energy decay for the neutrino equation in the exterior of a torus

Energy Decay for the Neutrino Equation

in the Exterior of a Torus

RALPH PHILLIPS • LEONARD SARASON

Communicated by J. SERRIN

Abstract Local energy decay is established for the solutions of the neutrino equation in the exterior G

of a torus for a class of boundary conditions, described as follows: To each energy conserving boundary condition at a point x on 3G there corresponds a vector in the tangent plane to t3G at x. The result has been proved when the torus and boundary conditions are axially symmetric and when the paths generated by this vector field are closed. What is novel about this problem is the fact that the boundary conditions are nowhere coercive.

1. The Neutrino Equation

In the theory of scattering for hyperbolic partial differential equations in an exterior domain, the local energy decay of the solution plays a central role. Previously this property has been proved f rom the coerciveness of the generator, and this in turn required the ellipticity not only of the corresponding differential operator but also of the boundary conditions (see Chapter VI, Part 1 of [4]). The neutrino equation can not be treated in this way even though the differential operator is elliptic because there are no boundary conditions which are both energy conserving and elliptic (see p. 206 of [4]).

We recall that a system is said to have the local energy decay property if for any solution u(x, t) (with initial data orthogonal to the null set of the generator) the energy in any bounded subdomain tends to zero as I t I ~ m. If the system is energy conserving on the entire domain, then it suffices to prove the local energy decay for a dense set of such solutions. To establish the local energy decay for a particular solution it is enough to show that the set of all data attained by the solution is locally compact, provided that the generator has no nonzero point spectra. Coercivity, of course, implies this and much more. In the present paper we are able to prove (in the absence of coercivity) the local energy decay for the neutrino equation in the exterior of a regular torus and for a certain class of energy-conserving boundary conditions. Unfortunately, our method does not handle all energy conserving boundary conditions; we feel, however, that the problem which we have solved is sufficiently interesting in itself and that the approach is sufficiently novel to warrant publication.

1 Arch. Rat ional Mech. Anal. , Vol. 41

2 R. PHILLIPS • L. SARASON"

The neutrino equation is of the form

3

(1.1) ut=Lu= ~ ~z~ ~iu, x i n G c R 3 i = 1

where u is a two-component vector and

1 0 = ( 1 ~ ) ' ~Z2=(?i 0 ) ' ~3=(10 g l ) "

The energy is defined in terms of the norm:

(1.2) Energy = 11 u ]l 2 _ ~ u . u d x - (u, u). G

We denote the Hilbert space of square integrable vector-valued functions on G by H. An integration by parts shows that

(1.3) d~Energy=(u, Lu)+(Lu, u)= ~ Bu. u dS ~G

where B denotes the "no rma l " matrix 3

(1.4) B(x) = ~ ~i ni i = l

and n=(nl, n2,/'/3) the outer unit normal to the boundary surface ~G. We note that for each unit vector n the matrix B is Hermitian with eigenvalues - 1 and 1.

We shall consider only local energy conserving homogeneous boundary conditions which are maximal with respect to being energy conserving. It follows from the relation (1.3) that this requires at each point x on OG that the solution u lie in some one-dimensional null space N(x) for which

(1.5) B(x)u. u = 0 for all u in N(x).

The condition (1.5) can be reformulated as follows: To each u we make correspond the real three-vector

q(u )= (~ 1 u . u , ~ z u . u , ~3 u . u ) - ~ u . u .

Under this correspondence a one-dimensional subspace M = {cu~} (not necessarily null) maps onto the ray {] c l2q(ul)). In particular, for the null space M=N(x) , x on aG, the condition (1.5) can be rewritten as q(ul) �9 n=0, which means that the ray corresponding to N(x) lies in the tangent plane to a G at x.

It is easy to see that the one-dimensional subspaces M = {cu I } are in one-to-one correspondence with the vectors

i e i~'/2 cos 0/2 ul= ~e_i~,/2sinO/2 ] for 0<0<zc and 0<~0<2zc.

A simple calculation shows that the ray corresponding to M is generated by

q (ul) = (cos ~0 sin0, sin~o sin0, cos 0).

Hence the mapping: M={cu~} ~ q(ul) defines a one-to-one bicontinuous transformation of one-dimensional subspaces onto $2. In particular each maximal

The Neutrino Equation 3

local energy conserving homogeneous boundary condition defines a tangent vector field on a G, and if we require N(x) to be smoothly varying, as we do, then this vector field will be continuous. Conversely, any continuous tangent vector field can be represented by a suitable continuous choice of one-dimensional null spaces N(x) on a G. Since the only surface in three space which supports a continuous vector field is homeomorphic to a torus, it follows that we can find suitable maximal local energy conserving homogeneous boundary conditions for the neutrino if and only if a G is homeomorphic to a torus. It is intriguing to speculate on the physical significance of this property.

