Uniqueness from discrete data in an inverse spectral problem for a pencil of ordinary differential operators Academic research paper on "Mathematics"

Uniqueness from discrete data in an inverse spectral problem for a pencil of ordinary differential operators

Malcolm Brown, Marco Marietta and Freddy Symons

Abstract

We prove a pair of uniqueness theorems for an inverse problem for an ordinary differential operator pencil of second order. The uniqueness is achieved from a discrete set of data, namely, the values at the points —n2 (n E N) of (a physically appropriate generalization of) the Weyl-Titchmarsh m-function m(A) for the problem. As a corollary, we establish a uniqueness result for a physically motivated inverse problem inspired by Berry and Dennis ('Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity', J. Phys. A: Math. Theor. 41 (2008) 135203).

To achieve these results, we prove a limit-circle analogue to the limit-point m-function interpolation result of Rybkin and Tuan ('A new interpolation formula for the Titchmarsh-Weyl m-function', Proc. Amer. Math. Soc. 137 (2009) 4177-4185); however, our proof, using a Mittag-Leffler series representation of m(A), involves a rather different method from theirs, circumventing the A-amplitude representation of Simon ('A new approach to inverse spectral theory, I. Fundamental formalism', Ann. Math. (2) 150 (1999) 1029-1057). Uniqueness of the potential then follows by appeal to a Borg-Marcenko argument.

1. Introduction: new definitions and problem statements

Let H denote the Hilbert space L2(0,1; r dr) = {u : (0,1) ^ C \ J^ r|u(r)|2 dr < to}. Suppose that q,w e L^c(0,1], with w > 0 almost everywhere and q real-valued. In the space H we examine the following operator pencil:

Lu(r; A) = APu(r; A) (r e (0,1)). (1.1)

Here L is a realization in H of the differential expression

tu(r) =--(ru'(r))' + q(r)u(r)

which we shall define precisely below, and P is the unbounded multiplication operator

Pu(r) = w(r)u(r)

with domain

D(P) = \ u e L2(0,1; rdr)

w(r)|u(r)|2 dr < to > = L2(0,1; w(r) dr),

in which the weight w is assumed to have the following singular behaviour:

w(r) = -1 (1 + 0(1)) (r — 0),

Received 27 October 2015; published online 18 September 2016.

2010 Mathematics Subject Classification 34L20, 34L25, 34L40, 35J75, 35P05 (primary), 30E05, 34B20 (secondary).

The authors thank the UK Engineering and Physical Sciences Research Council for funding the PhD studies of Freddy Symons.

© 2016 London Mathematical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

where v ^ 0 is fixed. Typically, one might treat equation (1.1) by writing it in the form

1 q(r)

-(ru'(r))' +--z—r u(r) = Xu(r)

rw(r) w(r)

and noting that the expression on the left-hand side is formally symmetric in the space L2 (0,1; rw(r)dr). In this paper, however, we work in the space H, which is the natural choice in the physical setting from which our problem arises. Briefly, if w(r) = r-2 and A = —An, then

(1.1) becomes the m = 0 case of

- ^(ru'n(r; m))' + q(r)un(r; m) + AAnUn(r; m) = M«n(r; m)- (1.2)

This is a Bessel-type equation with potential, and equations related to it have been studied quite extensively [2, 3, 9, 10, 19]. In addition, if An are the angular eigenvalues of a spherically symmetric time-independent Schrodinger equation in a sub-domain of R2, then separating the same equation into polar coordinates with radial component un yields precisely the system

(1.2). These eigenvalues are determined by the domain and boundary conditions. A particular choice of these is discussed later, in relation to a scenario first formulated in [7] and further explored in [22]. We refer the reader to Section 4 and to these two references for further details.

We now describe the domain of the pencil L — AP. It turns out that, in some cases, the natural choice of domain is A-dependent, and we require the following definition.

Definition 1.1. We say that the equation

--(ru'(r))' + q(r)u(r) = Xw(r)u(r) (1.3)

is in pencil-limit-point or pencil-limit-circle at 0, corresponding, respectively, to the L2 (0,1; r dr) solution space being one- or two-dimensional. We abbreviate these, respectively, by PLP and PLC.

Remark 1.1. For our example, with w(r) = r-v(1 + o(1)), it turns out that the problem is always in PLC at 0 if v G [0, 2), always in PLP if v > 2, and has A-dependent classification if v = 2; see Appendix C. In the case v = 2, (1.3) is in PLP at 0 if Im(VA) ^ 1 and in PLC at 0 if Im(%/A) < 1; we choose the branch of the square root with Im%/A > 0. The parabola ImVA = 1 divides C into components Qp and Qc, and the pencil is in PLP for A G Q.p, PLC for A G Oc; see Figure C.1 in Appendix C.

The definition of the domain of the pencil L — AP is slightly simplified by noting that, except when the problem is in PLP for all A, the point A = 0 always lies in the domain ilc in which the equation is in PLC: see Figure C.1. Thus, in the PLC case, we can take U to be any non-trivial real-valued solution of the equation £U = 0, and use it to define a boundary condition in the usual way for the classical limit-circle case. Let [•, •] denote the Wronskian.

Definition 1.2. In the PLC case let U be any non-trivial real-valued solution of £U = 0. Then the boundary condition at 0 defined by U is

[u, U](0+; A) := lim{u(r; A)rU'(r) — ru'(r; A)U(r)} = 0. (1.4)

Definition 1.3 (Domain of L — AP). In the PLP case,

D(L — AP) = {u G H | iv, — Awu G H, u(1; A) = 0}.

In the PLC case,

D(L — AP) = {u e H \ iu — Awu e H, u(1; A) = 0, [u,U](0+; A) = 0}.

To introduce the Weyl-Titchmarsh function m(A) for our pencil, we first remark that, owing to the asymptotics in Appendix C, equation (1.3) has at least one non-trivial solution in H. We can therefore make the following definition.

Definition 1.4. In the PLP case, let u(-; A) denote the unique (up to scalar multiples) solution of (1.3) in H. In the PLC case, let u(- ; A) denote the unique-up-to-multiples solution with [u, U](0+; A) = 0. Then the Dirichlet m-function is

m(A) = u'(1; A)/u(1; A). (1.5)

Using the variation-of-parameters formula, one may show that, with the domains as in Definition 1.3, the operator L — AP is invertible when m(A) is analytic, and that the eigenvalues of the pencil L — AP, which are poles of (L — AP)_1, are the poles of m(A). Note that, in the case v = 2, there is generally a discontinuity of m across the boundary curve between ilp and Qc, due to the freedom in choosing the boundary condition function U for Im(\/A) < 1.

