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NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2011, 2 (4), P. 71-77 UDC 517.958
THRESHOLD EIGENFUNCTIONS AND THRESHOLD RESONANCES OF SOME RELATIVISTIC OPERATORS
Y. Saito1, T. Umeda2 1 University of Alabama at Birmingham, Birmingham, USA 2University of Hyogo, Himeji, Japan saito@math.uab.edu, umeda@sci.u-hyogo.ac.jp
PACS 02.30.Jr, 02.30.Tb, 03.65.Pm
We give a review of recent developments on the study of threshold eigenfunctions and threshold resonances of magnetic Dirac operators and Pauli operators. Emphasis is placed on a proof of the non-existence of threshold resonances of the magnetic Dirac operators in a concise manner.
Keywords: Dirac operators, magnetic potentials, threshold energies, threshold resonances, threshold eigenfunctions, zero modes.
1. Threshold eigenfunctions
The relativistic operators we consider are magnetic Dirac operators
HA = a ■ - A(x)) +m/3, x e M3, (1.1)
and Pauli operators
3 -1 o
i dxi
3 =i 3
Here a = (a1, a2, a3) is the triple of 4 x 4 Dirac matrices
0 a
C o) Ü = 2' 3)
3
with the 2 x 2 zero matrix 0 and the triple of 2 x 2 Pauli matrices
_ '0 -
°"1 — I 1 n I 1 a2 = | „• n I , — 10 _1
(l o) ' a2 = o) , a3 = (o -1) ,
(12 0 ) Vo -h)•
and
V — (h 0 p 1 o -h/
The constant m is assumed to be positive. A(x) — (A1(x), A2(x), A3(x)) is a vector potential, and B = V x A is the magnetic field. By a ■ (4-Vx — A(x)) in (1.1), we mean
3 = 1
and similarly by a ■ B in (1.2) we mean J2j=1 Bj.
We need to make various assumptions on the vector potential in the present section. It is notable that all these assumptions share one common feature that each component of A is a
a.
real-valued function decaying at infinity in a certain sense. Therefore, any of these assumptions assures that Ha is essentially self-adjoint on [60°(R3)]4. The unique self-adjoint realization in the Hilbert space [L2(R3)]4 will be denoted by Ha again. Note that the domain of the self-adjoint operator Ha is given by the Sobolev space of order 1, i.e. Dom(fl^) = [H:(R3)]4. It is straightforward that the spectrum of the self-adjoint operator HA is given as follows:
a (Ha) = (—x>, —m] U [m, <x>).
By the threshold energies of HA, we mean the values m and —m.
It is a natural question whether these threshold energies become eigenvalues of Ha. It is well-known that ±m are generically not the eigenvalues of Ha. Precise description of this fact is given as follows.
Theorem 1.1 (Balinsky-Evans-Saito-Umeda). The set
{ A e [L3(R3)]3 | Ker(HA T m) = {0} } contains an open and dense subset of [L3(R3)]3.
For the proof of Theorem 1.1, see Balinsky-Evans [3, Theorem 2], together with Saito-Umeda [13, Corollary 2.1 and Theorem 4.2]. Theorem 1.1 says that the set of vector potentials which give rise to threshold eigenfunctions is sparse. A similar result in a different class of vector potentials also holds true. Actually, Elton [8] analyzed the structure of the set of vector potentials which produce threshold eigenfunctions.
Theorem 1.2 (Elton-Saito-Umeda). Let A be the Banach space defined by
A := { A e [C0(R3)]3 | \A(x)\ = o(\x\-1) as equipped with the norm
\\A\U = sup{<x)|A(x)|}.
x
Define
Z± = { A eA| dim(Ker(F t m)) = k }
for k = 0, 1, 2, •••. Then
(i) zk = z- for al1 h and A = Ufc^o z±.
(ii) Z0± is an open and dense subset of A.
(iii) For any k and any open subset Q(= 0) of R3 there exists an A e such that A e [qr>(fi)]3.
(iv) For k =1, 2 the set Z± is a smooth sub-manifold of A with co-dimension k2.
For the proof of Theorem 1.2, see Elton [8, Theorems 1 and 2], together with Saito-Umeda [13, Corollary 2.1 and Theorem 5.2].
By simple computations, one can see that PA = {cr ■ (-¿-V — A)}2. Hence one can define the Friedrichs extension in [L2(R3)]2 of Pa on [C0°(R3)]2 under appropriate assumptions on A and B. Balinsky-Evans [2, Theorem 4.2] showed the following result.
Theorem 1.3 (Balinsky-Evans). The set
{ B e [L3/2(R3)]3 | Ker(Pa) = {0} and Vx A = B } contains an open and dense subset of [L3/2(R3)]3.
