Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 5, 1–11; http://www.math.u-szeged.hu/ejqtde/

THE FRIEDRICHS EXTENSION OF CERTAIN SINGULAR DIFFERENTIAL OPERATORS, II

Grey M. Ballard and John V. Baxley

Department of Mathematics, Wake Forest University Winston-Salem, NC 27109 USA email: grey.ballard@gmail.com, baxley@wfu.edu

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract

We study the Friedrichs extension for a class of 2nth order ordinary differential operators. These selfadjoint operators have compact inverses and the central problem is to describe their domains in terms of boundary conditions.

Key words and phrases: Selfadjoint ordinary differential operators, Friedrichs extensions, boundary conditions. AMS (MOS) Subject Classifications: 47B25, 47E05, 34B05, 34L05

1 Introduction

An unbounded symmetric operator L with domain D(L) dense in a Hilbert space H with inner product (u, v) is called semibounded below if (Lu, u) ≥ c(u, u) for some constant c and every u ∈ D(L). By adding a multiple of the identity operator to L, without loss of generality it may be assumed that c > 0. The largest such c is called the lower bound. Such an operator has a particular selfadjoint extension F, called the Friedrichs extension [9]. A characteristic of the Friedrichs extension is that it preserves the lower bound: (F u, u) ≥ c(u, u) for every u ∈ D(F). Even a cursory examination of the development of the Friedrichs extension in [8, Section XII.5] shows that this extension depends critically on the domain D(L) of the operator. In fact, the domain of the Friedrichs extension of L is obtained by intersecting D(L^{∗}), the domain of the adjoint operator L^{∗}, with a subset of the Hilbert space H which is obtained by completing D(L), considered as an incomplete metric space, with the “new” inner product (Lu, u). The proof of the existence of the Friedrichs extension leaves open the possibility of the existence of additional selfadjoint extensions which preserve the lower bound.

For selfadjoint ordinary differential operators, domains are usually specified by the imposition of boundary conditions. Ever since [10], it has been of interest to find a boundary condition description of the domain of the Friedrichs extension of specific operators.

Let τ be the formal differential operator defined by

τu(x) = −\frac{1}{m(x)}(p(x)u′)′,

where m(x) > 0, p(x) > 0, m, p are infinitely differentiable on (0, 1] and

M = ∫_{0}^{1} m(x) [∫_{x}^{1} \frac{1}{p(t)} dt] dx < ∞.  (1)

By varying the initial domain, Baxley [2, 3] considered several different semibounded symmetric operators defined by the nth iterate τ^{n} in the Hilbert space L^{2}(0, 1; m) of functions u satisfying ∫_{0}^{1} |u(x)|^{2}m(x)dx < ∞ and with inner product

(u, v) = ∫_{0}^{1} u(x)\overline{v(x)}m(x) dx.

In each case, the Friedrichs extension was a selfadjoint operator with compact inverse. The main goal was to describe the domains of these Friedrichs extensions in the usual way in terms of boundary conditions. The motivation for this earlier work was an application to the theory of Toeplitz matrices [4].

We return again to this problem, with a similar motivation. Now the application is to the theory of Toeplitz integral operators associated with the Hankel transform (see [6, 7] for related earlier results), work which will appear elsewhere. The Friedrichs extension obtained here is most closely related to the one which was designated G_{n} in [3]. It is possible that the Friedrichs extension obtained here is the same as that G_{n}, but we have been unable to verify this conjecture. In any case, the initial domain here is different and is more convenient for our application; the development is also simpler and more self-contained.

In recent years, much attention has been given to characterizing the domains of Friedrichs extensions of ordinary differential operators (see [11, 13, 14, 15]), as well as partial differential operators (see [5], which contains further references).

