ′∥D_{r}w∥^{2} + const. ∫ rφ′ \frac{(V − q)^{2}}{q} ∥w∥^{2}.

Collecting terms, the assertion follows. □ Lemma 6.2. Let φ′ ≥ 0. Then

T := − ∫ 2r Re⟨g, D_{r}ξ⟩
≤ {const. ∫_{s−1}^{s} r^{2} [(1 + φ′)∥e^{φ}ζ∥^{2} + e^{2φ}∥D_{r}ζ∥^{2}]

+ ∫_{t_{k}}^{t_{k+1}} r^{1−δ_{0}}(1 + |φ′′|)∥e^{φ}ζ∥^{2}}

with a constant which is independent of φ and t_{k}. \underline{Proof}. Abbreviating ϕ := e^{φ}ζ, we have

T = − ∫ 2r Re⟨g, D_{r}ξ⟩ = ∫ 2r \frac{χχ′}{μ} Re⟨iα_{r}ϕ, D_{r}ϕ⟩.

On the interval [s − 1, s] the integral can be estimated by

∫_{s−1}^{s} 2r \frac{χ′}{μ} ∥ϕ∥ ∥D_{r}ϕ∥ ≤ const. ∫_{s−1}^{s} r^{2−δ_{0}}e^{2φ}∥ζ∥ ∥D_{r}ζ + φ′ζ∥,

using (6.3). On [t_{k}, t_{k+1}] we majorise the integral by

∫_{t_{k}}^{t_{k+1}} \frac{r}{μ} (−χ′)(∥ϕ∥^{2} + ∥D_{r}ϕ∥^{2})  (6.5)

and note that it is permitted to use (A.9), (A.11) with χ = 1 (i.e., g = 0) and ξ = ϕ, since

h = \frac{r}{μ} (−χ′), j = r (\frac{h}{r})′ = h′ − \frac{h}{r}

have compact support. Hence on [t_{k}, t_{k+1}]

∫ \frac{r}{μ} (−χ′) [∥D_{r}ϕ∥^{2} + ∥(\frac{S}{r} + φ′) ϕ∥^{2}]

= ∫ \frac{r}{μ} (−χ′)∥Qϕ∥^{2} + ∫ \frac{χ′}{μ} ⟨Aϕ, ϕ⟩ + ∫ [\frac{j′}{2} − (\frac{φ′}{r} + φ′′) h] ∥ϕ∥^{2}

+ ∫ j Im⟨Qϕ, α_{r}ϕ⟩.

Now, −φ′′h ≤ const.|φ′′|r^{1−δ_{0}} by (6.3), while −(φ′/r)h ≤ 0 on [t_{k}, t_{k+1}]. From h′ = o(1) = h′′ we conclude j = o(1) = j′. The integral (6.5) can therefore be estimated by

const. ∫_{t_{k}}^{t_{k+1}} r^{1−δ_{0}}(1 + |φ′′|)∥ϕ∥^{2},

which concludes the proof. □

Summing up, we have

∫ [δ_{0} + o(1)]∥χe^{φ}ζ∥^{2} ≤ ∫ (\frac{k_{φ}}{μ^{2}} + \frac{φ′}{μ} c + const.\frac{|φ′′|}{μ^{2}}) ∥χe^{φ}ζ∥^{2}

+ const. {∫_{s−1}^{s} r^{3}e^{2φ}[(1 + φ′ + |φ′′|)∥ζ∥^{2} + ∥D_{r}ζ∥^{2}]

+ ∫_{t_{k}}^{t_{k+1}} r^{1−δ_{0}}(1 + φ′ + r|φ′′|)e^{2φ}∥ζ∥^{2}} ,

which does not differ from (5.1) in any essential way. The previous bootstrap argument can therefore be repeated almost verbatim, proving ζ = 0 and so u = 0 a.e. on E_{R_{1}} for some R_{1} > R. □

Appendix

A Identities in Connection with the Virial Theorem

A.1 Algebraic Relations

The principal part in

(α · D + Q)u = 0, D := p − b, p = −i∇  (A.1)

can be decomposed with the operators

D_{r} := ∂_{r} − i\frac{x}{r} · b, α_{r} := α · \frac{x}{r},

S := \frac{n − 1}{2} − ∑_{1≤j<k≤n} iα_{j}α_{k}(x_{j}D_{k} − x_{k}D_{j})  (A.2)

as follows:

α · D = α_{r} (−ir^{(1−n)/2} D_{r}r^{(n−1)/2} + \frac{i}{r} S) .  (A.3)

S is a symmetric operator in L^{2}(S^{n−1})^{N} which commutes with every operator which solely depends on the radial variable r; it anticommutes with α_{r}. In two dimensions we have