The main objective of this paper is to obtain a pair of uniqueness theorems for the following.

Inverse Problem 1.1. Let w : (0,1] ^ (0, +œ) be locally bounded and suppose (1.1) is in PLP or PLC at 0. If in PLC, suppose that we have a boundary condition as in Definition 1.3. Now let S := ((-n2, m„))^=1 be a sequence of admissible points in the graph of a generalized Titchmarsh-Weyl m-function for (1.3). Recover the potential q from the sequence S under these conditions.

Our approach is firstly to show that, in both the PLP and PLC cases, the m-function is uniquely determined by its values at -n2 (n G N), then secondly to invoke the Borg-MarCenko-type theorem in Appendix A that uniquely determines a potential from its associated m-function. In the PLP case v ^ 2 (note v = 2 turns out to be treatable by a PLP technique), we will transform (1.3) to Liouville normal form on the half-line [0, ж), in PLP at ж, regular at 0, before utilizing the Rybkin-Tuan interpolation formula [24] for the classical limit-point m-function associated with such an equation. This is valid because the PLP and classical limitpoint m-functions are formally the same where their domains overlap, that is, all of C when v > 2 and ilp when v = 2; the Rybkin-Tuan interpolation holds in this region. However, when 0 ^ v < 2, the Liouville normal form of (1.3) holds on a finite interval; to our knowledge, there is no interpolation result for such a classical limit-circle problem.

To fill this gap, in Section 2 we will prove an interpolation result similar to that in [24], but which holds in a finite-interval limit-circle case. We will then argue using the same reasoning as in the PLP case that we may use the interpolation to prove our PLC uniqueness theorem. The uniqueness theorems will be stated and proved by the outlined methods in Section 3.

We will conclude the paper with an illustration of the relevance of this result in Section 4, where we explain how it proves a uniqueness theorem for the physically motivated Berry-Dennis PDE inverse problem, which involves boundary singularities and partial Cauchy data at the boundary.

2. Interpolation of a classical limit-circle m-function

We will use Theorem A.1 [24, Theorem 5] to prove our PLP uniqueness result. A drawback of the theorem is that it will not help for the PLC uniqueness, as it only applies to classical limit-point operators on the half-line. The purpose of this section is to establish an analogous result, for a particular finite-interval classical limit-circle Sturm-Liouville problem.

Suppose Q e L2(0,1) is real-valued. Then considered over L2(0,1) the differential equation

- u"(x; A)+^Q(x) - -^j u(x; A) = Xu(x; X), (x e (0,1)) (2.1)

is classical limit-circle non-oscillatory at 0 (see Lemma C.3 and note we may formally transform between (2.1) and (1.3) using the Liouville-Green transformation [13, equation (2.5.2)]). Hence, we require a boundary condition at 0: we will use the Friedrichs or principal one. Let Up be a principal solution of (2.1), that is, Up is non-trivial, and, for any linearly independent solution V, we have Up(x) = o(V(x)) as x ^ 0. The Friedrichs boundary condition at 0 is the requirement that a solution u satisfy

[u,Up](0+; A) = 0. (2.2)

Up to a scalar multiple, (2.1) and (2.2) uniquely specify a solution u (a simple consequence of Lemma C.3). Taking such a non-trivial solution u, we choose a purely Robin (-to-Robin) m-function, that is, for h = H both real, the unique mh,H (A) satisfying

u'(1; A) - Hu(1; A) = mhH(A)(u'(1; A) - hu(1; A)). (2.3)

We will interpolate this m-function, in the style of Theorem A.1.

The proof of Theorem A.1 in [24] relies fundamentally on the observation that the classical limit-point half-line Dirichlet m-function has a representation using a Laplace transform of the A-amplitude [4, 14]. This is used by first proving [24, Theorem 4] that a Laplace transform

F (z) = ^ [f ](z)

has representation

e"zxf (x) dx ( Re(z) > 1 + ß

O / .. \ n

F (z) = Y, cn (z + - - AY^n F (k + p), (2.4)

n=^ ' k=0

for cn,ank defined as in Theorem A.1 and fixed positive p, provided

e-Sx\f (x)| dx < to for every S > 0. (2.5)

Rybkin and Tuan then show [24, Theorem 5] that interpolation formula (2.4) applies to F(k) = m(-K2) - k.

We shall follow a similar line of attack, and eventually implement (2.4). Unfortunately, the A-amplitude Laplace transform representation in [14] is not valid in the classical limit-circle case at one endpoint of a finite interval, since one cannot transform such a problem to the half-line whilst retaining the Liouville normal form. Another approach must be used.

We will find a Laplace transform representation of mh,H (A) by showing that its so-called Mittag-Leffler series expansion (see, for example, [12, Chapter 8]) is simply related to a Laplace transform. We then prove that condition (2.5) holds, implying the validity of interpolation formula (2.4).

Self-adjoint operators associated with a classical limit-circle non-oscillatory Sturm-Liouville problem on a finite interval, with separated boundary conditions, have purely discrete spectrum comprising simple eigenvalues. One way to observe this is to use the Niessen-Zettl transformation [23] of such a problem to a regular problem on the same interval, then recall

that spectra of regular Sturm-Liouville problems comprise simple eigenvalues (see, for example, [11]). This holds under any choice of separated boundary conditions, whence we see that mh,H has, as its only singular behaviour, simple poles at the eigenvalues An of (2.1) and (2.2) with the further boundary condition

u'(1; A) = hu(1; A), (2.6)

since these are where the denominator of mh,H (A) vanishes.

In Lemma B.1, we show that in the classical limit-circle non-oscillatory case, enumerating the eigenvalues as An, n =1, 2, 3,..., we have

An = (n +1/4) V + 0(1),

\iXn = (n +1/4)n + 0(1/n) (n —>to ). (2.7)

For each n, the eigenfunction pn corresponding to An is defined by

Pn = p(-; An),

where p solves (2.1) with initial conditions p(1; A) = 1,p'(1; A) = h. Suppose that — is the linearly independent solution with -0(1; A) = 1, (1; A) = H, and

$(A) := p(0+; A), K(A) := — (0+; A).