2 Preliminary Results

For a given positive integer n, let C^{2n}(0, 1) denote the collection of all functions on (0, 1) with continuous derivatives up to and including 2n, and let

C_{n} = {u ∈ C^{2n}(0, 1) : u^{(j)}(1) = 0 for j = 0, 1, ⋯, n − 1, (τ^{n−1}u)′(x) = 0 near x = 0 ,

and for n ≥ 2, p(x)(τ^{j}u)′(x) → 0 as x → 0^{+}, for 0 ≤ j ≤ n − 2} .

It is clear that if u ∈ C_{n}, then τu ∈ C_{n−2}. The conditions u^{(j)}(1) = 0, j = 0, 1, ⋯, n−1 are equivalent to the conditions

τ^{j}u(1) = 0, (τ^{j}u)′(1) = 0, j = 0, 1, ⋯, \frac{n}{2} − 1, if n is even

and, if n is odd, u(1) = 0 together with

(τ^{j−1}u)′(1) = 0, (τ^{j}u)(1) = 0 for j = 1, 2, ⋯, \frac{n − 1}{2}, if n ≥ 3.

Lemma 2.1 If u ∈ C_{n}, then τ^{j}u(x) is bounded as x → 0^{+}, for j = 0, 1, ⋯, n − 1.

Proof: For u ∈ C_{n}, since (τ^{n−1}u)′(x) = 0 for x near 0, τ^{n−1}u(x) is constant near x = 0 and therefore bounded as x → 0^{+}. We complete the proof by showing that if g(x) = τ^{j}u(x) (1 ≤ j ≤ n − 1) is bounded as x → 0^{+}, then so is τ^{j−1}u(x). Since g is bounded on (0, 1] and

τ^{j−1}u(x) = − ∫_{x}^{1} \frac{1}{p(y)} [∫_{0}^{y} m(t)g(t) dt] dy,

then

|τ^{j−1}u(x)| ≤ K ∫_{x}^{1} \frac{1}{p(y)} [∫_{0}^{y} m(t) dt] dy.

Since the integral in (1) is finite, applying Fubini’s theorem, we see that

∫_{0}^{1} \frac{1}{p(y)} [∫_{0}^{y} m(t) dt] dy

is also finite and we are done.

Lemma 2.2 If u ∈ C_{n}, the following conditions hold:

1. For i, j ≥ 1, i + j ≤ n, then (τ^{i}u, τ^{j}u) = (τ^{i+j}u, u).

2. For 0 < x_{1} < x_{2} ≤ 1, |u(x_{2}) − u(x_{1})|^{2} ≤ (τu, u) ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt.

3. For n ≥ 2 and 0 < x_{1} < x_{2} ≤ 1, |p(x_{2})u′(x_{2})−p(x_{1})u′(x_{1})|^{2} ≤ (τ^{2}u, u) ∫_{x_{1}}^{x_{2}} m(t)dt.

Proof: Statement 1 follows by integrating by parts using Lemma 2.1 and the definition of C_{n} to see that all boundary terms vanish.

For 2, we use the Schwarz inequality and integration by parts to get, for 0 < x_{1} < x_{2} ≤ 1,

|u(x_{2}) − u(x_{1})|^{2} = |∫_{x_{1}}^{x_{2}} u′(t)dt|^{2} ≤ ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)}dt ∫_{0}^{1} p(t)|u′(t)|^{2}dt = (τu, u) ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt.

For 3, we similarly write

|p(x_{2})u(x_{2}) − p(x_{1})u(x_{1})|^{2} = |∫_{x_{1}}^{x_{2}} (p(t)u′(t))′ dt|^{2}
≤ ∫_{x_{1}}^{x_{2}} m(t) dt ∫_{0}^{1} \frac{1}{m(t)} |(p(t)u′(t))′|^{2} dt
= (τ^{2}u, u) ∫_{x_{1}}^{x_{2}} m(t) dt.