As was shown in [2, Lemma 2.2], there exists a unique vector potential A such that A e [L3(R3)]3, Vx A — B and divA — 0. Theorem 1.3 says that the set of magnetic fields which give rise to a zero mode of Pa (an eigenfunction of Pa corresponding to the eigenvalue 0) is sparse.
It is not only necessary but also important to mention examples of vector potentials A(x) which yield threshold eigenfunctions of Ha as well as Pa. For this purpose, it is convenient to introduce Weyl-Dirac operator
WA = a-(-Vx-A(x)). %
When A is sufficiently smooth, it is well-known (Thaller [16, p. 195, Theorem 7.1]) that
mv — 0 ^ Ha(*) — ^ Ha(I) — -m(l).
Since Pa — Wj2 (see the paragraph before Theorem 1.3), it follows that
PAy — 0 ^ WAy — 0,
provided that A is in [C1 (R3)]3 and satisfies an appropriate condition. Therefore, it is apparent that a zero mode of Wa provides a threshold eigenfunction of Ha corresponding to either one of the energies ±m, as well as a threshold eigenfunction of Pa corresponding to the energy 0.
Example 1.1 (Loss-Yau). Define
A(x) — 3(x)-4{(1 - \x\2)w0 + 2(w0 ■ x)x + 2w0 x x) (1.3)
where (x) = + \x\2, 4>0 = 0), and
Wo — 00 ■ :— ((0O, ^100)c2, (0o, ^20o)c2, (0o, ^3^o)c2).
Note that w0 ■ x and w0 x x denotes the inner product and the exterior product of R3 respectively. Then
y(x) :— (x)-:\h + ia ■ x)fo (1.4)
is a zero mode of the Weyl-Dirac operator Wa .
Following and developing the ideas in [9], Adam-Muratori-Nash [1] constructed a series of vector potentials which give rise to zero modes of the corresponding Weyl-Dirac operators. All of their vector potentials share the property that \A(x)\ ^ C(x)-2 with the one given by
(1.3).
Recently, Saito-Umeda [14] found an interesting connection between the series of the zero modes constructed in [1] and a series of solvable polynomials.
It follows that the zero mode defined by (1.4) has the asymptotic limits ip(x) x \x\-2 as \x\ ^ <x>. According to Theorem 1.4 below, every zero mode exhibits the same asymptotic limit.
Theorem 1.4 (Saito-Umeda). Suppose that \ A(x)\ ^ C(x)-p, p > 1. Let p be a zero mode of the Weyl-Dirac operator Wa. Then for any u e S2, the unit sphere of R3,
lim r2^(ru)
T—^ +
= T- [ { • A(y))l2 + i(T-{ux A(y))}ip(y) dy,
J R3
where the convergence being uniform with respect to u e S2.
See Saito-Umeda [11] for the proof of Theorem 1.4.
Theorem 1.4 excludes the case p = 1. In Balinsky-Evans-Saito [6], they were successful to derive estimates for zero modes of the Dirac operator of the form Hq = a ■ jV + Q(x) with Q(x) = 0(|x|-1) as ^ ^ w, where Q(x) is a 4 x 4-matrix-valued function. Their estimates are as follows:
A C4-valued function f is said to be an m-resonance (resp. a —m-resonance) if and only if f belongs to [L2'-S(R3)]4 \ [L2(R3)]4 for some s e (0, 3/2] and satisfies HAf = mf (resp. HAf = —mf) in the distributional sense. By a threshold resonance of HA, we mean an m-resonance or a -m-resonance.
Theorem 2.1 (Saito-Umeda). Suppose that ^(x^ ^ C{x)-p, p > 3/2. Let f = t(^>+, tp-) e [L2'-S(R3)]4 = [L2'-S(R3)]2 © [L2'-s(R3)]2 for some s with 0 < s < min(1, p — 1).
(i) If f satisfies HAf = mf in the distributional sense, then f e [H 1(R3)]4 and <p- = 0.
(ii) If f satisfies HAf = —mf in the distributional sense, then f e [H 1(R3)]4 and = 0.
Theorem 2.1 implies the non-existence of the threshold resonance of HA as far as p > 3/2 and 0 < s < min(1, p — 1). As was mentioned in Section 1, the vector potentials by Loss-Yau [9] and by Adam-Muratori-Nash [1] satisfy the inequality |A(x)| ^ C{x)-2. Therefore, these vector potentials do not yield ±m-resonances.
As an easy corollary to Theorem 2.1, one can get the following result, which seems a more natural formulation of the non-existence of threshold resonances from the physics point of view.
Corollary 2.1. Suppose that f e [L2oc(R3)]4 and that
f(x) = C^x- + C^x- + o^-2) as
If f satisfies either of HAf = ±mf in the distributional sense, then C1 = 0.
We shall give an outline of the proof of Theorem 2.1. Although the reader can find the proof in Saito-Umeda [13], it heavily relies on Saito-Umeda [12], hence the whole story of the proof is separated into two different papers. For this reason, we believe that it is beneficial to illustrate the whole story in the present article in a concise manner. Before giving the outline, we prepare two lemmas.