Then, by checking that f (x; A) := — (x; A) + mh,H(A)p(x; A) satisfies the 'boundary condition' in (2.3), it follows that

mh,H (A) = — fjg. (2.8)

Therefore, $(An) being 0 implies via integration by parts that

H — h = f '(1; A)pn(1) — f (1; A)pn(1)

(A - A„)

f(-; A)Vn

rl Ф(А) rl

= (A - ; A)^n - (A - Ап)ф(А) - ф(Ап) J

Ф(А„) r 1

Ф'(А„) J

< as A An. (2.9)

If we denote the norming constants associated with \n by an := JQ ^П, then we see from (2.8) and (2.9) that the residue of the m-function at its poles is given by

Res(mh,H; An) = -• (2.10)

Furthermore, in Lemma B.2 we prove that

an = 1/2 + O(1/n). (2.11)

The asymptotics (2.11) and (2.7) immediately imply that 1/an(A — An) is convergent,

uniformly for A in any compact set bounded away from {An}^=1. Furthermore,

^ ■ 0 (Im(A) —> ж). (2.12)

(A - An)

This will ultimately turn out to be the Mittag-Leffler series we seek, but we need to link this result to the m-function. We can achieve this via Nevanlinna-type properties of mh,H. For completeness, we briefly repeat here the following well-known calculation, showing that mh,H

is (anti-)Nevanlinna. Observe that, for any solution u of (2.1) and (2.2), we have

(A - A) u( ; A)u(-; A) = u'(0; A)u(0; A) - u(0; A)u'(0; A) Jo

= (h - H)(mh,H(A) - mh,H(A));

subtracting the complex conjugate of this whilst noting u(-; A) = u(-; A) shows that

If • \M2 (U mIm(mh,H(A))

|u(-; A)l =(h - H) Irn(A) •

Hence, if h > H, then mh,H is in the Nevanlinna class of functions that map the upper and lower half-planes to themselves, whilst if h < H, then mh,H is the negative of such a function, known as anti-Nevanlinna.

It is known that all (anti-)Nevanlinna functions have a Stieltjes integral representation; here

mh,H (A) = A + BX +

dp(t),

t - A 1+t2^

where p is the spectral measure associated with the problem (2.1), and

A = Re(mhH(i)), B = lim •

' t->IT

Note that p is increasing if and only if h > H. Furthermore, as a measure it assigns 'mass' only at points in the spectrum of the Sturm-Liouville operator associated with (2.1), that is, for any dp-integrable g,

g(t) dp(t) = 53 Yng(An),

Yn being the mass at An. Thus

-A + BA Y I

An - A 1 +A2,

/1 X \

mhH(X) = A + BX ^i; —X - TTXñ)

Integrating anti-clockwise along a sufficiently small, simple, closed contour around An and comparing with (2.10) shows that Yn = -Res(mh H; An) = (h - H)/an. Hence, we may split up the sum and write

mh,H (A) = A + BA +J2 —77-77 •

an(An - A)

n=1 v ^

To proceed, we need large-Im(A) asymptotics of mh,H(A). Expressing mh,H in terms of the Neumann m-function mN(A) := u(1; A)/u'(1; A) and using Lemma B.3, we see

1 - HmN (A) 1 - H/CVX

mh H(A) =---~-= —► 1 (Im(A) —> ^oo)^

h,H > 1 - hmN (A) 1 - h/CVA { A ^

From this and (2.12), we see that a! = 1 and B = 0. Thus we have proved the following lemma.

Lemma 2.1. Uniformly for A in any compact set that is non-intersecting with {An} have a Mittag-Leffler series representation for the Robin m-function given by

n}n=0,

mh,H (X) - 1 = Y" -^-77 •

' n=1 °n(Xn - X)

h- H (2.13)

an( An

Remark 2.1. Our calculations proving this result are adapted from parts of a calculation in [20, Chapter 3] for a regular Sturm-Liouville problem in normal form.

Lemma 2.1 gives us enough to deduce a Laplace transform representation of mh¡H, and hence our interpolation result. For the reader's convenience we state the theorem in full.

Theorem 2.1 (Classical limit-circle m-function interpolation). Under the hypothesis that Q G L2(0, b) is real-valued, the Robin m-function

u'(1; A) - Hu(1; A) mhH(A) - u'(1; A) - hu(1; A) ' for any square-integrable solution u of the limit-circle non-oscillatory problem

-u"(x; A)+ ^ Q(x) - u(x; A)- Au(x; A) (x G (0,1)), (2 14)

[Up,u](0+; A) — 0, x satisfies the interpolation formula

mh,H(A) - 1 — Cn(1/2 - 3 - tVA)^ ank{m(-(k + 3)2) - 1}.

n=0 k=0

Here 3 > 0 is fixed,

cn(z):—(2n +1)1(1/2,- Z)n (fora.e. z G C),

(1/2 + z)n+1

(z)n :— z(z + 1) *** (z + n - 1) (z G C),

. (-n)k(n +1)k , ank :—-(k]y2- (n,k > 0),

and the convergence of the series is uniform in any compact subset of Im(\/A) > 1/2 + 3.

The proof uses Lebesgue's dominated convergence theorem. We need the following lemma.

Lemma 2.2. Let e > 0 and define pn — yAn,. Then gN(t) :— e-et sin(pnt)/anpn is uniformly bounded, in t G (0, to) and N G N, by a fixed integrable function.

Proof. First note that the asymptotic expansion (2.7) may be written as pn — (n + 1/4)n + en, where en — 0(1/n). Then, for each fixed t ^ 0,

sin(pnt) — sin((n + 1/4)nt) cos(ent) + cos((n + 1/4)nt) sin(ent). (2.15)

Write e — 2a. It would be enough to find an L1(0, 2n) function that bounds, uniformly in N, the expression

sn(t):— e-<Jt f^ (t g (0, 2n)),

n=i anpn

so that gN(t) — e-JtsN(t) (t G (0, to)) is dominated by an L1(0, to) function, owing to the exponential decay of e-Jt. So, note that

if t G [0, a-1 log(n)), then \ent\ — O(log(n)/n),

if t > a-1 log(n), then e-Jt < e-JJ-1 log(n) — 1/n,

so that e-Jt sin(ent) — O(1/^/n); by a similar argument e-Jt(cos(ent) - 1) — O(1/n). Both estimates are uniform in t ^ 0. With (2.15), these are enough to ensure a constant bound for

sn(t) - e-Jt f sin((n +1/4)nt) (t g (0, 2n)).