Next we define the operator A ~ as L acting on all smooth functions with bounded support in G and satisfying the boundary condition: u(x) in N(x) for all x on OG. It can be shown (see [1], [3] and [6]) that the closure A of A ~ is a skew selfadjoint operator on H and hence the generator of a group of unitary operators {U(t)}. For the free space problem where G=R3, we denote the corresponding space, generator and group by Ho, Ao and {Uo(t)}, respectively.

2. The General Theory

Our approach to the energy decay problem for the neutrino equation is quite straightforward; it is complicated only by the fact that it requires the existence and smoothness of an essentially complete set of generalized eigenfunctions. In the present section we develop a suitable general theory for this problem based on the Lax-Phillips approach to scattering theory [4]; as we shall see the neutrino problem requires some interesting variations on [4]. The existence of generalized eigenfunctions will be established in Section 3.

We begin by defining incoming and outgoing subspaces D_ and D§ respectively, for the neutrino equation in the exterior of a torus; assuming that the torus is contained in the ball [I x l <p], we put specifically

D_=[f;[U(t)f](x)=O for Ixl<p-t, t<__0], (2.1)

D+--- [ f ; [U( t ) f ] (x)=0 for Ixl<p+t,t>O].

It can be shown that D_ and D+ are orthogonal subspaces (see Chapter VI, Section 1 of [4]). In terms of D_ and D+ we define

H_ =closed linear span of U U(t)D_,

H+ =closed linear span of U U(t)D+,

Z--- null space of A, H' = H O Z.

It is clear from these definitions that

1"

i)_ U(t)D_ cD_ for all t ~ 0 ,

i)+ U(t)D+ cD+ for all t > 0 ,

ii) NU(t)D_={O}=NU(t)D+.

The theory of scattering as developed in [4] requires in addition that

iii) H_=H'=H+.

4 R. PHILLIPS & L. SARA,SON:

(2.2)

where

The main ingredient for the proof of (iii) is the energy decay theorem. As a first step in this direction we prove

Theorem 1. H_ + H+ is dense in H'.

P r o o f . It suffices to show that any data f orthogonal to H_ and H+ necessarily belongs to Z. Since U(t) is unitary it is clear for such an f that U(t)f is orthogonal to D_ and D+ for all t. We denote the Radon transform of f ( t ) = U(t)f by h(s, co; t):

(s, co) = ~ 0~ m (s, co) h Z T ~

(2.3) m(s, co)= 2~xJ=J(x)dS.

Making use of the fact that f ( t ) is orthogonal to D_ and D+, it can be shown (see Chapter VI, Section 1 of [4]) that h(s, co; t) has its support in the interval ] s [ < p for all t.

The relation (1.26) of Chapter VI of [4] shows that

(2.4) [Uo(t-to)f(to)](x)= ~ hl(xco-~i(co)(t-to),co; to)rl(co)dco [,o1= t

where by definition

(2.5) - ( Z cz' coi)rj(co)=zj(co)rj(co) and hi=h . rj;

note that ~l(co)=l for the neutrino equation. On the other hand, a domain of dependence argument implies that

(2.6) [Uo( t - to)f (to) ] (x) = [U (t) f ] (x)

for [xl > p +6 if I t - t o 1<6. Hence if we follow the argument used in the proof of Theorem 3.3 of Chapter IV in [4] and employ (2.4) and (2.6) above, we see that the m th spherical harmonic coefficient of U(t)f, namely

u,,(r, t)=~ [U (t) f ] (rO) Y,,(O) dO,

can he extended to be an analytic function of t in the strip I t o - R e t 1<6 for each r>p +6. The argument then proceeds as in the proof of Lemma 2 on p. 258 of [4]. We conclude that [U(t)f](x) is constant in time for all x of absolute value greater than p + 6. Setting

g(q, t2)= U ( q ) f - U(t2)f

for fixed t~, t2, it follows that [U(t)g](x) vanishes identically for [x[>p+6 and all t; applying HOLMGREN'S uniqueness theorem we see that g vanishes through- out G. Since t~ and t2 were arbitrary, we conclude that U(t)f is constant in time and hence that A f = O, as was to be proved.

Corollary 1. The generator A has an absolutely continuous spectrum on H'.


Proof. This assertion can be verified directly for the free-space generator Ao by taking Fourier transforms (see p. 216 of [4]). Moreover, the wave operators:

(2.7) W• f = st. lim U ( - t) Uo (t) f t ~ :k o0

exist and are unitary mappings of H o onto H+ and H_, respectively, (see p. 5 of [4]). It therefore follows from the intertwining property of the wave operators:

W• Ao =AW• ,

that the spectrum of ,4 is absolutely continuous on H+ and H_ and hence on the closure of H_ + H+ = H'.

