Pauli operators and the ∂-Neumann problem Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 9. № 3 (2017). С. 165-171.

УДК 517.55+517.9

PAULI OPERATORS AND THE ^-NEUMANN PROBLEM

F. HASLINGER

Abstract. We apply methods from complex analysis, in particular the 9-Neumann operator, to study spectral properties of Pauli operators. For this purpose we consider the weighted 9-complex on Cn with a plurisubharmonic weight function. The Pauli operators appear at the beginning and at the end of the weighted 9-complex. We use the spectral properties of the corresponding 9-Neumann operator to answer the question when the Pauli operators are with compact resolvent. It is also of importance to know whether the related Bergman space of entire functions is of infinite dimension. The main results are formulated in terms of the properties of the Levi matrix of the weight function. If the weight function is decoupled, one gets additional informations. Finally, we point out that a corresponding Dirac operator fails to be with compact resolvent.

Keywords: 9-Neumann problem, Pauli operators, Schrodinger operators, compactness. Mathematics Subject Classification: 32W05, 32W10, 30H20, 35J10, 35P10

1. Introduction

Let p : R2n —> R be a C2-function. We consider the Schrodinger operators with magnetic field of the form

P± = -Aa ± V,

also called Pauli operators, where

^ 1 / dip dip dp dp

is the magnetic potential and

3 = 1

2\ dyi dxi""' dyn dxnj

/ d i dp \ 2 + / d + i dp \' V dxj 2 dyj V dyj 2 dxjJ

and V = iA<£. We write elements of R2n in the form (X\,y\,... ,xn,yn). We shall identify R2n with Cn, writing (zi,..., zn) = (x\,y\,..., xn, yn), this is mainly because we will use methods of complex analysis to analyze spectral properties of the above Schrodinger operators with magnetic field.

For n = 1, there is an interesting connection to Dirac and Pauli operators: recall the definition of A in this case and define the Dirac operator V by

d d V = (-i— - Ax) ai + (-i— - A2) @2 = Ax<J! + ^2, (1)

where

^ = ( 1 1 ) ' =(i 7)

F. Haslinger, Pauli Operators and the 9-Neumann Problem.

© F. Haslinger, 2017.

Partially supported by the FWF-grant P28154.

Поступила 1 июня 2017 г.

Hence we can write

v( 0 Ai -i A2 V Ai + i A2 0

We remark that i(A2A1 — A1A2) = V and hence it turns out that the square of V is diagonal with the Pauli operators P± on the diagonal:

V2

Af - i(A2Ai - AiA2) + A2

0

0

Af + i(Ä2Äi - A1A2) + A2

(0 A)•

where

=(-à-rf+(-1-4

±V = -Aa ± V

see [3] and [10].

Our aim is to investigate spectral properties of the Pauli operators P±. For this purpose we shall use methods from complex analysis, the weighted 9-complex. We suppose that <p : €n —> R is a plurisubharmonic C2-function.

Let

L2(€n, e-ip) = {g : €n —^ C measurable : \\g\\i = (g, g)v = j \g\2d\ < to}.

Jcn

Let 1 ^ q ^ n and

f = E ' fjdz'J,

IJ\=Q

where the sum is taken only over increasing multiindices J = (j1, and fj e L2(Cn, e-ip).

We write f e L20g)(€n, e-Lp) and define

., jq) and dzj = dzj1 A ■ ■ ■ Adz

3q

\J = 3=1

9f = E ' E ^.d~z3 A dzj

for 1 ^ q ^ n - 1 and

dom(9) = { fe L2m(€n, e~") : df G e-")},

where the derivatives are taken in the sense of distributions.

In this way d becomes a densely defined dosed operator and its adjoint dv depends on the weight

V.

We consider the weighted d-complex

L2o,q-i)(Cn, e~v) A L2{o,q)(Cn, e~*) L(0,q+i)(Cn, e-*)

kl2,

and we set

where

q™ = dd ;+9 *vd,

domp^) = {u G dom(d) n dom(d: du G dom(d^), d^u G dom(d)}.

It turns out that □{°'q) is a densely defined, non-negative self-adjoint operator, which has a uniquely determined self-adjoint square root (□^°'9))1/2. The domain of (□^°'9))1/2) coincides with dom( d) ndom(d(p), which is also the domain of the corresponding quadratic form

Qv(u, v) := (du, dv)ip + (d*vu, d*vv)lf>, and dom(^(°'9)) is a core of (□(°'?))1/2, see for instance [4].

Next we consider the Levi matrix

d2p \n dzj dzjjk=1

and suppose that the lowest eigenvalue ^ of satisfies

Mv = ( )

liminf ^(z) > 0. (2)

This inequality implies that □°,1) is injective and that the bottom of the essential spectrum CTe(D^),1)) is positive (Persson's Theorem), see [6]. Now this yields that has a bounded inverse,

which we denote bv

N<?,1) : L2)i)(Cn,e-^) ^0,1) (Cn, e-^). Using the square root of N^1 we get the basic estimates

Ml < c (\\Bu\\i + (3)

for all u G dom(9) n dom(9*), see [5] for more details.