Lemma 2.3 Let u ∈ C_{n} and 0 < x_{1} < x_{2} ≤ 1. Then for k odd, k = 2j + 1 ≤ n,

|τ^{j}u(x_{2}) − τ^{j}u(x_{1})|^{2} ≤ (τ^{k}u, u) ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt

and for k even, k = 2j ≤ n,

|p(x_{2})(τ^{j−1}u)′(x_{2}) − p(x_{1})(τ^{j−1}u)′(x_{1})|^{2} ≤ (τ^{k}u, u) ∫_{x_{1}}^{x_{2}} m(t) dt.

Proof: In the first case, τ^{j}u ∈ C_{n−2j} and n − 2j ≥ 1. In the second case, τ^{j−1}u ∈ C_{n−2(j−1)} and n − 2(j − 1) ≥ 2, so these statements follow quickly from Lemma 2.2.

Lemma 2.4 If u ∈ C_{n}, then (τ^{i}u, u) ≤ M(τ^{i+1}u, u) for 0 ≤ i ≤ n − 1.

Proof: Consider the i = 0 case. Since u ∈ C_{n}, from Lemma 2.2 (2 ) we have

|u(x_{2}) − u(x_{1})|^{2} ≤ (τu, u) ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt.

Letting x_{1} = x and x_{2} = 1 gives

|u(x)|^{2} ≤ (τu, u) ∫_{x}^{1} \frac{1}{p(t)} dt,

and integrating we have

(u, u) = ∫_{0}^{1} |u(x)|^{2} m(x) dx ≤ (τu, u) ∫_{0}^{1} m(x) [∫_{x}^{1} \frac{1}{p(t)} dt] dx = M(τu, u).

If i = 2j ≤ n − 1 is even, since u ∈ C_{n} implies τ^{j}u ∈ C_{n−2j} with n − 2j ≥ 1, then Lemma 2.2 (1 ) and the above argument give

(τ^{i}u, u) = (τ^{j}u, τ^{j}u) ≤ M(τ^{j+1}u, τ^{j}u) = M(τ^{i+1}u, u).

If i = 2j + 1 ≤ n − 1 is odd,

∥τ^{j}u∥^{2} = (τ^{j}u, τ^{j}u) ≤ M(τ^{j+1}u, τ^{j}u) ≤ M∥τ^{j+1}u∥ ∥τ^{j}u∥

which implies ∥τ^{j}u∥ ≤ ∥τ^{j+1}u∥, so

(τ^{i}u, u) = (τ^{j+1}u, τ^{j}u) ≤ M∥τ^{j+1}u∥^{2} = M(τ^{j+1}u, τ^{j+1}u) = M(τ^{i+1}u, u).

3 The Friedrichs Extension

Let C^{∞}(0, 1) be the collection of all infinitely differentiable functions on (0, 1) and let L_{n}u = τ^{n}u for u ∈ D(L_{n}) where

D(L_{n}) = {u ∈ C^{∞}(0, 1) : u(x) = 0 near x = 1, (τ^{n−1}u)′(x) = 0 near x = 0 ,
and for n ≥ 2, p(x)(τ^{j}u)′(x) → 0 as x → 0^{+}, for 0 ≤ j ≤ n − 2} .

Note that D(L_{n}) ⊂ C_{n}. Let C_{0}^{∞}(0, 1) consist of all functions in C^{∞}(0, 1) with compact support in (0, 1). The next lemma is obvious.

Lemma 3.1 C_{0}^{∞}(0, 1) ⊂ D(L_{n}) ⊂ D(L_{n+1}) for all n ≥ 1.

We view L_{n} as an operator in the weighted Hilbert space L^{2}(0, 1; m) defined earlier. From Lemma 3.1, D(L_{n}) is dense in L^{2}(0, 1; m) for each n.

Theorem 3.1 The operator L_{n} is symmetric and nonnegative for all n ≥ 1.

