Spectral function of Krein's and Kotani's string in the class [GAMMA].

1. Introduction. The aim of the present article is to improve one of the results in , where we discussed the asymptotic behavior of the spectral function of a generalized second-order differential operator.

By a string we mean a function

m : (-[infinity], +[infinity]) [right arrow] [0, + [infinity]]

which is nondecreasing, right-continuous and satisfies m(-[infinity] + 0) = 0. The Lebesgue-Stieltjes measure dm(x) describes the mass-distribution of the string. For a string m, we are interested in the spectral theory of the generalized Strum-Liouville operator

(1) [Laplace] = [d / dm(x)] (d / dx), -[infinity] < x < l,

where

l(= l(m)) = sup{x|m(x) < [infinity]} ([less than or equal to] + [infinity]).

Note that the operator

(2) [Laplace] = a(x) = [d.sup.2] / d[x.sup.2] + b(x)(d / dx) (a(x) > 0)

can be rewritten in the form (1) with a suitable change of the variable under mild conditions on a(x) and b(x). For example,

[Laplace] = 1/2 ([[d.sup.2] / d[x.sup.2]] + (1 / x)(d / dx)) = d / 2x dx (x [d / dx]), x > 0

can be written in the form

[Laplace] = (d / d[x.sup.2])(d / d log x) = (d / [de.sup.2s]) (d / ds), s [member of] R

with s = log x.

We say that a string m has left boundary of limit circle type if, for some c (< l),

(4) [[integral].sup.c.sub.-[infinity]] [x.sup.2] dm(x) < [infinity].

In  strings satisfying (4) are referred to as Kotani's strings. If m(-0) = 0 then (4) is trivially satisfied and such strings are called Krein's string. From the viewpoint of applications we are mainly interested in Krein's strings, but it is crucial that we adopt the framework of Kotani's strings. In what follows we denote by [M.sub.circ] the totality of Kotani's strings excluding the trivial case where m vanishes identically.

For each m [member of] [M.sub.circ], we can define [[phi].sub.[lambda]](x), (x < l), for every [lambda] [member of] C, as the unique solution of the following integral equation:

[[phi].sub.[lambda]](x) = 1 - [lambda][[integral].sup.x.sub.-[infinity] (x - y)[[phi].sub.[lambda]](y)dm(y), x < l.

Let [L.sup.2.sub.0]((-[infinity], l),dm) denote the space of all square integrable functions f such that Supp (f)[subset]([-infinity], l) and, for f [member of] [L.sup.2.sub.0](([-infinity], l), dm), we define the generalized Fourier transform by

[??]([lambda] = [[integral].sup.l.sub.-[infinity] f(x)[[phi].sub.[lambda]](x)dm(x), [lambda] > 0

Then a nonnegative Radon measure [sigma](d[xi]) on [0, [infinity]) is called a spectral measure if the Plancherel identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds. S. Kotani () proved a certain one-to-one correspondence between m [member of] [M.sub.circ] and the spectral measure [sigma](d[xi]) on [0, [infinity]) such that

[[integral].sub.[0, [infinity]]]([sigma]((d[xi])) / ([[xi].sup.2] + 1) < [infinity].

This correspondence is an extension of M. G. Krein's, which treats the case where

[[integral].sub.[0, [infinity]]] [[sigma](d[xi])] / ([xi] + 1) < [infinity].

We refer to  or  for details. The function [sigma]([xi]) := [[integral].sub.[0,x]] [sigma](d[xi]) will be referred to as the spectral function.

In  we studied conditions on m in order that

(5) [sigma]([xi]) ~ const x [[xi].sup.[alpha]] ([xi] [right arrow] +0).

('f(x) ~ g(x)' means that f(x) / g(x) [right arrow] 1.) The probabilistic meaning of the problem is as follows: Since the transition density of the diffusion process corresponding to (1) can be represented as

(6) p(t, x, y) = [[integral].sup.[infinity].sub.-0] [e.sup.t[xi]][[phi].sub.[xi]](x)[[phi].sub.[xi]](y)d[sigma]([xi]),

the condition (5) is equivalent to

p(t, x, y) ~ const x [t.sup.-[alpha]] (t [right arrow] [infinity])

by the well-known Tauberian theorem for Laplace transforms (see ).

