Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 483–488

SOLUTIONS OF f′′ + P (z)f = 0 THAT HAVE ALMOST ALL REAL ZEROS

Gary G. Gundersen

University of New Orleans, Department of Mathematics New Orleans, Louisiana 70148, U.S.A.; ggunders@math.uno.edu

Abstract. We show that for any given real constants a > 0 and b ≥ 0, there exist certain real constants λ such that the equation f′′ + (az^{4} + bz^{2} − λ)f = 0 possesses a solution f that has an infinite number of real zeros and at most a finite number of nonreal zeros. When b > 0, these equations are new examples of equations of the form f′′ +P (z)f = 0, where P (z) is a polynomial, that possess exceptional solutions of this kind. When b = 0, these equations are earlier examples of Titchmarsh.

1. Introduction

Consider a second order linear differential equation of the form

(1)  f′′ + P_{n}(z)f = 0,

where P_{n}(z) is a nonconstant polynomial of degree n ≥ 1. It is well known that every solution f of equation (1) is an entire function.

It is an interesting question to ask the following: For which nonconstant polynomials P_{n}(z) will equation (1) admit a solution that has an infinite number of real zeros and at most a finite number of nonreal zeros? (See [3], [4], and [2, Problem 2.71].) To the author’s knowledge, it appears that up until now, the only examples of this kind in the literature are Examples 1 and 2 below. The purpose of this paper is to give new examples of this kind.

Example 1. It is well known that the classical Airy differential equation

(2)  f′′ − zf = 0

possesses a special contour integral solution f(z) = Ai(z) called the Airy integral, where all the zeros of Ai(z) are real and negative, and where Ai(z) has an infinite number of zeros (see [8, pp. 413–415]). Thus if a ≠ 0 and b are any real constants, then from a suitable linear change of the independent variable in (2), we obtain that the equation

(3)  f′′ + (az + b)f = 0

possesses a solution with only real zeros and infinitely many.

2000 Mathematics Subject Classification: Primary 34M10; Secondary 30D35.

Example 2. This example comes from a simple transformation of an equation of Titchmarsh (see [10, pp. 172–173] and [3, p. 289] for the details). There exists an infinite sequence of positive constants λ_{k} satisfying 0 < λ_{0} < λ_{1} < λ_{2} < ⋯ < λ_{k} < ⋯, where λ_{k} → ∞ as k → ∞, such that for each λ_{k}, the equation

(4)  f′′ + (z^{4} − λ_{k})f = 0

possesses a solution f = f_{k}(z) that has an infinite number of real zeros and at most a finite number of nonreal zeros. Any nonreal zero of f_{k} must lie on the imaginary axis, and f_{k} possesses exactly k zeros that lie on the imaginary axis. When k is even, f_{k} is an even function, and when k is odd, f_{k} is an odd function. It follows that f_{0} and f_{1} each have only real zeros.

From equation (4) we see that if a ≠ 0 and b are any real constants, then the equation

(5)  f′′ + [a^{2}(az + b)^{4} − a^{2}λ_{k}]f = 0

possesses the solution f = f_{k}(az + b) which has an infinite number of real zeros and at most a finite number of nonreal zeros. Thus when k = 0 or k = 1, equation (5) possesses a solution that has only real zeros and infinitely many.

On the other hand, we now list several results which show that equations of the form (1) which admit the kind of solutions in Examples 1 and 2 are exceptional. In [5] it was shown that equation (1) cannot possess two linearly independent solutions that each have only real zeros. More generally, we have the following result.

Theorem A [3]. Let f_{1} and f_{2} be any two linearly independent solutions of equation (1). Then at least one of f_{1} , f_{2} has the property that its sequence of nonreal zeros has exponent of convergence equal to \frac{1}{2}(n + 2).

The next four results show that there are many equations of the form (1) which do not admit the kind of exceptional solutions in Examples 1 and 2. The following result shows that when the degree of the polynomial P_{n}(z) is one of the numbers 2, 6, 10, …, then equation (1) cannot possess a solution that has an infinite number of real zeros and at most a finite number of nonreal zeros.

Theorem B [3], [4]. Let P_{n}(z) be a polynomial of degree n = 2 + 4k for some integer k ≥ 0, and suppose that f ≢ 0 is a solution of equation (1). Then either f has only a finite number of zeros, or the exponent of convergence of the sequence of nonreal zeros of f is equal to \frac{1}{2}(n + 2).

The next result is an immediate corollary of Theorem 3 and Lemma 5 in [3]. Here we recall that an entire function is said to be real if it is real on the real axis.

Solutions of f′′ + P (z)f = 0 that have almost all real zeros 485

Theorem C [3]. If equation (1) possesses a solution f that has an infinite number of real zeros, then P_{n}(z) is a real polynomial and f is a constant multiple of a real solution of (1).

The next theorem and corollary show that when an equation of the form (1) possesses a solution that has only real zeros and infinitely many, then this puts a restriction on the number of real zeros that P_{n}(z) can have.