The above corollary furnishes us with the principal ingredient for a proof of a Rellich type uniqueness theorem for the reduced neutrino equation.

Coronary 2. I f f is a local solution satisfying the boundary condition of the eigenvalue equation:

(2.8) A f= i a f

for a ~0 and real, and if f is eventually outgoing or initially incoming, then f is identically zero in G.

Proof. Given that A has no point spectrum other than 0, an argument identical with that used in Theorem 2.3 of Chapter V in [4] suffices to prove the above assertion.

Remark. Essentially the same argument shows that Theorem 1 and its corol- laries continue to hold for all first order hyperbolic symmetric systems with constant coefficients and no zero velocities in an exterior domain in R,, n odd, provided of course that the boundary conditions are smooth and energy conserving as in Section 1.

It is convenient to introduce the notion of a translation representer for f in Ho; in terms of the function m(s, 09) of (2.3) and r 1 of (2.5) this is

(2.9) k(s, 09)=Osm(s, 09). r, (o~).

The mapping f -~ k is unitary from Ho onto L 2 (R, $2) (see p. 184 of [4]). The inverse map can be extended to distributions so as to include such non-Ho functions as fundamental solutions in these considerations (see pp. 191-197 of [4]).

Next we construct the outgoing spectral representation for A restricted to H+. We begin by setting

itr e -i~ r(09) ~o(X; ~, ~o) = (-T~)~ (2.10)

where

Thus -(E r(09)

(L- ia) ~Oo =0.

6 R. PHILLIPS ~; L. SARASON"

The vector-valued function r(~o) can be chosen to be Borel measurable and of unit norm on $2. The unitary mapping

(2.11) Fo: f i n Ho~fo=( f , 9o( . ;a , co)) in L2(R,N)

is readily verified to be the free space spectral representation of A o (see p. 216 of [4]); here N=L2(S2).

For the corresponding spectral representation of A restricted to H+ we need an initially incoming (see p. 194 of [4]) solution v_ (x; a, m) to the boundary value problem:

(2.12) ( L - i a ) v _ = O in G

such that

(2.13) tp_ =r + v-

satisfies the prescribed boundary condition on ~ G. In Section 3 we shall obtain v_ as a surface plus volume integral of the incoming fundamental solution over the scattering object, in this case the torus.

The incoming fundamental solution matrix 7 of L - i a can be constructed as on pp. 124-125 of [4], and it is clear from this construction that its translation representer will vanish for s>0 . It is somewhat simpler to define 7 in terms of the fundamental solution

70 = ~ - ~

of the reduced wave operator A +or 2, here r=lx-yl. Because of the fact that L 2= AI, we can see by inspection that

is a fundamental solution of L x - icr. To see that 7 is incoming we argue as follows: It is known that (70, ieTo) is incoming for the acoustic equation. Setting u ( t )= Uo(t)7, then we see as above that utt=Au and for r > 0 that u(0)= 7 and ut(0)= L 7 = io 7. Since 7 is a linear combination of 70 and its first derivatives, we conclude that u(x, t ) = 0 for Ix [< - t and t < 0. This proves that 7 is incoming, from which it follows that its translation representer vanishes for s > 0. As a consequence the translation representer of 7(x -a ) vanishes for s > l a l , and hence that of v_ vanishes for s > p.

Theorem 2. The outgoing spectral representation for A on H+ is given by the following unitary mapping:

F+: f in H+--*f+=(f,~o_(.;a,~o)) in Lz(R,N) ,

where N = L 2 ($2).

Proof. The proof of this theorem follows that of Theorem 5.3 of Chapter V in [4] (see also Theorem 2.6 of Chapter VI in [4]). Thus for f in Cd ~ (G), jr+ (a, o9) is well-defined and measurable in tr, 09. In particular if f is restricted to lie in


D+ c~ C~(G) (which is dense in D+), then f has a translation representer which vanishes for s< p; and since, as noted above, the translation representer for v_ vanishes for s > p we obtain

(f, v_)=0

from the Parseval relation for the translation representers. Thus in this case

L =(s, o)--fo and since the free space spectral representation Fo is known to be unitary, it follows that F+ is isometric on D+ c~ C~(G).

Again proceeding as in Theorem 5.3 of Chapter V in [4] we verify that

F+[U(t)f]=e~~

for f in C~ ~ (G); consequently F+ is isometric on U U(t)[D+ c~ C~(G)] and hence, by completion, on all of H+.

Since F+ coincides with Fo on D+ we may conclude that like Fo it takes D+ onto the Hardy class in L2(R, N). Recalling that the union of exp(iat) times the Hardy class (as t ranges over Rt) is dense in L2(R, N), we see that F+ is in fact a unitary transformation onto L2 (R, N).

Corollary 3. The inverse of F+ is given by

F+l: f+ in L 2 ( R , N ) ~ f = ( L , ~o_(x;., . ) ) in H+.