Hence, substituting the asymptotic expansions (2.7) and (2.11) into the second sum in the above expression means the following: if we can show that both ^N=1 cos(nx)/n and sin(nx)/n (x e (0, 2n)) are bounded, uniformly in N, by some fixed element of L1 (0, 2n), then it will follow that so is sN (t) (t e (0, 2n)), proving the lemma.

We will prove the uniform L1(0, 2n) bound for the cos-series; the same approach produces a similar bound for the sin-series. Denote by cN(x) the partial sum ^N=1 cos(nx)/n and note

cos((N + 1/2)x) - cos(x/2) 2sin(x/2) '

Thus c'N(x) is bounded by 1/sin(x/2). Noting \cN(n)| ^ 1, we see \cN(x)| ^ 1 + fX \c'N\ ^ 1 + 2 log \ cot(x/4) \, which is certainly integrable over (0, 2n) since to leading-order it is — log(x) for x near 0 and — log(2n — x) near 2n. □

c'N(x) = — ^^ sin(nx) =

Proof of Theorem 2.1. We first observe that the Mittag-Leffler series (2.13) may be written

mh,H (—K2) — 1 =

n=ían(P2n + k2)

(h — H )£

' e-Ktsin(p^ dt (Re(K) > 0).

(2.16)

Assuming that integration and summation may be interchanged (we show this below), we see that mhlH(—k2) — 1/(^ — H) is the Laplace transform L[/](«) of the series

f(t) :=

sin(pnt)

(t > 0).

(2.17)

We now prove the convergence of (2.17) and justify the interchange of summation and integration in (2.16).

From (2.15), we have sin(pnt) = cos(nt/4) sin(nnt) + sin(nt/4) cos(nnt)+ 0(1/n). Hence, by (2.7) and (2.11), the pointwise convergence of (2.17) is determined by that of Yj=1 eljx/j. But this is simply the Fourier series for the 2n-periodic extension of the expression — log \2sin(x/2)\ + i(n — x)/2 (x e (—n,n)) so the pointwise convergence of (2.17) is immediate.

We may now simply apply Lemma 2.2 to see that gN(t) := e~Re(K)í^N=1 sin(pnt)/anpn is dominated by an integrable function. Dominated convergence follows, and hence we may write

mh,H (—K2) — 1 = (h — H)

e-Ktf (t) dt (Re(K) < 0).

All that remains is to check condition (2.5). But this is obvious, since, by dominated convergence, e-St\f (t)\ is integrable for every S > 0. Therefore, by application of the interpolation result (2.4) to F(k) = mh,H(—k2) — 1, the theorem follows, with uniform convergence in any compact subset of the parabolic A-region Im(\/A) > 1/2 + ,3. □

3. Uniqueness theorems for the inverse problem

The main result of this paper is a pair of uniqueness theorems for Inverse Problem 1.1. We will state and prove these here, by means of Theorem A.1 and our interpolation result in Theorem 2.1. The uniqueness theorems are kept separate due to certain technical conditions in both being similar in representation, but fundamentally different in structure.

Theorem 3.1 (Uniqueness in the PLP case). Fix v > 2, c > 0 and a > v/2 - 1 > 0, and let w,q € L^c(0,1], with q real-valued and w ^ c almost everywhere (a.e ). Suppose that w,w' € AC(0,1] with w', w" € L^c (0,1]. Suppose also that, as r — 0,

(i) w(r) = (1/rv)(1 + O(ra));

(ii) q(r) = w(r)O(ra);

(iii) (w(r)rv)' = O(r-v/2), and (w(r)rv)'' = O(r-v).

If w is known, then the interpolation sequence ((-n2, mn))^=1, of values (in the graph) of the PLP Dirichlet m-function (1.5) for (1.3), uniquely determines the potential q.

Proof. We perform a Liouville-Green transformation:

t(r) =

z(t(r)) = r 1/2w(r)1/4u(r) (r € (0,1)).

This leads to the corresponding solution space L2(0, to; r(t)v dt) in which we seek z(-; A); further, over this space, the transformed equation is in PLP at to (or, in the case v = 2, has the PLP/PLC behaviour outlined in Appendix C, to which the reader is directed for details). That the domain in which t lies is (0, to) follows from the fact that, as r — 0, t(r) ~ jr s-v/2 ds — to.) The equation satisfied by z is

- z"(t; A) + Q(t)z(t; A) = Az(t; A) (t e (0, to)),

Q(t(r)) :=

-r-1/2,

—3/4_

r—(r—1/2 w(r)—1/4 )

rV—2(1+ Z(r))+ £2(r) (r e (0, 1)),

£l(r) = w(r)rv — 1,

£2(r) Z (r)

£1(r) 1+£i(r) \v — 2

5 r2e1(r)2

rV/(r) (1 + £i(r))2

2v re'i(r)

(v - 2)2 (1+ £i(r))3 (v - 2)2 (1+ £i(r))2'

We now want to apply Theorem A.1 to the m-function of equation (3.1); for this we need JK+1 |Q| to be a bounded expression in x € (0, to), that is, Q € lTO(L1)(0, to). It would suffice that Q € LTO(0, to). Note that

|Q(t)| dt =

r(t)V —2(1 + Z (r(t))) — £2(r(t))

By applying the hypotheses (i) and (iii) to (3.2), we easily observe that

ei(r) = O(r-v/2) C O(r1-v) and £?(r) = O(r-v) (r —► 0).

Thus Z(r) € LJOC(0,1] and is O(r2-V) as r — 0. Further, w,q € L^c(0,1] implies that £2(r) is bounded. Therefore Q € LTO(0, to) C lTO(L1)(0, to), so (3.1) is in classical limit-point at to. Formally the classical limit-point and PLP m-functions of (3.1), respectively, over the spaces

L2 (0, OO) and L2(0, to; r(t)v dt), have the same expression. Since the integral hypothesis of Theorem A.1 is satisfied, m( ) can be interpolated from its values at the points (— n?)'^=1.

In particular, given any non-real ray through the origin and the sequence of interpolation pairs

((—n2,m„))~=1, (3.6)

for any A on this ray we can calculate the value of m(A). Choosing any such ray in the first quadrant and applying Corollary A.1, we have immediately that Q is uniquely determined by the sequence (3.6), and by the reverse transformation it follows that q is as well. □

Theorem 3.2 (Uniqueness in the PLC case). Let 0 < v < 2,c> 0 and a > 3/2 - 3v/4 > 0, and fix w,q G L^c(0,1], with q real-valued and w ^ c almost everywhere (a.e.). Suppose that w, w' G AC(0,1] with w', w" G L£C(0,1], and that, as r ^ 0,

(i) w(r) = (1/rv)(1 + O(ra));

(ii) q(r) = w(r)O(ra-2);

(iii) (w(r)rv)' = O(ra-1), and (w(r)rv)'' = O(ra-2).