Although  treated the case 0 < [alpha] < 2, the case [alpha] = 1 was somewhat exceptional and the condition we gave there (see Theorem A below) is rather complicated. Therefore, in the present article we shall concentrate on the case [alpha] = 1 and simplify the condition given in : Since [sigma]([xi]) = [xi] corresponds to the string m(x) = [e.sup.x], which appeared in (3), up to a translation, we can expect that [sigma]([xi]) ~ [xi] holds if and only if m(x) [approximately equal to] [e.sup.x] in some sense. We shall see that such strings can be characterized by the class [GAMMA] which appears in the so-called de Haan theory.

2. Main results. In what follows we assume that m is an element of [M.sub.circ] such that l(m) = [infinity] and m([infinity] - 0) = [infinity]. Therefore, both m(x) and [m.sup.-1](x)(:= inf{u; m(u) > x}) are finite for all large x. Of course [sigma] denotes the spectral function of m.

Theorem 1. Let C > 0. Then,

(7) [sigma]([xi]) ~ C[xi] ([xi] [right arrow] +0)

holds if and only if

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A sufficient condition for (8) is that m(x) has continuous derivative such that

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, if m'(x) is eventually nondecreasing (i.e., nondecreasing on [A, [infinity]), [there exists]A > 0), then (9) is also necessary for (8).

The condition (8) implies

[m.sup.-1](x) ~ C log x (x [right arrow] [infinity])

but the converse does not hold in general.

We next extend Theorem 1 so that we can treat the case where, for example, [sigma]([xi]) ~ [xi] log(1 / [xi])([xi] [right arrow] +0): A function L : (A, [infinity]) [right arrow] (0, [infinity]) is said to be slowly varying (at [infinity])if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Typical examples are L(x) = const, log x, log log x, exp [square root of (log x)], etc. A function of the form f(x) = [x.sup.[rho]]L(x) with slowly varying L is said to be regularly varying with index [rho] [member of] R. Following  we denote by [R.sub.[rho]] the totality of regularly varying functions with index [rho]. Especially, [R.sub.0] is the totality of slowly varying functions. If L [member of] [R.sub.0], then [phi](x) := xL(x)[member of] [R.sub.1] and therefore [[phi].sup.-1](x) [member of] [R.sub.1]. This implies that [[phi].sup.-1](x) ~ x[L.sup.*](x) for some [L.sup.*] [member of] [R.sub.0]. Such [L.sup.*] is called a de Bruijin conjugate of L and is unique up to asymptotic equivalence (see [1,p.78]). In other words, [L.sup.*] is a function such that

xL(x)[L.sup.*](xL(x)) ~ x (x [right arrow] [infinity]).

For example, if L(x)= C then [L.sup.*](x) = 1/C, and if L(x)= log x, then [L.sup.*](x) = 1 / log x. The de Bruijin conjugate of [L.sup.*] is L itself.

Theorem 2. Let L [member of] [R.sub.0] and let [L.sup.*] be its de Bruijin conjugate. Then, the following conditions are equivalent:

(10) [sigma]([xi]) ~ [[xi] / L(1 / [xi])] ([xi] [right arrow +0);

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A sufficient condition for (11) is that [m.sup.-1] has continuous derivative such that

(12) ([m.sup.-1](x))' ~ [[L.sup.*](x) / x) (x [right arrow [infinity]).

If ([m.sup.-1])' is eventually nonincreasing, then (12 is also necessary for (11).

It is an easy calculus to see that (12) is equivalent to

(13) (log m(x))' ~ [1 / [L.sup.*](m(x))] (x [right arrow [infinity]).

Probabilistically the assertion of Theorem 2 can be written as follows by Karamata's Tauberian theorem and (6).

Corollary 1. Let X be a linear diffusion corresponding to (1) and let p(t, x, y) be its transition density with respect to dm(x). Then,

p(t,x,y) ~ [1 / tL(t)] (t [right arrow] [infinity])

if and only if (11).

The following theorem will be useful in applications.

Theorem 3. Let [m.sub.1], [m.sub.2] [member of] [M.sub.circ] and suppose that [m.sub.1](x) ~ [m.sub.2](x) (x [right arrow] [infinity]). Then [m.sub.2] satisfies (11) if so does [m.sub.1].

We postpone the proofs of Theorems 1-3 until Section 4 and give here a few examples, which are already proved in  but now the proofs are simplified greatly.