Proof: Symmetry follows directly from Lemma 2.2 (1 ). We note that for n = 2j even,

(L_{n}u, u) = (τ^{n}u, u) = (τ^{j}u, τ^{j}u) = ∫_{0}^{1} |τ^{j}u(x)|^{2}m(x) dx ≥ 0,

and for n = 2j + 1 odd,

(L_{n}u, u) = (τ^{n}u, u) = (τ^{j+1}u, τ^{j}u) = ∫_{0}^{1} p(x)|(τ^{j}u)′(x)|^{2} dx ≥ 0,

and so L_{n} is nonnegative for all k ≥ 1.

Iterating Lemma 2.4, we see that (u, u) ≤ M^{n}(L_{n}u, u) for any u ∈ D(L_{n}). Thus M^{−n} is a positive lower bound for L_{n}.

For each n, Theorem 3.1 shows that L_{n} has a Friedrichs extension, which we denote by L̃_{n}.

Lemma 3.2 Let u ∈ D(L̃_{n}). Then

1. u^{(i)}(1) = 0 for 0 ≤ i ≤ n − 1,

2. |u(x_{2}) − u(x_{1})|^{2} ≤ M^{n−1}(L̃_{n}u, u) ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt, for 0 < x_{1} < x_{2} ≤ 1,

3. (u, u) ≤ M^{n}(L̃_{n}u, u).

Proof: If u ∈ D(L̃_{n}), then from [8, p. 1242], there exists u_{i} ∈ D(L_{n}) such that ∥u_{i} − u∥ → 0 and (L_{n}u_{i}, u_{i}) → (L̃_{n}u, u) as i → ∞.

Since u_{i} ∈ D(L_{n}) implies τ^{k}u_{i} ∈ D(L_{n}) for any integer k ≥ 0, we can use Lemma 2.3 and Lemma 2.4 to conclude,

|τ^{j}u_{i}(x_{2}) − τ^{j}u_{i}(x_{1})|^{2} ≤ M^{n−k}(L_{n}u_{i}, u_{i}) ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt

for k = 2j + 1 ≤ n and

|p(x_{2})(τ^{j−1}u_{i})′(x_{2}) − p(x_{1})(τ^{j−1}u_{i})′(x_{1})|^{2} ≤ M^{n−k}(L_{n}u_{i}, u_{i}) ∫_{x_{1}}^{x_{2}} m(t) dt.

for k = 2j ≤ n, whenever 0 < x_{1} < x_{2} ≤ 1. Since τ^{j}u_{i} and (τ^{j}u_{i})′ vanish for x = 1, it follows that all the sequences {τ^{j}u_{i}}, for 2j + 1 ≤ n, and {p(τ^{j}u_{i})′}, for 2j ≤ n, are uniformly bounded and equicontinuous on any compact subset of (0, 1]. Using Ascoli’s theorem and a familiar diagonalization argument, by passing to a subsequence we can assume without loss of generality that all of these sequences converge uniformly on any compact subset of (0, 1]. Since ∥u_{i} − u∥ → 0, then u_{i} → u uniformly on any such compact subset of (0, 1]. Thus u(1) = 0. Letting v = lim_{i→∞} u′_{i}, we write −u_{i}(x) = ∫_{x}^{1} u′_{i}(t)dt and take limits using the bounded convergence theorem to get −u(x) = ∫_{x}^{1} v(t)dt. Differentiating gives u′(x) = v(x) for 0 < x ≤ 1. So u′ = lim_{i→∞} u′_{i}. Continuing in this way, we find that each of these convergent sequences converge to the appropriate derivative of the limit function u, and it is easy to see that u^{(j)}(1) = 0 for 0 ≤ j ≤ n − 1.

From Lemma 2.4, we have (τu_{i}, u_{i}) ≤ M^{n−1}(τ^{n}u_{i}, u_{i}) = (L_{n}u_{i}, u_{i}). Then from Lemma 2.2 (2 ),

|u_{i}(x_{2}) − u_{i}(x_{1})|^{2} ≤ M^{n−1}(L_{n}u_{i}, u_{i}) ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt

for 0 < x_{1} < x_{2} ≤ 1. Taking limits, we have

|u(x_{2}) − u(x_{1})|^{2} ≤ M^{n−1}(L̃_{n}u, u) ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt.