If w is known, then the interpolation sequence ((-n2,m„))^=1, of values (in the graph) of the PLC Dirichlet m-function (1.5) for (1.3) with boundary condition (1.4), uniquely determines the potential q.

Proof. First note that, under these assumptions, ^/w is integrable. All asymptotic estimates are as r or t ^ 0. We use a different transformation from that in the proof of Theorem 3.1, namely

W^JwWl1 (0,1)'

This gives rise to

where this time

z(t(r)) = r 1/2w(r)1/4u(r) (r G (0,1)). - z"(t; A) + Q(t)z(t; A) = Az(t; A) (t G (0,1)),

Q(t(r)) = -

-(2-v)

(1 + C (r)) + £2 (r),

with e2 and Z defined as in (3.3) and (3.4). Note e2(r) = O(ra), and that

— (2-v) = _

"4t2 :

whilst

(2 - v)||VW||li(O,I)

r1—v/2(1 + O(ra)).

Our aim is to apply Theorem 2.1, for which we need Q(t) := Q(t) + 1/4t2 G L2(0,1). Recalling (3.2), we use condition (iii) to observe e^r) = O(ra-1) and e'1'(r) = O(ra-2). Thus, by (3.4), C(r) = O(ra). Since f (t) G L2(0,1) if and only if f (t(r)) G L2(0,1; <Jw(r) dr) (easily checked), and 2(a — 2 + v) — v/2 > -1, we see Q G L2(0,1), as required.

Hence, by Theorem 2.1 the Robin m-function (and by a fractional linear transformation, any m-function) is uniquely determined by the sequence (3.6). Corollary A.1 concludes the proof. □

Corollary 3.1. Any finite number of values m(-n2) in the interpolation sequence may be discarded, yet the m-function, and hence the potential, will still be uniquely determined.

Proof. Since, in (2.4), the parameter ¡3 > 0 may be chosen freely, one may choose ¡3 to be any positive integer. The resulting interpolation formula does not require the values F(1),..., F(3 - 1), and so the values m(-1),..., m(-(3 - 1)2) are not needed. □

4. The Berry-Dennis problem

We now explain the claim made in the introduction, namely, that uniqueness for a physically inspired inverse problem is achieved as a corollary of the above result on pencils. The setup is that of [7, 22]. Consider the two-dimensional Schrodinger equation with spherically symmetric potential q € L^oc(0,1],

- AU(x) + q(|x|)U(x) = 0 (x € Q), (4.1)

where Q is the semi-circular region {x = (x, n) € R2 | x2 + n2 ^ 1,X ^ 0}. Let r := {(x,n) € R2 | x2 + n2 = 1, X > 0} C dQ,

and take 0 < £ < 1, g € H 1/2(r). Write x = (x, y) and assign to the differential equation (4.1) the boundary conditions

U(x)= g (x € r), (4.2)

U(x) + £y~dU(x) = 0 (x € dQ\r), (4.3)

where d/dv is the outward-pointing normal derivative on dQ. These considerations define an operator L over L2(Q), taking values LU = (-A + q)U and having domain D(L) := {U € L2(Q) | AU € L2(Q); (4.2), (4.3) hold}. In polar coordinates x = (r, 0), the action of L is that of

d2 1 d 1 d2

--—--—--+ q(r).

dr2 r dr r2 d02

Since on dQ\r the normal derivative is given by -d/dx = ±r-1 d/d0 (0 = ±n/2), we find that (4.3) becomes

dU, , „ / ..... , n

U(x)+ e — (x) = 0 (r e (0,1),e = ±2).

se v y v v ' " 2/

Hence, after performing the separation of variables U(r, e) = u(r)0(e), we arrive at the angular eigenvalue problem

—0" = A0 on (— n/2,n/2) 0(—n/2) + e0'(—n/2) = 0 = 0(n/2) + e0'(n/2),

which, it is easily calculated, has eigenvalues and eigenfunctions

Ao = — e2, An = n2 (n e N);

( e—e/£ (n = 0),

0n(e) = < cos(ne) — (ne) — 1 sin(ne) (n even), [cos(ne) + ne sin(ne) (n odd).

Feeding this information back into the problem, one can find (as remarked in [22]) that L is isometrically equal to the orthogonal direct sum of the ordinary differential operators

Ln (n = 0,1, 2,...) given by

1 dfdu\ An

Cnu(r) :=---— r — + q(r)u(r) + — u(r),

r dr \ dr J rr

and equipped with the domains

D(Ln) := {u e L2(0,1) | Lnu e L2r(0,1), u(1) = 0}.

We are concerned with an associated inverse problem. Consider the generalization L\ (A e C) of Ln:

L\u(r) := -1 — [r——J + q(r)u(r)--r u(r),

r dr dr rr

with domain

D(LX) := {u e ¿2(0,1) I Lxu e ¿2(0,1), u(1) = 0}.

The differential equation L\u = 0 is precisely (1.3) with w(r) = 1/rr, and hence displays the PLP/PLC behaviour outlined in Lemma C.1 and Figure C.1. We define the Dirichlet m-function m(A) as in (1.5).

Now recall, from the theory of inverse problems in PDEs, the Dirichlet-to-Neumann operator Ar : H 1/r(r) ^ H-1/r(r). This maps Dirichlet data U|r to Neumann data dU/dvlr for any solution U e H 1(Q) of (4.1) and (4.3). We may write any such solution using the generalized Fourier basis (un(r)©n(0))^Lo:

U(r,0) = Y, un(r)O(0). (4.4)

By differentiating, it follows that, in this basis, Ar takes the form of the diagonal matrix diag(m(-Ao), m(-Ai), m(-Ar),...).

Inverse Problem 4.1. Given an admissible Dirichlet-to-Neumann map Ar for

(-A + q(|x|))U(x) = 0 (x e 0), dU

U(x) + eyd- =0 (x e dQ\T), dv

recover the radially symmetric potential q.

Uniqueness for this inverse problem is immediate from Theorem 3.1, under the conditions q e L£Oc(0,1] and q(r) = O(ra-r) (r ^ 0) for some fixed a > 0. The uniqueness follows since, for positive n, the restrictions on q make the type (1.3) pencil, associated with each operator Ln, be in PLP at 0 (see [22]), whilst the sequences -An = -nr and m(-nr) form the interpolation sequence required in Theorem 3.1. Thus we have proved the following theorem.

