On the reality of the eigenvalues for a class of symmetric oscillators
Abstract.
We study the eigenvalue problem with the boundary conditions that decays to zero as tends to infinity along the rays
In particular, this implies that the eigenvalues are all positive real for the potentials when with and , and with the boundary conditions that decays to zero as tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and ZinnJustin.
Preprint.
1. Introduction
1.1. The main results
We are considering the eigenvalue problem
(1) 
with the boundary conditions that decays to zero as tends to infinity along the rays
(2) 
The boundary conditions here are those considered by Bender and Boettcher [1]. Note that the boundary conditions for are equivalent to the conditions that decays to zero as tends to infinity along the positive and negative real axes. If a nonconstant function along with a complex number solves (1) with the boundary conditions, then we call an eigenfunction and an eigenvalue.
Before we state our main theorem, we first introduce some known facts by Sibuya [19] about the eigenvalues of (1), facts that hold even when .
Proposition 1.
The eigenvalues of (1) have the following properties.

Eigenvalues are discrete.

All eigenvalues are simple.

Infinitely many eigenvalues exist.

Eigenvalues have the following asymptotic expression
(3) where the error term could be complex.
We will give precise references for Proposition 1 after Proposition 4 in Section 2. In this paper, we will prove the following theorem that says that the equation (1) with a polynomial potential in a certain class has positive real eigenvalues only.
Theorem 2.
Let ’s be the coefficients of the real polynomial . If for some , we have for all , then the eigenvalues of (1) are all positive real.
Corollary 3.
In particular, with the eigenvalues of
are all positive real, provided and .
Proof.
This is a special case of Theorem 2 with , and . ∎
We also mention that Delabaere et al. [8, 9] studied the potential and showed that a pair of nonreal eigenvalues develops for large negative . And Handy et al. [13, 14] showed that the same potential admits a pair of nonreal eigenvalues for small negative values of .
Remark.
By rescaling, the conclusion of Corollary 3 holds for the potential when and .
This paper is organized as follows. In the rest of Introduction, we will briefly mention some earlier work. Then in the next section, we state some known facts about the equation (1) and examine further properties. In Section 3, we prove Theorem 2, and in Section 4 we extend Theorem 2. Finally, in the last section we discuss some open problems for further research.
1.2. Motivation and earlier work
Around 1995, Bessis and ZinnJustin conjectured that eigenvalues of
(4) 
are all positive real. And later Bender and Boettcher [1] generalized the BZJ conjecture; that is, they argued that eigenvalues of
(5) 
are all positive real when . Notice this follows for by Theorem 2 with . The case is open, except for which are covered by Theorem 2.
Recently, Dorey et al. [10, 11] have studied the following problem
(6) 
with the boundary conditions same as those of (1), and being all real. They proved that for , eigenvalues are all real, and for , they are all positive. A special case of (6) is the potential (when ), which is the version of the BZJ conjecture, but their results do not cover the version. (Suzuki [20] also studied the whole version of (6) under different boundary conditions.)
The proof of our main theorem, Theorem 2, has two parts. The first part follows closely the method of Dorey et al. [11, 12], developing functional equations for spectral determinants, expressing them in factorized forms and then studying an “associated” eigenvalue problem. We also introduce a symmetry lemma that is required by our more complicated potentials. The second part builds on earlier work of the author in [18], estimating eigenvalues of the “associated” problem by integrating over suitably chosen halflines in the complex plane. Of course both this paper and [11, 12] are indebted to the work of Sibuya [19].