S = \frac{1}{2} − iσ_{1}σ_{2}(x_{1}D_{2} − x_{2}D_{1}) = \frac{1}{2} + σ_{3}(x_{1}D_{2} − x_{2}D_{1}),

where σ_{1}, σ_{2}, σ_{3} are the Pauli matrices. For n = 3 it is convenient to define σ := (−iα_{2}α_{3}, −iα_{3}α_{1}, −iα_{1}α_{2}). Then

S = 1 + σ · L with L := x × D,

but the operator K := βS is also used instead of S. We notice that α · Bx anticommutes with α_{r}, since B is skew-symmetric. A longer but completely elementary calculation shows

A := [D_{r}, S] = −iα_{r}(α · Bx).

Since α_{r}^{2} = 1, this implies A^{2} = |Bx|^{2} and

[D_{r}, iα_{r}S] = α · Bx.  (A.4)

Furthermore,

T := (β + η α · Bx)(1 + ζ iα_{r})

is an Hermitian N × N matrix with square

T^{2} = (1 + η^{2}|Bx|^{2})(1 + ζ^{2}),

if η, ζ ∈ R. Finally we note that the (n+1) Dirac matrices can be given the following block structure :

α_{j} = (\begin{matrix}0 & a_{j} \\ a_{j}^{∗} & 0 \end{matrix}) (j = 1, 2, ⋯, n), β = (\begin{matrix}I_{N/2} & 0 \\ 0 & I_{N/2} \end{matrix}) .  (A.5)

The a_{j} are (N/2) × (N/2) matrices (Hermitian if n is odd) which satisfy the following commutation relations :

a_{j}a_{k}^{∗} + a_{k}a_{j}^{∗} = 2δ_{jk}I_{N/2}, a_{j}^{∗}a_{k} + a_{k}^{∗}a_{j} = 2δ_{jk}I_{N/2}.

For n = 2, 3 this is of course well-known; the Pauli matrices take the role of the α_{j} if n = 2 and of the a_{j} if n = 3. For general n see [KY].

A.2 Analytic Tools

Let φ, χ, q, h and j be smooth real-valued functions which depend only on r; we assume that χ has compact support and q is positive. When u is a solution of (A.1), then (A.3) implies that

ξ := χe^{φ}U with U := r^{(n−1)/2}u

and

w := q^{−(1/2)}ξ

are solutions of

[−iα_{r}D_{r} + iα_{r} ( \frac{S}{r} + φ′) + Q] ξ = g := −iα_{r}χ′e^{φ}U,  (A.6)
[−iα_{r}D_{r} + iα_{r} ( \frac{S}{r} + φ′) + Q − iα_{r} \frac{q′}{2q} ] w = f := q^{−(1/2)}g. (A.7)

Splitting Q into an Hermitian part Q_{1} and Q_{2} := Q − Q_{1}, the following virial relation holds :

∫ ⟨[∂_{r}(rQ_{1}]ξ, ξ⟩ = − ∫ ⟨(α · Bx)ξ, ξ⟩  (A.8)
+ ∫ 2 Re⟨r Q_{2}ξ, D_{r}ξ⟩ + ∫ 2rφ′ Re⟨−iα_{r}D_{r}ξ, ξ⟩ − ∫ 2r Re⟨g, D_{r}ξ⟩.

Norm and scalar product in L^{2}(S^{n−1})^{N} are denoted by ∥ · ∥ and ⟨·, ·⟩, respectively. We write ∥ξ∥ rather than ∥ξ(r·)∥ and similarly for the scalar product. Integration is over (0, ∞). Starting from

∫ 2r Re⟨g, D_{r}ξ⟩

(A.8) can immediately be verified, using (A.6) and (A.4) as well as an integration by parts in the term containing Q_{1} (see [KOY], Proposition 3.1 and the remark on p. 40). In case χ can be replaced by 1, the virial theorem in its familiar form follows from (A.8) by setting φ = 0, Q_{2} = 0 and observing that g is zero. As a consequence of (A.6) – (A.7) we note the following energy relations:

∫ h [∥D_{r}ξ∥^{2} + ∥(\frac{S}{r} + φ′) ξ∥^{2}] = ∫ h∥g − Qξ∥^{2}  (A.9)

− ∫ \frac{h}{r}⟨Aξ, ξ⟩ − ∫ r (\frac{h}{r})′ ⟨ \frac{S}{r} ξ, ξ⟩ − ∫ (hφ′)′∥ξ∥^{2},

∫ 2h Re⟨−iα_{r}D_{r}w, Qw⟩  (A.10)

= ∫ h(∥f∥^{2} − ∥D_{r}w∥^{2} − ∥f + iα_{r}D_{r}w∥^{2}) − ∫ \frac{h}{r}⟨Aw, w⟩

− ∫ r (\frac{h}{r})′ ⟨\frac{S}{r} w, w⟩ + ∫ [(h\frac{q′}{2q} )′ − (hφ′)′] ∥w∥^{2}.