Proof. For a unitary mapping F -1 =F*; the above characterization of Fu t is clearly the adjoint of the transformation defined in Theorem 2.

We denote by HI(GR) the Hilbert space of square integrable functions on GR=Gc~ [Ixl<R] with square integrable first derivatives (in the sense of distributions);

Ilgll~<R)={ $ l-lgl2+lVgl2]dx} �89 G(R)

In the next section we shall prove the following basic result for the neutrino equation in the exterior of a regular torus subject to a certain class of boundary conditions.

Theorem 3. There exist subsets (K,} of Rt, K~K~§ meas(RI-UK~)=0, such that the functions ~o_ (x; a, co) exist and are uniformly bounded in 1-11 (GR) for cr in K~ and all 09 of length one, n = 1, 2, ....

Assuming Theorem 3 we can now prove the local energy decay for solutions of the neutrino equation in the exterior of a regular torus, again subject to a certain class of boundary conditions.

Theorem 4. lim [[U(t)fil~(g)=O for all f in H'. [tl-,oo

Proof. Since the solution operators {U(t)} are unitary it suffices to prove the assertion of the theorem for a dense subset of H'; hence, applying Theorem 1, we see that it suffices to prove the assertion for dense subsets of H_ and H+. We shall restrict our considerations to H+ ; an analogous argument holds for H_.

8 R. PHILLIPS • L. SARASON:

Now for f in H+ it follows from Theorem 2 and Corollary 3 that

[ U (t) f] (x) = (e't~f+, (p _ (x; . , .)).

Next we choose f so that f+ (a, co) is bounded with a-support in the subset K. of Theorem 3; the union of all such data f is dense in H+. For such data f

(2.15) 1[ U(Ofll = II/ll

and ]Ox[U(t)f ] (x)[ 2< Ill+ II = [[lOx ~o_ ( x , . , .)ILK"] 2 ,

from which we obtain

(2.16) IIV[U(Of](x)II~(R)<IIflI { 5 Y X[Vq'-(x; a, co)12dcodadx} ~. G(R) Kn

According to Theorem 3

IIq~-(", or, co)ll~ ~n)

is uniformly bounded for co in Sz and o- in/,2, so that the right member in (2.16) is finite and obviously independent of t. The relations (2.15) and (2.16) together imply by the Rellich compactness theorem that the functions {[U(t)f](x); - oo < t < oo} are compact in L2 (G(R)). On the other hand, we see by Corollary 1 that U(t) restricted to H+ has an absolutely continuous spectrum so that by the Riemann-Lebesgue Lemma

(2.17) (U(t) f, g)=(e"" f+ , ~+)~0

as It[ ~ ~ for all g in H+. Since U(t)fremains in H+ for all t, the relation (2.17) is valid for all g in H. Thus {U(t)) c} are locally compact and converge weakly to zero; we conclude that they converge locally to zero in the norm topology. This completes the proof of Theorem 4.

An important consequence of the energy decay theorem is

Corollary 4. H _ = H' = H + .

The argument here is precisely the same as the proof of Lemma 2.2 of Chapter V in [4].

We have now established the fact that D_ and D+ are indeed orthogonal incoming and outgoing subspaces in the sense of [4] for the neutrino equation in the exterior of a regular torus subject to a certain class of boundary conditions. This suffices for the basic scattering theorems of [4] to hold. In particular the scattering matrix exists and is unitary on N for real a, and it is the boundary value of an operator-valued function analytic for Im z < 0 (see Chapter II of [4]).

3. Existence of Generalized Eigenfunctions

By a regular torus we shall mean a torus generated by rotating a smooth, closed, non-self-intersecting curve about an axis lying in the plane of the curve but which does not intersect the curve. We shall take 0 G to be a regular torus with the x3-axis as its axis of symmetry.


The class of boundary conditions under which we shall prove Theorem 3 is determined by the following restrictions:

(i) The integral curves on d G generated by the vector field q(x) should be c lo sed .

(ii) The family of such integral curves should be invariant under rotations about the xa-axis, and

(iii) q(x) is never parallel to ( - s i n ~, cos ~, 0), where ~ is the polar angle in the xl , x2, plane:

tan �9 = x2/xl.

Before proving Theorem 3 we make three

Remarks. (a) The neutrino equation is semi-invariant under a rotation of the independent spatial variables. If x ~ y is such a rotation (with determinant + 1), then there is a unitary mapping of the dependent variable u which brings the equation into its original form; moreover this can be done continuously.

(b) If u leN on OG, then Bul. B(Bul)=O, and q(Bul)=-q(ul ) . For a normalized choice of ul, we shall denote Bul (x) by a(x). (Because of Remark (a), it is sufficient to check Remark (b) with n=(O, O, 1).)