Theorem 4.1 (Uniqueness for the Berry-Dennis inverse problem). Any given Dirichlet-to-Neumann map Ar for the system (4.1) and (4.3) may have arisen from at most one radially symmetric potential q e L£Oc(0,1] n O(ra-1; r ^ 0).

The 0th term (1/er,m(1/er)) is superfluous for our needs. However, we can go farther. Following Corollary 3.1, we may discard arbitrarily many of the diagonal terms of A and still retain uniqueness of q.

Remark 4.1. Theorem 4.1 is markedly different from existing results for inverse problems involving partial-boundary Dirichlet-to-Neumann measurements in two-dimensional domains. Such existing results, for example, [16-18] all deal with problems in which the portion of

the boundary where the measurements are not made, has a homogeneous Dirichlet or

Neumann condition assigned; the Berry-Dennis setup has a singular boundary condition here.

Appendix A. Limit-point interpolation and a Borg-Marcenko theorem

We collect here some useful theorems. The first is the interpolation result from [24] mentioned in Section 2 and applied in the proof of Theorem 3.1, whilst the second is a general Borg-Marcenko uniqueness result and a simple corollary, the latter being what we need in Section 3. We state the first in full to highlight its similarities with Theorem 2.1.

Theorem A.1 (Rybkin-Tuan; classical limit-point m-function interpolation). Let Q be a real-valued function in l^(L1)(0, to), that is,

||Q|| := sup

•x+1

|Q| < m.

Suppose that m is the Weyl-Titchmarsh m-function associated with the limit-point Schrodinger operator

S := -d2/dx2 + Q(x) (x e (0, to))

on L2(0, to), that is, m(A) := u'(0, A)/u(0, A) (ImA > 0) for any square-integrable solution u of Su(- ; A) = Au(- ; A). If A is from the parabolic domain with (ImA)2 > 4/2ReA + 4/34, then

m (A) - iVA = ^ Cn(-i"/A - /q + 1) ank (m(-^l) + ),

n^Q k=0

where e > 0 is a fixed parameter, /Q := max^2||Q||,e||Q||} + 1 + e,

cn(z):=(2n +1)n(1/2,- Z)n (fora.e. z e C),

(1/2 + z)n+1

(z)n := z(z + 1) *** (z + n - 1) (z e C),

. (-n)k(n +1)k , ank :=-- (n,k ^ 0),

^k := k + /q - 2.

Remark A.1. The parabolic domain (ImA)2 > 4^^ReA + 4/3^ may be written more succinctly as Im%/A > /0, where arg(vA) G [0,n). Furthermore, it is concave, and its intersection with any non-real ray through the origin is an infinite complex interval.

Theorem A.2 (Simon, Gesztesy-Simon, Bennewitz; Borg-Marcenko-type uniqueness). Let Qj G Lloc[0, b) (j = 1, 2) be real-valued, b G (0, to], and mj(A) (A G C\R,j = 1, 2) be the Titchmarsh-Weyl m-functions associated, respectively, with the differential expressions

-d2/dx2 + Qj(x) (x G (0, to), j = 1, 2)

(with self-adjoint boundary conditions at b if needed). In addition, let a G (0,b), 0 < e < n/2 and suppose that as A ^ to along the ray arg(A) = n — e, we have

|m1(A) — m2(A)| = 0(exp(—2Im(vA)a)).

Then Q1 = Q2 a.e. in [0, a].

Theorem A.2 was originally stated in a slightly weaker form (without the ray condition) by Simon in 1999 [25]; the above improvement was first published, with a shorter proof, by Gesztesy and Simon in 2000 [15]. An alternative, even shorter, limit-point proof was found by Bennewitz in 2001 [6]. All are generalizations of the original, much-celebrated uniqueness theorem proved separately in 1952 by Borg [8] and Marchenko [21]. As an immediate consequence we have the result we need in this paper.

Corollary A.1. If m\ = m2 in an infinite sub-interval of the ray \rei(n-e') | r e (0, to)} with fixed 0 < e < n/2, then Qi = Q2 a.e. in [0, b).

Appendix B. Various asymptotics for a Bessel-type equation

In this appendix, we collect some necessary results on the large-n asymptotics of the eigenvalues and norming constants defined in Section 2, as well as a result on asymptotics of the m-function, needed in the same section.

The eigenvalues of the Bessel equation of zeroth order, with Dirichlet and Neumann boundary conditions at the left and right endpoints, respectively, of (0, 1), are well-studied, and are algebraically equivalent to the positive zeros of the Bessel function J1. This information is enough to determine the eigenvalues An for the boundary value problem (2.1), (2.2) and (2.6), asymptotically to order 1/n. We calculate these first for the unperturbed equation, and then use a result from [10] to move to the perturbed version.

Lemma B.1. Let Q e Lr(0,1), h e R and denote by Up(-; A) the principal solution at 0 of

- u"(x; A) + |Q(x) - u(x; A) = Au(x; A) (x e (0,1)), (B.1)

that is, Up is non-trivial, and for all linearly independent solutions V we have Up(0+) = o(V(0+)). When ordered by size and enumerated by n = 1, 2, 3,... the eigenvalues An of the above differential equation with the boundary conditions

( [u, Up](0+; A) = 0, \ u'(1; A) = hu(1; A),

satisfy the asymptotics

y/An = (n + 1/4)n + O(1/n).

Proof. Suppose firstly that Q = 0, and denote the corresponding eigenvalues by An. The boundary condition at 0 allows us to choose any constant multiple of x1/r J0(y/~Ax) as our solution. The condition at 1 then forces the eigenvalues to be the positive zeros of

VaJ^vA) + (h - 1/2) Jo(VA).

Thus, for each fixed c, we seek asymptotics for the zeros of

f (z) := zJ1(z) - cJo(z).