In the following it will be important to know conditions on p implying that the Bergman space of entire functions

A2(Cn, e-ip) := L2(Cn, e-v) n 0(Cn) is infinite dimensional. This space coincides with kerd, where

d : L2(Cn, e-Lp) ¿2°,1)(Cn, e-<p).

If n = 1, we can use the Mowing concept. Let D(z,r) = {w : \z - w\ < r}. A non-negative Borel measure ^ on C is doubling, if there exists a constant C > 0 such that for any z G C and any r > 0

/j.(D(z, r)) < Cp(D(z, r/2)). (4)

It can be shown that

D(z, 2r)) > (1 + C-3)/j.(D(z,r)), (5)

for each z G C and for each r > 0 in particular ^(C) = x>, unless ^(C) = 0 (see [9]). Example: if p(z, z) is a polynomial on C of degree d, then

d^(z) = \p(z,z)\a d\(z), a > -1

C,

Theorem 1.1. [2], [7] Let p : C —> R+ be a subharmonic C2-function. Suppose that d^ = ApdX is a non-trivial doubling measure.

Then the weighted space of entire functions

A2(C,e-v) = {f entire : \\f \\2 = i \f\2e-^ d\ < }

J C

is of infinite dimension.

More general, in Cn, Hormanders L2-estimates for the solution of the inhomogeneous Cauchv-Riemann equations yield

Theorem 1.2. [8], [5] Suppose that the lowest eigenvalue satisfies

lim \z\2^v(z) = +&>. (6)

Then the weighted space of entire functions

A2(Cn,e-n = {f entire: \\f \\2 = / \f \2e-^ d\ < }

jcn

is of infinite dimension.

Concerning compactness of the 9-Neumann operator we have the following result:

Theorem 1.3. [5] Let 1 ^ q ^ n. Suppose that the sum sg of the smallest q eigenvalues of the Levi matrix Mv satisfies

lim sq(z) = +to. (7)

Then the d-Neumann operator

K°'q) : Lfo,q)(Cn, e-v) Lf0,q)(Cn, e-^)

is compact.

The next result asserts that compactness percolates up the d-complex.

Theorem 1.4. [5] Let 1 ^ q ^ n — 1. Suppose that N^'^ is compact. Then N^'q+1 is also compact.

We will also consider special weight functions, the so-called decoupled weights, and, using the tensor product structure of the essential spectrum ae (D^'^) we get the following (see fl])

Theorem 1.5. Let pj G C2(C, R) for 1 ^ j ^ n with n > 2, and set

p( zi,..., zn) := pi( zi) +-----h pn( zn).

Assume that all pj are subharmonic and such that Apj defines a nontrivial doubling measure. Then

(i) dim{ker(D^'0)) = dim(A2(Cn, e-^)) = to, where D^0'0 = d* d,

(ii) ker(di0'q)) = {0}, for q> 1,

(in) N*0'q) is bounded for 0 ^ q ^ n,

(iv) N*0'^ with 0 ^ q ^ n — 1 is not compact, and

(v) N^°'n) = dd* is compact if and only if

lim / tr(Mp) d\ = to, where B\(z) = {w e Cn : lw — zl < 1}.

2. Pauli operators

Now we apply the results on the weighted d-Neumann operator to derive spectral properties of the Pauli operators and discuss some special examples.

Theorem 2.1. Let p : Cn —> R be a plurisubharmonic C2-function. Suppose that the smallest eigenvalue of the Levi matrix Mp satisfies

lim ^¡p(z) = to. (8)

1 i dp dp dp dp \

Let

2 V dyi' dxi''"' dyj dxnJ and V = l Ap. Then the Pauli operator P- = — A^ — V fails to have a compact resolvent, whereas the Pauli operator P+ = — A^ + V has a compact inverse operator acting on L2(R2n).

Proof. For the proof we first consider the complex Laplacian □ = d*v d, which acts on L2 (Cn, e-ip) at the beginning of the weighted d-complex as a non-negative self-adjoint, densely defined operator, we take the maximal extension from C0^(Cra), as is essentially self-adjoint, there is only one

self-adjoint extension. For f G C0f(Cn) we get

□(o,o) f = g* Qf = — y( A — □ ( ; 0(0î j^Kdz, dzjdzy

Now we apply the isometry

U* : L2(Cn) —> L2(Cn,, e-()

defined by Uv(g) = e'p/2g, for g G L2(Cn), and afterwards the isometry

U-v :L2(Cn, e-Lp) L2(Cn) defined by U-V(f) = e-v/2f, for f G L2(Cn, e-v). Hence we get

n /

e-,/2D(0)0)(ep/2 g) = £ (

3 = 1 ^ '

d2g 1 dp dg 1 dp dg 1 dp dp 1 d2p

d Zjd Zj

+ 2 d Zj d Zj

2 dzj dzj + 4 dzj dz j

2 d Zjd Zj

5

and separating into real and imaginary part

d 1 id d

dz i

2 \dxi

A) JL = 1fJL +

d y J ' d~Zj 2 \dx3 d Vi)

we obtain

where

1

e-(/2D(°'0)(ep/2 g) = 4(-Aa -V)g,

(9)

a =

2 \ dy 1 ' dxi

d p d p dyn ' dxn

and

V = 2tr(M() = -Ap.