Theorem D [9]. Suppose that equation (1) possesses a solution f that has only real zeros and infinitely many. Then the number of real zeros of P_{n}(z) counting multiplicities is less than \frac{1}{2}(n + 2).

Corollary [9]. If equation (1) possesses a solution f with only real zeros and infinitely many, and if P_{n}(z) has only real zeros, then equation (1) is an equation of the form (3) and f is a constant multiple of Ai(αz + β), where Ai(z) is the Airy integral and α and β are real constants.

2. The new examples

We now state the new examples. For any given real constants a > 0 and b ≥ 0, we will show that there exists an infinite sequence of real constants λ_{k} satisfying λ_{0} < λ_{1} < λ_{2} < ⋯ < λ_{k} < ⋯, where λ_{k} → ∞ as k → ∞, such that for each λ_{k}, the equation

(6)  f′′ + (az^{4} + bz^{2} − λ_{k})f = 0

possesses a solution f = f_{k}(z) that has an infinite number of real zeros and at most a finite number of nonreal zeros. By making a linear change of the independent variable in equation (6), we can obtain more equations that possess solutions that have an infinite number of real zeros and at most a finite number of nonreal zeros.

Remark. When b > 0, equation (6) does not have the form of equation (5). When b = 0, we see that equation (6) is Example 2. Hence these examples in equation (6) extend Example 2.

3. Proof of the new examples

We will prove the existence of these examples in equation (6) by combining the classical theory of eigenvalues and eigenfunctions with the asymptotic integration theory of equation (6).

Hille used his theory of asymptotic integration together with the Liouville transformation to obtain many basic properties of the solutions of equations of the form (1); see Chapter 7.4 in [6]. The following theorem contains some of these properties.

Theorem E [6, Chapter 7.4]. In equation (1) we set

P_{n}(z) = a_{n}z^{n} + a_{n−1}z^{n−1} + ⋯ + a_{0},

where a_{0}, a_{1}, …, a_{n} are constants with a_{n} ≠ 0.

The critical rays θ_{0}, θ_{1}, …, θ_{n+1} of equation (1) are defined as follows:

θ_{j} = \frac{2πj − arg a_{n}}{n + 2} for 0 ≤ j ≤ n + 1.

For convenience with notation, we set θ_{n+2} = θ_{0} and θ_{n+3} = θ_{1} .

Let f ≢ 0 be a solution of equation (1), and let ε > 0 be a small fixed constant. Then the following statements hold:

(i) For each j = 0, 1, 2, …, n + 1, we have either

f(z) → ∞ as z → ∞ in θ_{j} + ε ≤ arg z ≤ θ_{j+1} − ε,

or

f(z) → 0 as z → ∞ in θ_{j} + ε ≤ arg z ≤ θ_{j+1} − ε.

(ii) For each j = 0, 1, 2, …, n + 1, f(z) has at most a finite number of zeros in θ_{j} + ε ≤ arg z ≤ θ_{j+1} − ε.

(iii) If for some particular j ∈ {1, 2, …, n + 1, n + 2}, we have either

f(z) → 0 as z → ∞ in θ_{j−1} + ε ≤ arg z ≤ θ_{j} − ε,

or

f(z) → 0 as z → ∞ in θ_{j} + ε ≤ arg z ≤ θ_{j+1} − ε,

then f has at most a finite number of zeros in θ_{j} − ε ≤ arg z ≤ θ_{j} + ε.

Although [6, Chapter 7.4] contains several more properties of the solutions of (1), the properties in Theorem E are enough for our purposes.

We now prove the existence of the examples in equation (6). To this end, we consider the differential equation

ψ′′ + (λ − az^{4} + bz^{2})ψ = 0,

where λ, a, and b are real constants such that a > 0 and b ≥ 0. From the classical theory of eigenvalues and eigenfunctions, it is well known that for any given real constants a > 0 and b ≥ 0, there exists an infinite sequence of real constants λ_{k} (the eigenvalues) satisfying λ_{0} < λ_{1} < λ_{2} < ⋯ < λ_{k} < ⋯, where λ_{k} → ∞ as k → ∞, such that for each λ_{k}, the equation

(7)  ψ′′ + (λ_{k} − az^{4} + bz^{2})ψ = 0

possesses a solution ψ = ψ_{k}(z) (an eigenfunction) that is real on the real axis and has exactly k real zeros, such that ψ_{k}(z) is an even function when k is even and is an odd function when k is odd, and where the set of eigenfunctions ψ_{0}, ψ_{1}, ψ_{2}, … form a complete orthonormal set for L^{2}(R). Proofs of these statements are contained in Chapters 2 and 5 in [10].