Letting x_{1} = x and x_{2} = 1 and integrating, we have (u, u) ≤ M^{n}(L̃_{n}u, u).

Theorem 3.2 All eigenvalues of L̃_{n} are strictly positive and L̃_{n} has a compact inverse.

Proof: Let u be an eigenfunction of L̃_{n}. Then from Lemma 3.2, (u, u) ≤ M^{n}(L̃_{n}u, u) = M^{n}λ(u, u) where λ is the corresponding eigenvalue. Hence all eigenvalues of L̃_{n} are strictly positive. Thus L̃_{n}^{−1} exists. To show L̃_{n}^{−1} is compact, suppose there is a sequence {L̃_{n}u_{i}} such that ∥L̃_{n}u_{i}∥ ≤ K for n = 1, 2, · · ·. We will show that {L̃_{n}^{−1}(L̃_{n}u_{i})} = {u_{i}} has a convergent subsequence. From Lemma 3.2 (3 ), ∥u_{i}∥ ≤ M^{n}∥L̃_{n}u_{i}∥, and so we have

(L̃_{n}u_{i}, u_{i}) ≤ M^{n}K^{2} for all n = 1, 2, · · · .

Then from Lemma 3.2 (2 ),

|u_{i}(x_{2}) − u_{i}(x_{1})|^{2} ≤ M^{2n−1}K^{2} ∫_{x_{1}}^{x_{2}} \frac{1}{p(t)} dt, for 0 < x_{1} < x_{2} ≤ 1

and so with x_{2} = 1, x_{1} = x

|u_{i}(x)|^{2} ≤ M^{2n−1}K^{2} ∫_{x}^{1} \frac{1}{p(t)} dt, for 0 < x ≤ 1.

Thus the functions {u_{i}} are equicontinuous and uniformly bounded on compact subintervals of (0, 1]. From Ascoli’s theorem and a diagonalization argument we may assume the sequence converges uniformly to a limit function u on each compact subinterval of (0, 1]. Since the last inequality also holds for the limit function u, the dominated convergence theorem guarantees u_{i} converges to u in L^{2}(0, 1; m), so L̃_{n}^{−1} is compact.

The following lemma is immediate since 0 is not in the spectrum of L̃_{n}.

Lemma 3.3 The range of L̃_{n} is all of L^{2}(0, 1; m).

Because of the previous three lemmas, it follows from the theory of compact selfadjoint operators that all eigenvalues of the operator L̃_{n} are positive and have finite multiplicity, and the eigenfunctions span L^{2}(0, 1; m).

To prepare for the next lemma, for 0 ≤ x ≤ 1, we define Q_{0}(x) = 1 and

Q_{j+1}(x) = ∫_{0}^{x} \frac{1}{p(y)} [∫_{0}^{y} m(t)Q_{j}(t) dt] dy for j = 0, 1, ⋯, n − 2,

and we define R_{0}(x) = ∫_{x}^{1} \frac{1}{p(t)} dt and

R_{j+1}(x) = ∫_{x}^{1} \frac{1}{p(y)} [∫_{y}^{1} m(t)R_{i}(t) dt] dy, for j = 0, 1, ⋯, n − 2.

Just as in the proof of Lemma 2.1, it follows by induction that each Q_{j} is well-defined and bounded.

Lemma 3.4 If w ∈ N(L_{n}^{∗}), then for some constants a_{j}, w = ∑_{j=0}^{n−1} a_{j}Q_{j}(x).