(c) At each point of OG, the vectors ul(x) and a(x) form an orthonormal basis for E 2. Consequently the boundary condition (r is equivalent to the condition v �9 a = -~Po" a.

Proof of Theorem 3. Denote the function q~_ -~Po of Theorem 3 by v and set -q~o" a=g. We seek an initially incoming solution to the equation

(3.1) (L - io )v=O in G, o real,

with boundary condition

(3.2) v. a = g on ~G.

If K is any bounded subset of R~, then for each real r the functions g = -r (a, co). a are uniformly bounded in 11,(0 G) as the vector (a, co) ranges over K• $2. Hence it suffices to require of each K, that for some r the map E: g ~ v , considered as a map from H,(OG) to H1 (G~) is uniformly bounded.

We take as an ansatz for our solution of (3.1) and (3.2),

(3.3) v(x) = T~ f + V ~ h, x q~ OG

where

(3.4) TO f = I [(L, - i a) 7o (x, y)] a (y)f(y) d r S, 0G

with f an unknown scalar function defined on d G, and

(3.5) V ~ h = I [ ( L , - i t r) 7o (x, y)] h (y) d y D

with h an unknown vector-valued function defined on the interior D of the torus. We also require of h that it satisfy the equation

(3.6) - h + l t V ~ 1 7 6 in D

10 R. PHILLIPS ~r, L. SARASON:

where/~ is an arbitrary but fixed scalar function, chosen such that

/t~C~~ ~=0 in G, and p > 0 in D.

Note that both (3.4) and (3.5) are linear combinations of the incoming fundamental solution expressed in (2.14) with LxTo instead of L r 7o = - L x 70.

Now for x on 0 G we have by the usual jump condition

v(x) = �89 B (x) a (x) f(x) + ~ [ (Ly - i or) 70] a (y) f(y) dy S + V~ h. OG

Since Ba. a = 0 , (3.2) takes the form

Tf+ Vh = g on OG, (3.7)

where

and (T f ) (x) = ( T~ f ) (x) " a (x)

(Vh)(x)=(V ~ h)(x), a(x).

We shall treat T as a pseudo-differential operator (see [2]). The difficulty with this approach comes from the fact that the term of highest order is not elliptic; however, the sum of the two highest-order terms is invertable modulo a compact operator for most values of a. Our analysis depends on the fact that on a two-dimensional compact manifold, an integral operator of the form

f"* I k(x, y) r-S f(y) dy S

with k smooth, has order s - 2 , that is, it maps H r boundedly into H,+2-~, if s < 2 or when s = 2 and j" k(x, y)dyS=O for all x.

Since the highest order term in T comes from the tangential derivatives of l/r, T has order zero. We expand T in terms of decreasing order, looking at the first two terms explicitly. Within this order of accuracy, we can replace a(x) by a ( y ) + ( x - y ) . ~a, and we can replace Ly by L t, the tangential part of Ly; the latter substitution is justified because the normal derivative of r is d)(r) and B(y) a(y). a(x) is (9(r), and hence the error introduced has order - 2 . Further,

1 we can replace 7o by 4-~r- ' since

( L t - i t r ) ~ - = ( L t - i a ) l+d~( l ) .

Thus the significant part of T (the part of order greater than - 2 ) is given by

1 4re S (Ltr-1)a(Y) " a(y)f(y)dyS

OG (3.8)

+ ~ (Ltr-1)a(y) �9 [ ( x - y ) . Vya(y)]f(y)dyS- ~ i a r - t f ( y )dyS . OG ~G

In order to use the theory of pseudo-differential operators it is convenient to introducc local coordinates on a G. Let C c ~ G bc a closed integral curve of the field q(x). In a neighborhood No of C on aG, let e(x) be a field of unit tangent vectors orthogonal to q, and let 2 be an integral curve of the field e(x). Then the


integral curves of the fields q(x) and e(x), together with a parameterization of C and of ). determine a (local) coordinate net for some neighborhood N c N o of C. We denote the coordinates by 0 and q~, respectively, and define

s=lOx/~OI, and t=lOx/d4,1.

With V. q and 17. e denoted by a~ and de, respectively, we have

d d aq=s - l ~ and de= t -1 dq~ "

Further, we define ~0=~. q---~ ~jqs and ~§ �9 e.

Except in the proof of Lemma 1, we can and will assume that the above coordinatization extends to all of ~ G in such a way as to cover it exactly once; the fact that the coordinate curves of constant 0 are not in general closed does not impair the argument below.1

Using the coordinates 0, ~b to describe the torus, we have

L t = s - l otot3o+ t - t t ~ t3~ ,

and with the help of remark (b),

~ , a . a = ( ( ~ , e) a . a ) = - q , e=O,

ctoa . a = ( ( ~ , q)a . a ) = - q . q = - l .

dy Also the Jacobian ~ of the transformation equals s t.