Recall that J0 and J1 have only simple positive zeros [1, Subsection 9.5], and note f (j0n) = j0,nJ1(j0,n), which alternates in sign as n is incremented because j0 n interlace with j1n. The intermediate value theorem then gives a zero zn e (j0,n,j0,n+1) for f, whilst the fact that J0 and J1 oscillate with asymptotically the same 'period' [1, Subsection 9.2] means zn is unique. Since j0,n = (n - 1/4)n + O(1/n) and j1jn = (n +1/4)n + O(1/n) [1, equation 9.5.12], the positive

zeros of f are

zn = (n + 1/4)n + en

with the leading-order behaviour following from lenl ^ n/2 + 0(1/n). We now use the asymptotic expansion [1, equation 9.2.1] of the first-order Bessel function JM(x) = y/(2/nx)(cos(x -pn/2 - n/4) + 0(1/x)) (x ^ +m) to observe that

0(1/n) 3 Jo(zn) = ^/znJi(zn)

= -J 2(cos(zn + n/4) + 0(1/n)).

Taylor-expanding around the zeros of cosine implies ■sfAAn = zn = (n + 1/4)n + 0(1/n). Finally, the second equation of [10, p. 17] is precisely \An = vAi + 0(1/n). □

The next lemma provides a powerful asymptotic representation of the norming constants in Section 2. For its proof we will relate our notation to that of [10] and then utilize some results from the same paper.

Lemma B.2. Let Q g L2(0,1), h g R and suppose that y>(-; A) solves (2.1) with initial conditions ^(1; A) = 1, y'(1; A) = h. Then the norming constants an := JQ ; An) satisfy

an = 1/2 + 0(1/n).

Proof. By checking the boundary conditions, one may easily see that yn = y2(- ; An)/ y2(1; An), where y2(- ; A) is the solution of the differential equation (B.1) satisfying the boundary condition t-1/2y2(t; A) ^ 1 (t ^ 0). In the second-to-last equation of [10, p. 16] it is observed that, as p ^ +m, we have

№(• ; p2)2 = -Qp

1+ 0( log(p)

Defining pn = vA", we see that the lemma would follow if y2(1; An) 2 = pn(1 + 0(1/n)). To justify this we appeal to [10, Lemma 3.2], which implies that

y2(1; An) - ^ JQ(Pn)

J(n) _

where (using Cauchy-Schwarz for the third line)

0 < I(n) :=

< — pn

Q 1+ Pnt

(1 - log(t))|Q(t)| dt

(1 - log(t))|Q(t)| dt

1 X 1/2

3||Q||LW)

(1 - log(t))2 dt ||Q||lW)

Owing to (B.3), (B.4), and Lemma B.1, we find

y2(1; An) = ^lJ0(Pn) + O(n-3/2). (B.5)

Lemma B.1 shows, furthermore, that pn = j1jn + O(1/n) = (n + 1/4)n + O(1/n) which, owing to J0 = - J1, are asymptotically the local extrema of J0. Hence, by expanding the cosine part of [1, equation (9.2.1)] in a first-order Taylor approximation around nn, it follows that

Jo(Pn) = \/ — npn

(-1)" + O( -n

Upon substitution into (B.5), this yields the desired result. □

The large-imaginary-part asymptotics of m-functions is also a well-studied topic. The result we use in Section 2 is an application of the very general Theorem 4.1 of [5] to (2.1); we state it as a lemma.

Lemma B.3. Let mN be the Neumann m-function for (2.1), (2.2) with Q e L2(0, b), that is, mN(A) := u(1; A)/u'(1; A) for a non-trivial solution u( -; A). Then, as A along any non-real ray through 0,

mN (A) = -1;= (1 + o(1)).

Appendix C. PLP and PLC behaviour; dimension of solution space

We will analyse here the dimension of the solution space of (1.3) with w(r) ~ r-v and v ^ 0. It will be helpful to treat the two cases v ^ 2 and 0 ^ v < 2 separately, respectively, in Lemmas C.1 and C.3. The first analysis is by transforming the problem to Liouville normal form on the half-line and using known large-x asymptotics of solutions. The second follows a different approach, using asymptotic analysis and variation of parameters to build recursion formulae that can be used to construct a pair of linearly independent solutions.

Lemma C.1. Suppose v ^ 2 and a> (v - 2)/2, and, furthermore, let q,w e L^c(0,1] be real-valued with w > 0 a.e., satisfying, as r ^ 0,

(i) w(r) = (1/rv)(1 + O(ra));

(ii) w(r) is a.e. bounded away from 0; and

(iii) q(r) = w(r)O(ra).

Then equation (1.3) is in

(i) PLP at 0 when v > 2 or ImVA ^ 1, and in

(ii) PLC at 0 when v = 2 and 1 > ImVA > 0.

To prove this, we will use a result given by Eastham [13, Ex. 1.9.1], which, by providing asymptotic expressions for the solutions of equation (1.3), will give us the means to determine when any solution is in L2(0,1; rdr). For convenience and completeness we state the form of this result, which provides the most generality when applied here.

Lemma C.2 (Eastham; one-dim. Schrodinger equation solution asymptotics). Let c be non-real and R e L2(a, to). Then the differential equation

-y' + Ry = c2y on (a, to)

has solutions y± asymptotically given, as x ^ œ, by

y±(x) = exp ( ±i\cx - —

With this in mind, we proceed with the proof. Proof of Lemma C.1. We write

fl (1 + o(1)).

w(r) = —(1 + £l(r)),

= e2(r) (r G (0,1)),

£j(r) = O(ra) (r 0, j = 1, 2).

By performing a Liouville-Green-type transformation, with

p-v/2 dp (r G (0,1)),

(C.1) (C.2)

z(t) = r(t)(2-v)/4u(r(t)) (t G (0, œ)), we arrive at the following equation, for t G (0, œ),

- z"(t; A) +

(ei£2 - Aei + e2)(r) -

„v-2

(t) z(t; A) = Az(t; A).

= :Q(t;A)

Note that if v> 2, then we have t(r) = ((v - 2)/2)(r1-v/2 - 1), whereas if v = 2, then t(r) = — log(r). For conciseness, we will treat both cases v = 2 and v > 2 at the same time, as the only difference between them arises near the end of the reasoning, and will be highlighted clearly.

We want to apply Lemma C.2 to equation (C.3), for which we need Q(-; A) G L2(0, to). Since Q(-; A) G L~[0, to), the large-t behaviour of Q(t; A) determines its square-integrability. Hence, Q( ; a) g L2(0, to) if and only if

|(£1£2 - Aei + £2 )(r (t )) |2 dt

r-v/2\(£i£2 - Aei + £2)(r)|2 dr. (C.4)

But this holds automatically, due to (C.1). Thus we have a pair of solutions z±(- ; A) for equation (C.3) given, as t ^œ, by

^ Q(- ; AUV1 + °(1)).

z±(t; A) = exp -

Now, the integral in the argument of this exponential is easily calculated to be

(£i£2 - Aei + £2)(p) -

pV-H p-v/2 dp.