Note that our result Corollary 3 proves the full BZJ conjecture; that is, eigenvalues of (4) are all positive real. Also Theorem 2 contains the polynomial potential case () of problem (6), though only with , whereas Dorey et al. handle . (Our proof in the case can be seen to reduce to that of Dorey et al.) In Theorem 11 we do manage to handle the case , by using also the harmonic oscillator inequality, which is a different approach from that used in [11, page 5701].
In a related direction, Bender and Boettcher [2] found a family of the following quasiexactly solvable quartic potential problems
(7) 
with the same boundary conditions as those of (1), where and . Note here that the positive integer denotes the number of the eigenfunctions that can be found exactly in closed form. However, for the purpose of studying the reality of the eigenvalues, we can allow . Our results in Theorem 2 confirm that if for any , we have either and , or and , then eigenvalues of (7) are all positive real.
The above Hamiltonians are not Hermitian in general. However, according to Bender and Weniger [6], Hermiticity of traditional Hamiltonians is a useful mathematical constraint rather than a physical requirement, in order to guarantee real eigenvalues. All Hamiltonians mentioned above are the socalled symmetric Hamiltonians. A symmetric Hamiltonian is a Hamiltonian which is invariant under the product of the parity operation (an upper bar denotes the complex conjugate) and the time reversal operation . These symmetric Hamiltonians have arisen in recent years in a number of physics papers, see [7, 13, 14, 16, 17, 21] and other references mentioned above, which support that some symmetric Hamiltonians have real eigenvalues only. In general the symmetric Hamiltonians are not Hermitian and hence the reality of eigenvalues is not obviously guaranteed. But the important work of Dorey et al. [11], and results in this paper, prove rigorously that some symmetric Hamiltonians indeed have real eigenvalues only.
As a final remark of the introduction, we mention that if is symmetric, then and so is an even function and is an odd function. Hence if is a polynomial, then for some real polynomial . Certainly (1) is a symmetric Hamiltonian.
2. Properties of the solutions
In this section we will introduce some definitions and known facts related with the equation (1). One of our main tasks is to identify the eigenvalues as being the zeros of a certain entire function, in Lemma 6. But first, we rotate the equation (1) as follows because some known facts, which are related to our argument throughout, are directly available for this rotated equation.
Let be a solution of (1) and let . Then solves
(8) 
where and is a real polynomial (possibly, ) of the form
Next we will rotate the boundary conditions. We state them in a more general context by using the following.
Definition .
The Stokes sectors of the equation (8) are
See Figure 1.
It is known that every nonconstant solution of (8) either decays to zero or blows up exponentially, in each Stokes sector . Thus the boundary conditions on in (1) become that decays in .
Before we introduce Sibuya’s results, we define a sequence of complex numbers in terms of the and , as follows. For fixed, we expand
(9)  
Note that do not depend on . We further define if is odd, and is even. if
Now we are ready to introduce some existence results and asymptotic estimates of Sibuya [19]. The existence of an entire solution with a specified asymptotic representation for fixed ’s and , is presented as well as an asymptotic expression of the value of the solution at as tends to infinity. These results are in Theorems 6.1, 7.2 and 19.1 of Sibuya’s book [19]. The following is a special case of these theorems that is enough for our argument later. The coefficient vector
is allowed to be complex, here.
Proposition 4.
The equation (8), with , , admits a solution with the following properties.