In terms of 0 and q~ we have

(Ltr-1)a �9 a f drS=~ ~ s- l (~or-1)~oa . a f st dO d(a dG

= - S ~ (aor-t) f tdOd~b=~ ~r - l do( t f )dOd~

= ~ r - l ~ q f d r S + i ~ r - l k l f d r S c~G 8G

where ik i = (ts)- 1 do t. Hence if C is a closed integral curve of the field q,

~k~ I d x l = ~(ts)-~(~ot)sdO -- ~do(logt) dO=O. C C C

Thus the first integral in (3.8) can be written as

1 (3.9) 4-'-r~ o~ r- ' ~gqf(y)drs+lo~ iki(y)f(y)d~,S where ~ k~ds=O; here ds is the element of arc length along C.

c Next we consider the second integral in (3.8). Since x - y and raLtr - i are,

to highest order, both homogeneous of degree one in x - y , the leading term in the

1 One can avoid introducing global coordinates by introducing a partition of unity {Pi} such that 8oPi=O for all i.

12 R. PHILLIPS t~, L. SARASON:

kernel has the form r - 3 Q ( x , x - y )

where Q is quadratic in its second argument x - y .

Lemma 1. The second integral in (3.8) can be written in the form

(3.10) i ~ r -1 k2(y)f(y)dyS+ T_2(Oqf)+Rof dG

where T_ 2 and R o are pseudo-differential operators of order -2 , and where the line integral ~ k2(x)ds along any of the closed integral curves of the field q(x) is

C

real (here ds is again arc length).

Lemma 1 will be proved in the Appendix.

Inserting (3.9) and (3.10) in (3.8), we find

(3.11) Tf= ~ r - l [Oqf- i ( t r+v(y)) f ]dyS+ T_2(O, f )+Rl f OG

where R1 has order - 2 and ~ vds is real. c

With the operator F-} ~ I x - y l - l F ( y ) d y S

OG

denoted by K, (3.1 l) can be rewritten in the form

(3.12) Tf =(K + T-2)(dqf - i {a + v(y)) f ) + R f

where R has order - 2 .

Because of the semi-invariance of the neutrino equation and properties (i) and (ii) of the field q(x), the operator Oq-iv(y), considered as an operator from L2(0G ) to itself, has a discrete point spectrum {2,), determined by the equation ~{v(y)+A.}ds=2rcn. Thus if ~ v(y)ds=m and if C has length l, then C C

(3.13) 2 , = l - l ( 2 n n - m )

and the operator D~=dq-i(a+v(y))

is uniformly invertable as tr ranges over any compact set which is uniformly bounded away from the set { -2 . } (note that m is real). In particular, if tr+ - 2 , , n =0, ___ 1, _+ 2 . . . . . then D~ is a Fredholm operator and has index zero from L2(OG ) to L2(OG ). From this point on we shall assume that trr

The operator K is an elliptic pseudo-differential operator of order - 1 with non-negative leading symbol, as can be checked by using an orthogonal coordinate system. Further, K has index zero as an operator from L2 (a G) to H~ (c~ G), since it is homotopic to the operator K 0 on S 1 x S ~, given by

Ko(Za,,.ei~"~176176 0<0_<2rr, 0<q~<2~r.


Since the operator T_2: L2(tgG ) is compact, K - i - T _ 2 also has index zero. Therefore the product

(K + T_ 2) D~: L 2 (~G)--*H 1 (OG)

has index 0 + 0 =0. Finally, since R is compact as an operator from L 2 (a G) to H l (~G), T and (K+ T_2)D ~ have the same index. We conclude that T: L2(dG) H1 (a G) has index zero.

Our aim is to show that the equation pair (3.6) and (3.7) has a unique solution, and that it is equivalent with an equation

(3.14) 7"(f, h)=F

such that 7"has index zero. It will then follow that a solution exists.

To this end we substitute for h in (3.7) its value as given by (3.6). This gives us (3.14) with F={g, 0}, and where

T: L2(aG ) x L2(D)--.H 1 (~G) x L2(D ) is defined by

~'{f, h}= {(T+ Vt~ T~ f + V~ V~ h, - h + p V~ h + lt T~ f } .

To complete the proof that T has index zero it suffices to prove the following lemma.

Lemma 2. The following operators are compact:

(a) #V~ L2(D)~L2(D),

(b) # TO: L 2 ( a a ) ~ L 2 (D),

(c) VpV~ L2(D)~HI(t~G),

and

(d) V#T~ L2(dG)--',HI(tgG ).