Proof: It is easy to verify that

τ^{i}Q_{j} = (−1)^{i}Q_{j−i}, for i ≤ j, and τ^{i}Q_{j} = 0 for i > j  (2)

and

p(τ^{i}R_{j})′ = (−1)^{i}pR′_{j−i} for i < j and p(τ^{j}R_{j})′ = (−1)^{j+1}.  (3)

Thus, τ^{n} maps all of the functions Q_{0}, R_{0}, ⋯, Q_{k−1}, R_{k−1} to 0 and it is easy to show that the functions Q_{j} and R_{j} are linearly independent. Since w ∈ N(L_{n}^{∗}), it follows as in [8, pp. 1291-1294], that w ∈ C^{∞}(0, 1) and τ^{n}u = 0. Thus the 2n functions {Q_{0}, R_{0}, ⋯, Q_{k−1}, R_{k−1}} must span N(L_{n}^{∗}). Therefore for w ∈ N(L_{n}^{∗}),

w = ∑_{j=0}^{n−1} (a_{j}Q_{j} + b_{j}R_{j})

for some constants a_{j} and b_{j}. We need to show that b_{j} = 0 for all j = 0, 1, ⋯, k − 1. Suppose b_{γ} ≠ 0 where b_{j} = 0 for γ + 1 ≤ j ≤ k − 1. Choose u ∈ C^{∞}(0, 1) such that u(x) = Q_{n−γ−1}(x) for 0 ≤ x ≤ 1/4 and u(x) = 0 for 3/4 ≤ x ≤ 1. Then u ∈ D(L_{n}) and

(τ^{n}u, w) = (L_{n}u, w) = (u, L_{n}^{∗}w) = (u, τ^{n}w).

Repeated integration by parts gives

(τ^{n}u, w) = ∑_{i=0}^{n−1} (τ^{i}u)p(x)(τ^{n−i−1}w)′|_{0}^{1} − ∑_{i=0}^{n−1} p(x)(τ^{i}u)′(τ^{n−i−1}w)|_{0}^{1} + (u, τ^{n}w)

and thus

∑_{i=0}^{n−1} (τ^{i}u)p(x)(τ^{n−i−1}w)′|_{0}^{1} − ∑_{i=0}^{n−1} p(x)(τ^{i}u)′(τ^{n−i−1}w)|_{0}^{1} = 0.

Since u(x) ≡ 0 for 3/4 ≤ x ≤ 1 our equation simplifies to

lim_{x→0} [∑_{i=0}^{n−1}(τ^{i}u)p(x)(τ^{n−i−1}w)′ − ∑_{i=0}^{n−1} p(x)(τ^{i}u)′(τ^{n−i−1}w)] = 0.

For 0 ≤ x ≤ 1/4, u(x) = Q_{n−γ−1}(x) so from (2), τ^{n−γ−1}u = (−1)^{n−γ−1}Q_{0} ≡ (−1)^{n−γ−1}. Thus we have

lim_{x→0} [ ∑_{i=0}^{n−γ−1}(τ^{i}Q_{n−γ−1})p(x)(τ^{n−i−1}w)′ − ∑_{i=0}^{n−γ−2} p(x)(τ^{i}Q_{n−γ−1})′(τ^{n−i−1}w)] = 0. (4)

Substituting for w, we see that any term with a product of the form (τ^{i}Q_{j})p(x)(τ^{k}Q_{m})′ tends to 0, so all terms in w involving the Q_{j}’s vanish and we may as well assume that

w = ∑_{j=0}^{γ} b_{j}R_{j}.

In the second sum in (4), we encounter only factors of the form τ^{k}R_{j} where k ≥ γ + 1 and j ≤ γ. So by (3), all these terms tend to zero. In the first sum in (4), we encounter only factors of the form p(x)(τ^{k}R_{j})′ where k ≥ γ and j ≤ γ. By (3), the only term which does not tend to zero is the one for which k = j = γ so i = n − γ − 1 and (2),

(3) give

lim_{x→0}(τ^{n−γ−1}Q_{n−γ−1})p(x)(τ^{γ}b_{γ}R_{γ})′ = (−1)^{n}b_{γ} = 0.