The critical rays of equation (7) are ±π/6, ±π/2, and ±5π/6. We now fix a nonnegative integer k and consider the solution ψ_{k} of (7). Let ε > 0 be a small fixed constant. Since ψ_{k} is in L^{2}(R), we can deduce from Theorem E(i) that ψ_{k}(z) → 0 as z → ∞ in −π/6+ε ≤ arg z ≤ π/6−ε. Thus from Theorem E(iii), we obtain that ψ_{k}(z) has at most a finite number of zeros in each of the following two angles: (A) −π/6−ε ≤ arg z ≤ −π/6+ε, and (B) π/6−ε ≤ arg z ≤ π/6+ε. From Theorem E(ii), ψ_{k}(z) has at most a finite number of zeros in each of the following three angles: (A) −π/2 + ε ≤ arg z ≤ −π/6 − ε, (B) −π/6 + ε ≤ arg z ≤ π/6 − ε, and (C) π/6 + ε ≤ arg z ≤ π/2 − ε. Therefore, ψ_{k}(z) has at most a finite number of zeros in −π/2 + ε ≤ arg z ≤ π/2 − ε.

Now suppose that ζ = Re^{iβ} is a zero of ψ_{k}(z), where π/2 − ε < β < π/2. Arguing as on pp. 172–173 of [10], we start with the Green’s transform of equation (7) (see [7, p. 509]):

(8) ∫_{0}^{ζ}(λ_{k} − az^{4} + bz^{2})|ψ_{k}(z)|^{2} dz = ψ′_{k}(0)\overline{ψ_{k}(0)} − ψ′_{k}(ζ)\overline{ψ_{k}(ζ)} + ∫_{0}^{ζ} |ψ′_{k}(z)|^{2} \overline{dz}.

If k is even, then ψ_{k}(z) is an even function, and so ψ′_{k}(0) = 0, while if k is odd, then ψ_{k}(z) is an odd function, and so ψ_{k}(0) = 0. Since we also have ψ_{k}(ζ) = 0, we obtain from (8) that

∫_{0}^{ζ}(λ_{k} − az^{4} + bz^{2})|ψ_{k}(z)|^{2} dz = ∫_{0}^{ζ} |ψ′_{k}(z)|^{2} \overline{dz}.

By integrating along the ray arg z = β, we obtain

∫_{0}^{R}(λ_{k} − ar^{4}e^{i4β} + br^{2}e^{i2β})|ψ_{k}(re^{iβ})|^{2} dr = e^{−i2β} ∫_{0}^{R} |ψ′_{k}(re^{iβ})|^{2} dr.

Taking imaginary parts gives

(9) ∫_{0}^{R}(−ar^{4} sin 4β + br^{2} sin 2β)|ψ_{k}(re^{iβ})|^{2} dr = − sin 2β ∫_{0}^{R} |ψ′_{k}(re^{iβ})|^{2} dr.

Since π/2 − ε < β < π/2, the left-hand side of (9) is positive and the right-hand side of (9) is negative. Hence we have a contradiction. Therefore, ψ_{k}(z) cannot have any zeros in π/2 − ε < arg z < π/2. Since we have already shown that ψ_{k}(z) has at most a finite number of zeros in −π/2 + ε ≤ arg z ≤ π/2 − ε, we obtain 488 Gary G. Gundersen

that ψ_{k}(z) has at most a finite number of zeros in −π/2 + ε ≤ arg z < π/2. Since ψ_{k}(z) is real on the real axis, it follows that ψ_{k}(z) has at most a finite number of zeros in −π/2 < arg z < π/2. Furthermore, since ψ_{k}(z) is either an even function or an odd function, it follows that ψ_{k}(z) has at most a finite number of zeros that do not lie on the imaginary axis.

Since ψ = ψ_{k}(z) is a solution of equation (7), it follows that ψ_{k}(z) has order 3 (see [1]). If k is even, then ψ_{k}(z) is an even function, and therefore ψ_{k}(\sqrt{z} ) is an entire function of order \frac{3}{2} , which implies that ψ_{k}(z) has an infinite number of zeros. If k is odd, then ψ_{k}(z) is an odd function, and therefore \sqrt{z} ψ_{k}(\sqrt{z} ) is an entire function of order \frac{3}{2} , which also implies that ψ_{k}(z) has an infinite number of zeros. It follows that ψ_{k}(z) has an infinite number of zeros that lie on the imaginary axis and at most a finite number of zeros that do not lie on the imaginary axis.

Now set f_{k}(z) = ψ_{k}(iz). Since ψ = ψ_{k}(z) is a solution of equation (7), we obtain that f = f_{k}(z) is a solution of equation (6), and f_{k}(z) has an infinite number of real zeros and at most a finite number of nonreal zeros. This completes the proof of the existence of the examples in equation (6).

References

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[4] Hellerstein, S., and J. Rossi: Zeros of meromorphic solutions of second order linear differential equations. - Math. Z. 192, 1986, 603–612.

[5] Hellerstein, S., L.-C. Shen, and J. Williamson: Real zeros of derivatives of meromorphic functions and solutions of second order differential equations. - Trans. Amer. Math. Soc. 285, 1984, 759–776.

[6] Hille, E.: Lectures on Ordinary Differential Equations. - Addison-Wesley, Reading, Mass., 1969.

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[10] Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-order Differential Equations, Part I, second edition. - Oxford University Press, London, 1962.

Received 28 March 2000