Example 4. Let m [member of] [M.sub.circ] and suppose that m(x) ~ A[x.sup.[gamma]][e.sup.Bx] (x [right arrow] [infinity]). Then we have (7) with C = 1/B. Indeed by Theorem 3 we may assume that m(x) ~ A[x.sup.[gamma]][e.sup.Bx] for all sufficiently large x and then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, our assertion follows from Theorem 1.

Example 5. If m(x) ~ A[e.sup.Bx + [square root of x]] (x [right arrow] [infinity]), then we have (7) with C = 1/B.

Example 6. Suppose that m(x) ~ A[x.sup.[gamma]][e.sup.[square root of Bx]] as x [right arrow] [infinity], where A, B > 0 and [gamma] [member of] R. Also let L(x) = B/(2 log x) so that [L.sup.*](x) = 1/L(x) = (2log x)/B. Then we may assume that m(x) = A[x.sup.[gamma]][e.sup.[square root of Bx]] for sufficiently large x, and we have

(log m(x))' ~ [square root of B] / 2[square root of x].

Since

[L.sup.*](m(x)) = 2/B log m(x) ~ [2 / [square root of B]][square root of x] (x [right arrow] [infinity]),

we have

(log m(x))' ~ [1 / [L.sup.*](m(x))] (x [right arrow] [infinity].

Thus (13) is satisfied, and by Theorem 2 we conclude that

[sigma]([xi]) ~ [xi / [L(1 / [xi])] = 2 / B [xi] log 1 / [xi] ([xi] [right arrow] +0).

3. Preliminaries. We prepare some results on so called de Haan theory.

Definition 7. (i) The de Haan class [[PI].sub.+] is the totality of eventually finite functions f : (0, [infinity]) [right arrow] [-[infinity], [infinity]) for which there exists an L [member of] [R.sub.0] such that

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) The class [GAMMA] is the totality of eventually positive functions F : R [right arrow] [0, [infinity]), nondecreasing and right-continuous, for which there exists a measurable function g : R [right arrow] (0, [infinity]) such that

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For example, log x [member of] [[PI].sub.+] and [e.sup.x] [member of] [GAMMA] (with g(x) = 1) . These two classes [[PI].sub.+] and [GAMMA] are closely related as follows:

Proposition 8. (i) If f [member of] [[PI].sub.+], then [f.sup.-1] [member of] [GAMMA]. (ii) Conversely, if F [member of] [GAMMA], then [F.sup.-1] [member of] [[PI].sub.+].

For the proof we refer to [1, Thm.3.10.4]. (When f (or F) is strictly increasing and continuous, then the assertion is almost clear.)

Lemma 1. Let m be a string given at the beginning of Section 2 and let L [member of] [R.sub.0]. Also let [L.sup.*] be its de Bruijin conjugate. Then the following conditions are equivalent:

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some q([lambda]).

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Consider the inverse functions of the both sides of (16). Then it can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for some q(x). This is also equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[phi]]([lambda]) = [lambda]L([lambda]). In other words

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is the same as (17) because [[phi].sup.-1]([lambda]) ~ [lambda][L.sup.*]([lambda]). []

Lemma 2. Let F : R [right arrow] [0, [infinity]] be a nondecreasing function such that F([infinity]) = [infinity], and let L [member of] [R.sub.0]. Also let 1 / [L.sup.#] be its de Bruijin conjugate of 1 / L (i.e., if [psi](x) = x / L(x), then [[psi].sup.-1](x) ~ x / [L.sup.#](x)). Then the following conditions are equivalent:

(18) [L([lambda]) / [lambda]] F ([x / L([lambda])] + q([lambda])) [right arrow] [e.sub.x] ([there exists]q([lambda])),

(19) [[F.sup.-1]([lambda]x) - [F.sup.-1]([lambda])] / [L.sup.#]([lambda])) [right arrow] log x.

Proof. The proof is essentially the same as that of Lemma 1.

(18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. []

4. Proofs of Theorems 1-3. For the given m [member of] [M.sub.circ] let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Of course these functions exist under the assumption (4) when l(m) = [infinity].

The proofs of Theorems 1 and 2 are based on the following result in .

Theorem A. Let L [member of] [R.sub.0]. Then, (10) holds if and only if

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some q([lambda]).

Proof of Theorem 2. We first prove the Tauberian implication. Suppose that (10) holds.