Proof. (a) holds because V ~ has order - 1 a n d / ) is compact. To prove (b), we observe that if x has distance 6 to ~3 G, then the kernel of T O is tV(t~-2), and its derivatives of order < m are r Since p~C~~ the kernel of pT ~ has bounded derivatives of all orders, and #T ~ maps L2(aG) boundedly into H,,,(D) for all m. This last fact (giving (b)), together with the fact that V is a bounded map from H 2 (D) to H 2 (O G) shows that V# T O is a bounded map from L2 (~ G) to HE (~ G) and hence is a compact map from L2 (t~ G) to H 1 (~ G).

To prove (c), observe that V ~ and V are each of order - 1, that is, they are bounded operators from H,,,(D) to H,,,+I(D). Therefore V#V ~ maps L2(D ) boundedly in to / /2 (D), and its restriction to a G therefore maps L2 (D) boundedly into H~(aG), and hence compactly into HI(t3G). This completes the proof of Lemma 2.

We have shown that 7" has index zero. To show that (3.14) has a solution for all F, it remains to show that the kernel of Tis {0, 0}, that is T{f, h} = 0 implies that f = 0 and h = 0.

Suppose now that g = 0 , trr and that the pair {f,h}eL2(OG)xL2(D ) and satisfies (3.14). Let v be defined by (3.3). Then V~ and the

14 R. PmLLIVS & L. SARASON:

restrictions of T~ to D and to GR are in / / i (to justify the statements about T~ we refer to AGMON, DOUGLIS & NmENBERG [5], who show that near a G the tangential derivatives of T~ are in L2; that the normal derivative is also in L2 follows from the fact that T~ satisfies the reduced neutrino equation.)

It follows that v is a local, initially incoming solution of the reduced neutrino equation (3.1), satisfying the smooth conservative boundary condition v - a = 0 . According to Corollary 2, then, v - 0 in G. The jump in v across 0 G is B(x)a (x)f(x), so the jump in v-a equals 0. It follows that the restriction of v to D satisfies the boundary condition v- a = 0 on 0 G.

Because of the ansatz (3.3), v satisfies in D the equation

(3.15) (L- ia ) v+h=O.

From (3.6) and (3.3) we see that h=#v in D. Therefore (3.i5) implies

(3,t6) Lv+(kt+ia)v=O in D.

Since v satisfies the dissipative system (3.16) together with the conservative boundary condition v . a = 0 on OG, and since w i l l (D), we conclude that v = 0 in D. I Therefore, as h=l~V in D, we conclude that h=0 . Finally, from the facts that v = 0 both in D and in G, and that the jump in v across 0G, namely B(x)a(x)f(x), also vanishes, we conclude that f = 0 . Hence f , h =0 , as was to be shown.

To complete the proof of Theorem 3 it suffices to show that 'F depends continuously on a, for then so does 7"-1 wherever it exists. In this case the subsets K, of R 1 defined by the conditions

(i) l a l < n , and

(ii) la -2 j t> l /n for all integersj,

dearly satisfy the conditions of Theorem 3. In particular, it suffices to show that O~/Oa is bounded. But

Now

aT lOT ~ OV o V#~f+__dZ_l~VOh~ {f, h}=l_~_aj+_.~_#T f+ OT ~ OV

_ O V ~ , ~ V ~ . OT ~ _') + v # # n , l t~ t t+~ t - - f~a J ~.

1 As in (1.3) we now have

O=(Lv+ (g--ia) v, v)D+ (v, Lv+(#-- ia) v)o= 2Otv, v) o - f By" v dS= 2(gv, ~)0 OG

so that v----O in D .


One sees by inspection that O T/8 a is an operator of order - 1 , and therefore is a bounded operator from L 2 (0 G) to H 1 (c3 G). Similarly, ~ V~ is a bounded operator from L2(D) to H2(D). Arguments similar to those in the proof of Lemma 2 combined with differentiation under the integral sign show easily that ~3 V OT ~ 0 a : Lz (D) ~ H 1 (a G) and it ~ : L z (0 G) ~ H x (D) are bounded. Combining

these facts with the proof of Lemma 2 now shows that d 7']Oa is bounded.

We now know that {f~, ho}=T- l{g~ , 0} are uniformly bounded for a in K,. We need only recall that T ~ and V ~ are uniformly bounded on LI(aG) and L 2 (D) to Hj (GR), respectively, for a in K,; this completes the proof of Theorem 3.

Appendix

Proof of Lemma 1. We shall use the coordinates 0, ~b introduced in the previous section. We suppose that the point x has coordinates (0', ~b'), and that the point y has coordinates (0, ~). To the lowest order of approximation, we have

r 2 ,~ s 2 (0 - 0') 2 + t 2 (~b - ~b') 2, (A.1)

s - l gor- l~s(O'- -O)r -a, and t - l ~4, r - l , , , t ( r 1 6 2 -3.