Lemma 2.1. Let K be an integral operator define by
for any k e [1, 10/3).
2. Threshold resonances
To define threshold resonances, we introduce a weighted Hilbert space
L2'-s(R3) = {u W^uW^ < w}.
Then K(a ■ -V)ip = (p if (p E [L2'~3/2(R3)]2 and (a ■ -V)<p G [L2'f (R3)]2 for some t > 1/2.
1
1
Proof. We give a formal proof. A rigorous proof can be found in [12, Section 4].
Note that К = (a ■ -¡-V)/2 = • where I2 denotes the Riesz potential (cf. Stein [15, Chapter V]). It follows that
К {a • -V)p = I2(a ■ ~V)2p = h{~^)p = V, г г
since (a- 7 V)2 = — Д. Here the anti-commutation relations a,at, I rr/,ay- = 25jkh were used. □ Lemma 2.2. If t ^ 1, then the Riesz potential I\ is a bounded operator from L2'1 (R3) to L2 (R3). Proof. Let и E C0°(R3), where и denotes the Fourier transform of u. Then
4^2 Jr3 I2 JR3 v2
where we have used the Hardy inequality with respect to £ variable. See, e.g., [5, p. 19] for the Hardy inequality. □
Outline of the proof of Theorem 2.1. We shall give the proof only for an m-resonance. The proof for a —m-resonance is similar and shall be omitted.
Let f = f'(p+, p-) be in [L2—s(R3)]4 and satisfy HAf = mf in the distributional sense. We then have
rmp+ + a ■ (—V — A{x))p~ = rmp+ г
a ■ (—V — A(x))'p+ — mp~ = mp~ i
in the distributional sense, which is equivalent to
a ■ (-V - A(x))<p~ = 0 г
a ■ (—V - A(x))p+ = 2rrup~ г
(2.1)
in the distributional sense. Since p e [L2' S(R3)]2, it follows from the first equation in (2.1) that
(a • -V)p~ = (a • A)p~ E [L2'p"s(R3)]2, (2.2) 1
where p — s > 1. Hence Lemma 2.1 is applicable to p~, and we have
p~ = K(a ■ —V)p~ = K(a ■ A)p~. (2.3) %
It follows from (2.3) that
[ a | 1 ,2 \(a-A)(y)p-(y)\dy=^h{\(a-A)p-\)(x). (2.4)
Jr 3 ^X — y|2 2
The inequality (2.4), together with Lemma 2.2, implies that p~ e [L2(R3)]2. Noting that (a ■ \V)<p~ £ [L2(R3)]2 by (2.2), we can conclude that <p~ e [^(R3)]2.
On the other hand, it follows from the second equation of (2.1) that
(a • -V)p+ = 2rmp~ + {a ■ A)p+ e [L2(R3)]2. (2.5) i
To conclude that p = 0, we need to show that
1
a ■ p )[L2]2 = ((<7 • A)p+, p )[L2]2, (2.6)
[L2]2 V^Vf > Y 7[L2]2'
(see Remark 2.1 below). In fact, combining (2.5) with (2.6) and noting that m > 0, we can conclude that p- = 0.
The fact that ip~ = 0, together with the second equality in (2.1), implies that a ■ (4-V — A)^>+ = 0. It is now evident that one can repeat the same arguments above for у- to conclude that e [H 1(E3)]2. □
Remark 2.1. A rigorous proof of (2.6) can be found in [13, Lemma 6.1], where the condition s < 1 is used. Since the proof of (2.6) is lengthy, we prove it by a formal argument as follows:
((a • <p~)[L2]2 = (o- • |v)^")[l2]2
= (p+, (a • A)V-)[l2]2 (2.7)
= ((a • , y-) [l2]2.
Here we have made integration by part in a formal manner in the first equality in (2.7), and in the second equality in (2.7) we have used the first equality in (2.1).
As for the non-existence of zero-resonances of Pauli operators, Morita [10] recently obtained the following result.
Theorem 2.2 (Morita). Suppose that A e [C~(R3)]3 and that
\Aj(x)\ + \VAj(x)\ ^ С(x)-p, p ^ 2
for j = 1, 2, 3. If p e [H 1,-s(R3)]2 for some s with 0 < s ^ 1 and satisfies Pa^> = 0 in the distributional sense, then p e [H 1(R3)]2. Here
H ^(R3) = { и \ \\(x)-su\\L2 + \\(x)-sVu\\L2 < ж].
One should note that L2(R3) £ H 1'-S(R3), but H 1'-S(R3) \ L2(R3) = 0. Obviously, there is room for improvement in Theorem 2.2.
3. Acknowledgements
TU is supported by Grant-in-Aid for Scientific Research (C) No. 21540193, Japan Society for the Promotion of Science.
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