Thus b_{γ} = 0, a contradiction.

Since 0 is a point of regular type of L_{n} [1, pp. 91-94], the deficiency indices of L_{n}^{∗} are equal to the dimension of the null space of L_{n}^{∗}. By Lemma 3.4, these deficiency indices are each n. The following theorem almost follows from the general theory [8] which states that every selfadjoint extension of L_{n} comes by the imposition of n boundary conditions, all at the regular endpoint x = 1. The problem is that the general theory in [8] is all developed by starting with the initial domain C_{0}^{∞}(0, 1) and one would have to re-do that theory in this new context. It is likely that this approach would work, but it is not hard to give a direct proof.

Theorem 3.3 D(L̃_{n}) = {u ∈ D(L_{n}^{∗}) : u^{(i)}(1) = 0 for i = 0, 1, ⋯, n − 1}.

Proof: Since D(L̃_{n}) ⊂ D(L_{n}^{∗}) and Lemma 3.2 (1 ) implies the boundary conditions are satisfied for u ∈ D(L̃_{n}), we need only show the set described is a subset of D(L̃_{n}).

Let u ∈ D(L_{n}^{∗}) such that u^{(i)}(1) = 0 for i = 0, 1, ⋯, n − 1. From Lemma 3.3, the range of L̃_{n} is all of L^{2}(0, 1; m), so there exists v ∈ D(L̃_{n}) such that L̃_{n}v = L_{n}^{∗}u. We let w = u − v and we will show that w = 0. Since D(L̃_{n}) ⊂ D(L_{n}^{∗}), w ∈ N(L_{n}^{∗}), and from Lemma 3.4, w has the form

w = ∑_{j=0}^{k−1} a_{j}Q_{j}(x).  (5)

It is easy to verify that w satisfies the conditions required for membership in C_{n} as x → 0^{+}. Since u satisfies the boundary conditions by hypothesis and v satisfies the boundary conditions by Lemma 3.2 (1 ), we have w^{(i)}(1) = 0 for i = 0, 1, ⋯, n − 1, and so w ∈ C_{n}. Iterating Lemma 2.4, we find

(w, w) ≤ M^{n}(τ^{n}w, w),

so the Schwarz inequality gives ∥w∥ ≤ M^{n}∥τ^{n}w∥, implying w = 0 or u = v ∈ D(L̃_{n}).

A reader for whom the theory in [8] is not well-known might be worried that some functions in D(L_{n}^{∗}) would lack sufficient smoothness for the boundary conditions in Theorem 3.3 to make sense. Although it follows from [8, Theorem 10, p. 1294] that all such functions have 2n − 1 continuous derivatives, it is easy to see without this general theory that any u ∈ D(L_{n}^{∗}) has at least n − 1 continuous derivatives. Returning to the proof of Theorem 3.3, one sees that u = w + v, where w ∈ C^{∞}(0, 1) (as noted in the proof of Lemma 3.4), and v ∈ D(L̃_{n}), which (as shown in the proof of Lemma 3.2) has at least n − 1 continuous derivatives.

We now identify the eigenfunctions and eigenvalues of L̃_{1} in one special case that arises in the case of Toeplitz integral operators associated with the Hankel transform.

Theorem 3.4 Suppose m(x) = p(x) = x^{2ν} where ν > 0 is a constant. Then the eigenvalues Λ_{k} of L̃_{1} have multiplicity one and Λ_{k} = z_{k}^{2}, where z_{k} is the k^{th} positive zero of the Bessel function J_{ν−1/2}. An eigenfunction corresponding to Λ_{k} is x^{1/2−ν}J_{ν−1/2}(z_{k}x).