Since the kernel of 9 : L2(Cn, e-^) —> L2° 1)(Cn, e-^) coincides with the Bergman space A2(Cn, e-v) we get from (9) and the fact that (8) implies that A2(Cn, e-v) is infinite dimensional (see Theorem 1.2) that 0 G ae(^°,0)). Hence □(°,0) fails to be with compact resolvent.

In order to show that the Pauli operator P+ has a compact inverse we look at the end of the weighted 9-complex.

Let u = u dz1 A ■ ■ ■ A dzn be a smooth (0, n)-form belonging to the domain of . For 1 ^ j ^n

denote by Kj the increasing multiindex Kj := (1,

J- 1J + 1,

du \

' n) of length n — 1. Then

d > = £ (-1y+1( dp-u -^)d-ZKi.

3 = 1

d Zj)

Hence

ddpu =

Ë M

■hdtA

3 = 1

n / ^

3=1 v

3 \d Z3

d p d u u —

d Zj

d2 p d p d u

* u + ^

dz1 A - ■ ■ A dzn

d2 u

d zjdz j

d Zj d Zj

d Zjd Zj /

dz1 A ■ ■ ■ A dzn.

Conjugation with the unitary operator U-v : L2(Cn,e v) ^ L2(Cn) of multiplication by e ip/2 gives

e-(/2a(°'n) e(/2 g =

n

3 = 1 v

d2g

d Zjd Zj

1 d p d 1 d p d 2 dzj dZj "

1 dp dp 1 d2p

2 dZj d~2j 4 d~Zj dZj^ 2 dZjd~2j^

'

where g G L2(Cra) and we just wrote down the coefficient of the corresponding (0, n)-form. This operator can be expressed by real variables in the form

e-p/20$'n) ep/2g = Vaa + V )g

with

Aa = E

3 = 1

d

(10)

dxj

d p 2 d d p 2 2 dyJ V dVj 2 dx3 J

and V = 2tr (My). It follows that -A^ + V is a Schrodinger operator on L2(R ) with the magnetic vector potential

1 / dp dp dp dp \ 2\ dy 1, 8x1 , dyn, dxn) ,

where Zj = Xj + iyj, j = 1,...,n, and non-negative electric potential V in the case where p is plurisubharmonic.

From (8) we get that N^'11 is compact (Theorem 1.3) and by Theorem 1.4 that is compact.

Finally (10) implies that the Pauli operator P+ has a compact inverse.

□

For decoupled weights p(zi,..., zn) = pi(zi) + ■ ■ ■ + pn(zn) even more can be said. Theorem 2.2. Let pj e C2(C, R) for 1 ^ j ^ n with n > 1, and set

p( zi,..., Zn) := pi( zi) +-----+ pn( Zn).

Assume that all pj are subharmonic and such that Apj defines a nontrivial doubling measure. Let

1 i dp dp dp dp \

2\ dy i' dxi''"' dyn dxnJ

and V = 2Ap. Then the Pauli operator P- = — A^ — V fails to have a compact resolvent, the Pauli operator P+ = — A^ + V has a compact inverse if and only if

lim / tr(Mp) d\ = то, where Bi(z) = {w G Cn : lw — zl < 1}.

Proof. By Theorem 1.1 we obtain that A2(Cn,e~v) is infinite dimensional. So, P_ fails to be with compact resolvent. The assertion about P+ follows from Theorem 1.5. □

Example: For p(zi,..., zn) = |zil2 + ■ ■ ■ + |znl2 both Pauli operators P_ and P+ fail to be with compact resolvent.

Finally, we get the following result for Dirac operators (1).

Theorem 2.3. Let n = 1 and let p be a subharmonic C2-function such that Ap defines a nontrivial doubling measure. Then the Dirac operator

x>-( — 1 dpp\ ( -dL 1dpp\

\ dx 2 dy J ai \ dy 2 dx J a2,

dx 2 dy J \ dy 2 dx

where

' 0 M ( 0 -

= ( 0 1 ) , = ( 0 0 ) ,

x 1 0

fails to be with compact resolvent.

Proof. By spectral analysis (see [5]) it follows that V2 has compact resolvent, if and only if V has compact resolvent. Suppose that V has compact resolvent. Since

I P- 0 2-

(P-PJ

this would imply that both Pauli operators P_ and P+ have compact resolvent, contradicting Theorem 2.2.

□

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Friedrich Haslinger,

Fakultat fur Mathematik, Universitat Wien,

Oskar-Morgenstern-Platz 1,

A-1090 Wien, Austria

E-mail: friedrich.haslinger@univie.ac.at