By (C.1), the first part of this integral is convergent to a finite limit as t ^œ. The second part is 0 if v = 2 and convergent if v > 2, since v - 2 - v/2 > -1. Thus, in fact, we have the

leading-order asymptotics

z±(t; A) ~ e

For any solution u(- ; A) of equation (1.3) and its corresponding transformed solution z(- ; A) of (C.3), we have J0 r|u(r; A)|2 dr = r(t)2|z(t; A)|2 dt. But the leading-order asymptotics (C.5)

Figure C.1. A partition of the X-plane for equation (1.3) with v = 2.

show that r(t)2\z±(t; A)|2 dt < to if and only if

2 i +iV\t |2 ,, _

I2 dt =

2 „±2lmVxt

When v > 2, the transformation (C.2) simplifies to r(t) = (1 - ((2 - v)/2)t)2/(2_v), which will not affect the exponential large-t asymptotics of the integrand in (C.6). This implies that precisely one solution of equation (1.3) (up to scaling by a constant), namely, u_(; A), is in L2 (0,1; rdr). In other words, for v > 2, (1.3) is in PLP at 0.

On the other hand, when v = 2, we find r(t)2 = e_2t, which when multiplied with the other exponential factor e±2lm^'^t in (C.6) means that Im\/A ^ 1 makes (1.3) in PLP at 0, whilst if ImVA < 1, the latter must be in PLC at 0. □

Remark C.1. When v = 2, we may represent graphically the L2(0,1;rdr) nature of the solutions of (1.3); see Figure C.1. Here, Qp := {A £ C \ ImVA > 1} and Qc := {A £ C \ ImVA < 1}, so that if A £ Qp or Qc, then equation (1.3) is, respectively, in PLP or PLC.

Lemma C.3. Consider equation (1.3) with real-valued w,q £ L11oc(0,1]. Let 0 ^ v< 2. Define £1(r) = rvw(r) — 1 and e2(r) = q(r)/w(r), and suppose that

£j(r) = 0(1) (r —^ 0, j = 1, 2). (C.7)

Then there is a fundamental system {u1( ; A),w2(-; A)} satisfying u1(r; A) ^ 1,u2(r; A) ~ log(r) as r ^ 0, and both u1 and u2 are in L2(0,1; r dr).

Proof. Transform by v(r) = r1/2u(r), so that (1.3) becomes

—v"(r; A)---v(r; A) = (Aw — q)(r)v(r; A)

= r_v(1 + £1(r))(A — £2(r))v(r; A). (C.8)

Consider the sequences (vk(•; A))cj=0 and (yk(•; A))'f=0 defined by

—vk+1(r;A) — 44rj vk+1(r;A) = (Aw — q)(r)vk(r;A), and an equation of the same form for yk, satisfying v0(r; A) = r1!2 and y0(r; A) = r1!2 log(r).

We now suppress the A-dependence to simplify notation. If we can show that the series (note, starting from k = 1) V := J2fc=1 vk and Y := ^converge uniformly near 0, and satisfy the asymptotics

V(r) = o(r1/2), Y(r) = o(r1/2 log(r)) (r —► 0), (C.9)

then r-1/2v = r-1/2v0 + r-1/2V and r-1/2y = r-1/2y0 + r-1/2Y is the required solution pair.

Note that v0, y0 form a fundamental system in the kernel of the left-hand side of (C.8), and their Wronskian is 1. Therefore, by variation of parameters, vk (and yk in place of vk) must satisfy

Vh+i(r) =

(vo(r)yo(s) - yo(r)vo(s))(Aw - q)(s)vk(s) ds

si/2-V(log(s) - log(r))(1 + ei(s))(A - £2(s))vfc(s) ds.

We want to estimate this integrand. By (C.7), this is straightforward. For each fixed A there is ¿1 > 0 such that

|(1 + £1(r))(A — £2(r))| < 2|A| (0 < r < ¿1). (C.10)

Furthermore, there is ¿2 > 0 with

|log(r)| <r-1+v/2 (0 < r < ¿2). (C.11)

Take ¿(e) = min^, ¿2}, where e = 1 — v/2 > 0. We first consider vk. By the triangle inequality, (C.10) and (C.11), we have the estimate

f r r r r \

|vfc+1(r)| < 2|A|r1/2 s^^v (s)| ds + r- s2e-3/2 v (r)| ds \ (0 < r < ¿, k = 0,1, 2,...). Uo Jo J

From this and |v0(r)| ^ r1/2, we derive inductively that

\vu (r) | < ^k+lfeT r1/2(' (0 <r<¿,k = 1, 2, 3,...), (C.12)

where (z)k = z(z + 1) ■■■ (z + k — 1) is the Pochhammer symbol. But, for all j ^ 0, we have

JZI < 1 (k g n).

Thus (C.12) simplifies to

vk(r)| < r1/21 (i^ll^l (0 <r<^k G N), (C.13)

k! \ e J

implying, by Weierstrass' M-test for convergence of functional series, that V is uniformly convergent on the interval (0^). Furthermore, by (C.13), all terms in V are O(r1/2+e) = o(r1/2), so one-half of (C.9) is satisfied; it follows that v(r) = r1/2(1 + O(re)) (r ^ 0), as required.

We appeal to a similar argument in the case of y, using (C.10) alongside the slightly different estimates

|log(r)| <r-E'2 (0 < r < S(e/2)),

101 X\r1/2+e |yi(r)| < - (0 <r<S(e/2)),

( fr fr ^ |yfc+i(r)| < ^X^l s^l^ykXs^ ds + r-e ss2^312^^ ds\ (0 < r < 5(e), k e N). Uo Jo )

These can be used inductively to show that

^ «i < (®

(0 < r < 5(e/2),k e N).

Thus, as with vk, the series Y is uniformly convergent on (0,S(e/2)), and the estimates show that the remaining half of (C.9) is satisfied: Y(r) = O(r1/2+£) = o(r1/2 log(r)).

The last claim is that both u1(r) = r-1!2v(r) and u2(r) = r-1/2y(r) are in L2(0,1;rdr). Clearly, for any 5 > 0, on the interval (5, 1) the equation (1.3) is regular, so its solutions are all continuous. We now see that u1(r) ^ 1,u2(r) ~ log(r) as r ^ 0, so the claim follows immediately. □

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Malcolm Brown

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Marco Marletta and Freddy Symons

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