Proof: If Λ is an eigenvalue of L̃_{1} and u is a corresponding eigenfunction, then Λ > 0, u is in the null space of L̃_{1} − ΛI and we have the equation

L_{1}^{∗}u − Λu = L̃_{1}u − Λu = 0.

Again, using [8, pp. 1291-1294], u is infinitely differentiable and τu = Λu. This equation reduces to

u′′ + \frac{2ν}{x} u′ + β^{2}u = 0

where Λ = β^{2}. It is easy to verify that the general solution of this equation is

u = c_{1}x^{1/2−ν}u_{1}(βx) + c_{2}x^{1/2−ν}u_{2}(βx),

where u_{1}, u_{2} are any two linearly independent solutions of Bessel’s equation. We may choose u_{1}(x) = J_{ν−1/2}(x) and u_{2}(x) = Y_{ν−1/2}(x). Let w_{2}(x) = x^{1/2−ν}u_{2}(βx). Known behavior (see [12]) of u_{2}(x) as x → 0 shows that

lim_{x→0} x^{2ν}w′_{2}(x) = d ≠ 0

for some constant d. Choosing v in the domain of L_{1} so that v(x) = 1 in a right neighborhood of x = 0, we can integrate by parts to get

(L_{1}v, w_{2}) = (τv, w_{2}) = −d + (v, τw_{2}),

so that w_{2} is not in the domain of L_{1}^{∗}. Thus c_{2} = 0. A similar calculation shows that x^{1/2−ν}u_{2}(βx) is in the domain of L_{1}^{∗}. Then u is in the domain of L̃_{1} if and only if u(1) = 0, which occurs if and only if u_{1}(β) = J_{ν−1/2}(β) = 0. Thus β must be a zero z_{k} of this Bessel function.

References

[1] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. I, Ungar, New York, 1961.

[2] J. V. Baxley, The Friedrichs extension of certain singular differential operators, Duke Math. J. 35 (1968), 455-462.

[3] J. V. Baxley, Eigenvalues of singular differential operators by finite difference methods, II, J. Math. Anal. Appl. 38 (1972), 257-275.

[4] J. V. Baxley, Extreme eigenvalues of Toeplitz matrices associated with certain orthogonal polynomials, SIAM J. Math. Anal. 2 (1971), 470-482.

[5] J. F. Brasche and M. Melgaard, The Friedrichs extension of the Aharonov-Bohm Hamiltonian on a disc, Integr. Equ. Oper. Theory 52 (2005), 419436.

[6] J. R. Davis, Extreme eigenvalues of Toeplitz operators of the Hankel type I, J. of Math. and Mech. 14 (2) (1965), 245-275.

[7] J. R. Davis, On the extreme eigenvalues of Toeplitz operators of the Hankel type II, Trans. Amer. Math. Soc. 116 (1965), 267-299.

[8] N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Interscience, New York, 1963.

[9] K. O. Friedrichs, Spektraltheorie halbbeschrankter Operatoren, I, Math. Ann. 109 (1934), 465-487.

[10] K. O. Friedrichs, Uber die ausgezeichnete Randbedingung in der Spektraltheorie der halbbeschrankten gewohnlichen Differential-operatoren zweiter Ordnung, Math. Ann. 112 (1935), 1-23 .

[11] H. G. Kaper, M. K. Kwong, and A. Zettl, Characterizations of the Friedrichs extensions of singular Sturm-Liouville expressions, SIAM J. Math. Anal. 17 (1986), 772-777.

[12] N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, New Jersey, 1965.

[13] M. Marletta and A. Zettl, The Friedrichs extension of singular differential operators, J. Differential Equations 160 (2000), 404-421.

[14] H.-D. Niessen and A. Zettl, Singular Sturm-Liouville problems: the Friedrichs extension and comparison of eigenvalues, Proc. London Math. Soc. 64 (1992), 545-578.

[15] A. Zettl, On the Friedrichs extension of singular differential operators, Commun. Appl. Anal. 2 (1998), 3136.