Denote the second integral in (3.8) by I 2. Then the highest order term in the kernel of 4rrI 2, expressed as an integral with respect to dO dd?, is given as

(A.2) s t[r-as(O'-O)ctoa+r -s t(~b'-~b)c% a] �9 [(O'-O)~oa+(r

Denote by fl the angle between the vectors x - y and q(y). To the lowest order of approximation the expression (A.2) can be written in the form

(A.3) r- 1 (cl (x) cos 2 fl + c2 (x) sin fl cos fl) + i r - 1 s t k(x)

where ik is the coefficient of str -1 sin2fl, that is, of st a r-3(~b-~b') 2, in (A.2), namely

(A.4) i k = t - 1 ~ a . O~ a .

The terms involving fl in (A.3) form the kernel of a singular integral operator whose symbol is

(A.5) const, p - ' ( c , (x) cos 2 f i+ c 2 (x) sin ficos/~);

here p=(~2+r/2)~, cosj~=~/p, s in~=q/p , and (~,~/) are the variables dual to (0, ~). This can be verified by a straightforward computation, using the fact that the symbol of an operator with kernel k(x, x - y ) is k(x, 4), where ~ denotes Fourier transformation with respect to the second variable.

According to the theory of pseudo-differential operators, if F and G are operators with symbols f and g, then FG differs from the operator with symbol f g by an operator of order one less than that of FG. We write the symbol (A.5) as such a product f g with

f = const, p - 2 S (C 1 (X) COS/~ "~- C 2 (X) sin fl) and

g=s -a pcosf i .

16 R. PHILLIPS • L. SARASON"

Since f is the symbol of an operator T_ 2 of order - 2 and g is the symbol of the operator aq, it follows that an operator with symbol (A.5) can be written as

T-2 t3#+ R1,

where T_ 2 and R1 have order - 2 . To complete the proof of Lemma l it suffices to show that w k(x) ldx[ is real,

that is, that c

(A.6) Re ~t -1 ctea. Or Idx l=0 . C

Since q= - a - fia, we have

a .~z,~a=a. (o~. e )a=(a . ~a). e=0

so that a§ %a)=0. Hence

2 Re (0r a) . ~r a + a . ( ~ ~t~) a = 0 ,

and (A.6) is equivalent to the vanishing of

s a . (ar %) a d0. (A.7) Re c~ t

Now ar247162 �9 e)=~- dee, so (A.7) can be written as

S Re ~ - - a . (o~ . c3r adO ,

c t which equals

S (A.8) - R e ~r'-f" q" ~ edO.

Since te= t34,x, t~r e = ~ x .

Combining this with the identities aex=te , q. e=0, and s q= ~ox, we put (A.8) into the form

(A.9) - Re ~ t - 2 (~0 x) (t~ x) d 0. C

From q. e = 0 we have dox. c~r Differentiating with respect to q~, we find

(~0 ~r x). ~ x + 090 x). ~ x = 0.

Thus (A.9) can be written as

(A.10) Re ~ t- 2 (~0 c3~ x)- ~ x d 0, C

which equals . c~ot 2

Re c~ t - 2 6qO(6q 0 X) 2 dO= Re c ~ - ~ dO= ~c O~176 dO=O.

This concludes the proof of Lemma 1.


Remark. The introduction of local coordinates in a neighborhood of the entire closed curve C in the above manner is possible because the integral curves of q(x) near to C are also closed. If instead, for example, they spiraled in towards C, the argument would fail. Note that the argument applies if C is replaced by any segment of C such that t is the same at both end points. This remark allows us to obtain some information in case the integral curves of q are not closed. For example, one can show that the operator D o has a point spectrum when restricted to functions which are rotationally invariant about the axis of symmetry of ~ G, and a continuous spectrum when restricted to the orthogonal complement of that subspace.

Note. This research was sponsored in part by the National Science Foundation under Grants GP 8857 and GP-17526, and by the United State Air Force under contract F 44620-68-C-0054.

References

1. FRIEDRICHS, K.O., Symmetric positive linear differential equations. Comm, Math. 11, 333-418 (1958).

2. HORMANDER, LAP.S, Pseudo-differential operators. Comm. Pure Appl. Math. 18, 269-305 (1965).

3. LAX, P.D., & R. S. PmLLIPS, Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math. 13, 427-455 (1960).

4. LAx, P. D., & R. S. PHILLIPS, Scattering Theory. New York: Academic Press 1967. 5. AGMON, S., A. DOUGLIS, & L. NIRENBERG, Estimates near the boundary for solutions of

elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17, 35-92 (1964).

6. SARA.SON, LEONARD, On weak and strong solutions of boundary value problems. Comm. Pure Appl. Math. 15, 237-288 (1962).

Stanford University and

The University of Washington Seattle

(Received October 16, 1970)

2 Arch. Rational Mech. Anal,, VoL 41

energy decay for the neutrino equation in the exterior of a torus

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