(A.9) follows from

∫ h∥iα_{r}(g − Qξ)∥^{2} = ∫ h ∥D_{r}ξ − (\frac{S}{r} + φ′) ξ∥^{2}

by undoing the square on the right-hand side and integrating by parts (cf. [KOY], Proposition 3.3 and the remark on p.40). Similarly, (A.10) can be proved by inserting into the left-hand side the expression for Qw which arises from (A.7) (cf. [KOY], Proposition 3.2 and the remark on p.40). Since S is unbounded from above and from below, the corresponding term on the right-hand side of (A.9)–(A.10) has to be eliminated. This can be done with the help of the auxiliary identities

∫ j ⟨(\frac{S}{r} + φ′) ξ, ξ⟩ + \frac{1}{2} ∫ j′∥ξ∥^{2} + ∫ j Im⟨Qξ, α_{r}ξ⟩

+ ∫ jχχ′∥e^{φ}U∥^{2} = 0,  (A.11)

∫ j ⟨(\frac{S}{r} + φ′) w, w⟩ + \frac{1}{2} ∫ (j′ − j \frac{q′}{q} ) ∥w∥^{2} + ∫ j Im⟨Qw, α_{r}w⟩

+ ∫ \frac{j}{q} χχ′∥e^{φ}U∥^{2} = 0.  (A.12)

(A.11), for example, results at once from

∫ Im ⟨iα_{r} (\frac{S}{r} + φ′) ξ, jα_{r}ξ⟩ ,

inserting (A.6) and integrating by parts (cf. [KOY], p.23). Let F = F (r) > 0 be a smooth function, m_{0} > 0 and λ ∈ R. When V is a scalar function and Q = V + m_{0}β − λ, it may be advantageous to split a solution u of (A.1) into two vectors u_{1}, u_{2} with N/2 components and use the block structure (A.5) of the Dirac matrices jointly with the transformation

ζ := (\begin{matrix}F U_{1} \\ (1/F) U_{2} \end{matrix}) , U_{j} = r^{(n−1)/2}u_{j}  (A.13)

(cf. [KOY], p.37). Then α_{r} = (\begin{matrix}0 & a_{r} \\ a_{r}^{∗} & 0 \end{matrix}) where a_{r} := ∑_{j=1}^{n}(x_{j}/r)a_{j}. With a

smooth function q = q(r) and

P := \frac{F′}{F} iα_{r}β +  (A.14)

(\begin{matrix}(1/F^{2}) (V − q + q + m_{0} − λ)I_{N/2} & 0 \\ 0 & F^{2}(V − q + q − m_{0} − λ)I_{N/2} \end{matrix}) ,

we then have

(−α_{r}D_{r} + \frac{i}{r} α_{r}S + P ) ζ = 0.  (A.15)

If, for example, q − m_{0} − λ > 0, requiring

\frac{1}{F^{2}} (q + m_{0} − λ) = μ = F^{2}(q − m_{0} − λ),  (A.16)

leads to

μ = [(q − λ)^{2} − m_{0}^{2}]^{1/2}, F = (\frac{q + m_{0} − λ}{q − m_{0} − λ})^{1/4}  (A.17)

Since

\frac{F′}{F} = −\frac{m_{0}}{2} \frac{q′}{μ^{2}} ,

the potential (A.14) becomes

P = μI_{N} + (V − q) (\begin{matrix}F^{−2}I_{N/2} & 0 \\ 0 & F^{2}I_{N/2} \end{matrix}) − \frac{m_{0}}{2} \frac{q′}{μ^{2}} iα_{r}β.  (A.18)

B Asymptotic Behaviour of Solutions

In case Q in

(−iα · ∇ + Q)u = 0  (B.1)

is Q = m_{0}β + λ − q and q is a rotationally symmetric scalar function, it suffices to discuss the ordinary differential equation

u′ = (\begin{matrix}−(k/r) & −q + m_{0} + λ \\ q + m_{0} − λ & (k/r) \end{matrix}) u,  (B.2)

where k is an eigenvalue of the angular momentum operator S in (A.2) with b = 0. k is an integer or half-integer such that |k| ≥ (n − 1)/2. (For general n, the reduction of (B.1) to (B.2) can be found in [KY].)