Then by Theorem A (10) we have (20). By the monotone density convergence theorem (see Theorem B in Appendix), (20) implies

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By the same argument (21)implies

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now (22) is equivalent to (11) by Lemma 1.

Next let us prove the converse. Suppose that (11) holds. Then [m.sup.-1] [member of] [[PI].sub.+] and therefore by Proposition 8 we see m [member of] [GAMMA], which implies M [member of] [GAMMA] (see [1, Cor. 3.10.7]). Repeating the same argument we also have N [member of] [GAMMA] so that [N.sup.-1] [member of] [[PI].sub.+]: i.e.,

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some [L.sub.0] [member of] [R.sub.0]. Let [L.sub.1] := [L.sup.#.sub.0]([member of] [R.sub.0]). That is 1 / [L.sub.1] is the de Bruijin conjugate of 1 / [L.sub.0]. Note that [L.sub.1] = [L.sup.#.sub.0] implies [L.sup.#.sub.1] = [L.sub.0]. So, (23) is written as

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, by Lemma 2 this implies

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for some q(x). By Theorem A, this implies

(26) [sigma]([xi]) ~ [[xi] / [L.sub.1](1 / [xi])], ([xi] [right arrow] +0).

Now it remains to show that [L.sub.1](x) ~ L(x). Recall that we have already proved the Tauberian implication. Therefore, (26) implies

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Comparing this with (11) we see that [L.sup.*.sub.1](x) ~ [L.sup.*](x) and hence [L.sub.1](x) ~ L(x), which completes the proof of the Abelina implication.

Let us next see the latter half of the theorem. Although this fact might be familiar to some people, we give the proof for the convenience of the reader: If (12), then, for every fixed x > 0,

([m.sup.-1]([lambda]x))' ~ ([L.sup.*]([lambda]x) / [lambda]x) ~ ([L.sup.*]([lambda]) / [lambda]x) ([lambda] [right arrow] [infinity])

so that

(28) [[lambda] / [L.sup.*]([lambda])] ([m.sup.-1])'([lambda]x) [right arrow] 1 / x ([lambda] [right arrow] [infinity]),

the convergence being locally uniform in x > 0 by the well-known property of regularly varying functions. Since

[[m.sup.-1]([lambda]x) - [m.sup.-1]([lambda]) / [L.sup.*]([lambda])] = [[integral].sup.x.sub.1] [[lambda] / [L.sup.*]([lambda]) ([m.sup.-1])' ([lambda]u) du,

we can deduce from (28) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus we have (11). To see the converse, use the monotone density convergence theorem (see Theorem B in Appendix). []

Since Theorem 1 is just a special case of Theorem 2, we omit the proof.

Proof of Theorem 3. In view of Theorem 2 and Lemma 1 it is sufficient to show that m2 satisfies (16) if so does [m.sub.1] when [m.sub.1](x) ~ [m.sub.2](x). However, this is almost trivial. []

5. Appendix.

Theorem B. Let [F.sub.[lambda]]x) and F(x) be absolutely continuous functions on an interval I with nondecreasing (or nonincreasing) derivatives [f.sub.[lambda]]x) and f(x), respectively. If [F.sub.[lambda]]x) [right arrow] F(x) ([lambda] [right arrow] [infinity]) for all x [member of] I, then [f.sub.[lambda]]x) [right arrow] f(x) ([lambda] [right arrow] [infinity]) at all continuity points x of f.

Proof. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which implies, for every [epsilon] > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. []

doi: 10.3792/pjaa.88.173

References

 N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge Univ. Press, Cambridge, 1987.

 Y. Kasahara and S. Watanabe, Brownian representation of a class of Levy processes and its application to occupation times of diffusion processes, Illinois J. Math. 50 (2006), no. 1 4, 515 539 (electronic).

 S. Kotani, Krein's strings with singular left boundary, Rep. Math. Phys. 59 (2007), no. 3, 305 316.

 S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, in Functional analysis in Markov processes (Katata/Kyoto, 1981), 235 259, Lecture Notes in Math., 923 Springer, Berlin, 1982.

Yuji KASAHARA

Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan (Communicated by Masaki KASHIWARA, M.J.A., Nov. 12, 2012)

2000 Mathematics Subject Classification. Primary 47E05, 34L05, 60J55, 60G51.
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