B.1 Case m_{0} = 0 = λ

In the new variables

t = log r, v(t) = u(e^{t})  (B.3)

(B.2) reads

v′ = \{(\begin{matrix}−k & 0 \\ 0 & k \end{matrix}) + (\begin{matrix}0 & −e^{t} q(e^{t}) \\ e^{t} q(e^{t}) & 0 \end{matrix})\} v.  (B.4)

If e^{t} q(e^{t}) → 0 as t → ∞, then (B.4) has a fundamental system of solutions v_{±} with the property

lim_{t→∞} \frac{1}{t} log |v_{±}(t)| = ±k.

(We refer to the references in [AKS] for this theorem which goes back to Perron, Lettenmeyer and Hartman-Wintner.) Hence (B.2) has a solution which is in L^{2} at infinity if rq(r) → 0 as r → ∞ and 2|k| > 1. Note that (B.4) has a fundamental system of solutions

v_{±}(t) = e^{±\sqrt{k^{2}−c^{2}} t}v_{±}^{0}

if q = c/r. Hence (B.2) has an L^{2}-solution at infinity if |k| > [(1/2) + c^{2}]^{1/2}.

B.2 Case λ = −m_{0} < 0

In this case (B.2) reads

u′ = \{(\begin{matrix}0 & 0 \\ 2m_{0} & 0 \end{matrix}) + (\begin{matrix}−(k/r) & −q \\ q & (k/r) \end{matrix})\} u =: (J + R)u.  (B.5)

The Jordan matrix J can be removed by observing that

ϕ := (\begin{matrix}1 & 0 \\ σ & 1 \end{matrix}) with σ := 2m_{0} r

has the properties ϕ′ = J = Jϕ. Hence z := ϕ^{−1}u satisfies

ϕ′z + ϕz′ = (J + R) ϕz or z′ = ϕ^{−1}R ϕz

(see [Ea], p.43 for this trick). Since the asymptotically leading term in ϕ^{−1}Rϕ still has the double eigenvalue zero, a second transformation is required. With

D := (\begin{matrix}1 & 0 \\ 0 & σ \end{matrix}) , w := D^{−1}z

we obtain

w′ = [(D^{−1})′D + (ϕD)^{−1}RϕD]w  (B.6)
= \{(\begin{matrix}−k & 0 \\ 2k & k − 1 \end{matrix}) + (\begin{matrix}−σrq & −σrq \\ σrq + rq/σ & −σrq \end{matrix})\} \frac{1}{r} w.

The constant matrix in (B.6) has the eigenvalues

μ_{±} := −\frac{1}{2} ± \sqrt{\frac{1}{4} + k(k − 1)}.

So, if k ≠ 0, 1, and if r^{2} q → 0 r → ∞, (B.6) has a fundamental system of solutions w_{±} with

|w_{±}(r)| = r^{μ_{±}+o(1)} as r → ∞.

Since u = ϕDw and |ϕD| ≤ const.r, (B.5) has a solution which is of integrable square at infinity if 2μ_{−} + 3 < 0, i.e., |k − (1/2)| > 1. For q = c/r^{2} system (B.6) reads

w′ = \{A + (\begin{matrix}0 & 0 \\ \frac{c}{2m_{0}} & 0 \end{matrix}) \frac{1}{r^{2}} \} \frac{1}{r} w,  (B.7)

where

A := (\begin{matrix}−(k + σ_{0}) & −σ_{0} \\ 2k + σ_{0} & k − 1 − σ_{0} \end{matrix})

and σ_{0} := 2mc. Introducing new variables as in (B.3), (B.7) becomes

w̃′(t) = \{A + e^{−2t} ( \begin{matrix}0 & 0 \\ \frac{c}{2m_{0}} & 0 \end{matrix})\} w̃(t).  (B.8)

Since the eigenvalues of A are

μ_{±} := −\frac{1}{2}(1 + 2σ_{0}) ± [\frac{1}{4} + k(k − 1 − 2σ_{0}) − σ_{0}^{2}]^{1/2} ,

it follows from the Levinson theorem ([Ea], Theorem 1.8.1) that (B.8) has a solution which is in L^{2} at infinity if |k| is sufficiently large. The same is therefore true of (B.5)

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Hubert Kalf Mathematisches Institut der Universität München Theresienstr. 39 D-80333 München, Germany hubert.kalf@mathematik. uni-muenchen.de Takashi Okaji Department of Mathematics Graduate school of Science Kyoto University Kyoto 606-8502 Japan okaji@math.kyoto-u.ac.jp

Osanobu Yamada Faculty of Science and Engineering Ritsumeikan University Kusatsu, Shiga 525-8577 Japan yamadaos@se.ritsumei.ac.jp