is an entire function of .

and admit the following asymptotic expressions. Let . Then
as tends to infinity in the sector , uniformly on each compact set of values . Here

Properties (i) and (ii) uniquely determine the solution of (8).

For each fixed and , and also admit the asymptotic expressions,
(10) (11) as tends to infinity in the sector , where
(12)
Proof.
In Sibuya’s book [19], see Theorem 6.1 for a proof of (i) and (ii); Theorem 7.2 for a proof of (iii); and Theorem 19.1 for a proof of (iv). And note that properties (i), (ii) and (iv) are summarized on pages 112–113 of Sibuya’s book. ∎
We now give references for the proof of Proposition 1. We use the number
Proof of Proposition 1.
See Theorem 29.1 of Sibuya [19] for a proof which says that eigenvalues are simple, and
(13) 
where is given by (12). Note that Sibuya studies the equation (8) with the boundary conditions that decays in , while in this paper we consider the boundary conditions of the rotated equation (8) that decays in . The factor in our formula (13) is due to this rotation of the problem.
The remaining two claims (I) and (III) are easy consequences of the asymptotic expression (13).
Also one can compute directly or see the equation (2.22) in [12], which says
So this along with (13) and the identity implies (3). Note that the asymptotic expression (3) of the eigenvalues agrees with that of Bender and Boettcher [1] obtained by the WKB calculation for the eigenvalue problem (5), after an index shift.
We mention that the simplicity of the eigenvalues can be proved by using the fact that for each Stokes sector, there exist two solutions of (8) with no boundary conditions imposed such that one decays to zero and another blows up as tends to infinity in the sector. ∎
The next thing we want to introduce is the Stokes multiplier. First, we let
Let be the function in Proposition 4. Note that decays to zero exponentially as in and so blows up in . Then one can see that the function
which is obtained by rotating , solves (8). It is also clear that decays in and blows up in since decays in . Then since no nonconstant solution decays in two consecutive Stokes sectors, and are linearly independent and hence any solution of (8) can be expressed as a linear combination of these two. Especially, for some coefficients and ,
(14) 
These and are called the Stokes multipliers of with respect to and .
We then see that
where is the Wronskian of and . Since both are solutions of the same linear equation (8), we know that the Wronskians are constant functions of . Since and are linearly independent, for all . Moreover, we have the following which is needed in the proof of our main theorem.
Lemma 5.
The Stokes multiplier is independent of . Moreover, if then .
Proof.
First note that Sibuya’s multiplier in [19] is while we use . Since , we see that
Hence using , we see that
(15) 
which is the equation (26.28) of [19].
So using the equation (26.29) on page 117 of [19], one can get
(16)  
where
From (16), it is clear that is independent of . We want to be real if , so that . Suppose . Since 19] (or can be directly verified from (9)), it is sufficient to show that is real when . Since ’s are all real, from (9) we conclude that must be real. This completes the proof. ∎ as noted on page 117 of [
Thus from the proof of Lemma 5 we get for some and hence from (14) we have
(17)  
(18) 
From this, for each we can relate the zeros of with the eigenvalues of (1) as follows.
Lemma 6.
For each fixed , a complex number is an eigenvalue of (1) if and only if is a zero of the entire function .
Hence, the eigenvalues are discrete because they are zeros of a nonconstant entire function. Note that the Stokes multiplier is called a spectral determinant or an Evans function, because its zeros are all eigenvalues of an eigenvalue problem.
Proof.
Suppose that is an eigenvalue of (1) with the corresponding eigenfunction . Then we let , and hence solves (8) and decays in . Since is another solution of (8) that decays in , we see that is a multiple of . Similarly is a multiple of . Hence the righthand side of (17) decays in . But blows up in , and so (17) implies .
Next we examine (18) and its differentiated form at , which are,
(19)  
(20) 
The righthand sides of these are given by differences of two functions of . We will express these righthand sides with single functions, respectively. To this end, we prove that and both have some symmetry as follows.
Lemma 7.
Let . Then we have
(21) 
Especially, we have that if is real, then
(22) 
Proof.
Let , which is the entire function in Proposition 4 and hence decays in . Then solves
Next we take the complex conjugate of this and replace by . Then we see that is entire and solves the following equation
(23) 
Since the entire functions and are solutions of (23) that decay in , we see that these two are linearly dependent. So one is a constant multiple of the other. Moreover, from (9) we see that for all where we used instead of to indicate its dependence on and . Also we have in Proposition 4. Hence the entire functions and along with their first derivatives satisfy the same asymptotic expressions in Proposition 4 (ii), so we conclude that
(24) 
by Proposition 4 (iii). Next substituting in (24) gives the first equation in (21). Also we differentiate (24) with respect to and substitute to get the second equation in (21). For (22), just note that ∎
Next we want infinite product representations of and , with respect to . But first, we recall the definition of order of an entire function, which will be needed in the proof of the next lemma. Let for . Then the order of an entire function is
If for some positive real numbers , we have for all large , then the order of is finite and less than or equal to .
Lemma 8.
Suppose . The functions and have infinitely many zeros and , respectively. They admit the following infinite product representations for each fixed :
Moreover, these infinite products converge absolutely.
Proof.
If we show that both and have orders (with respect to ) strictly less than one, then this lemma is a consequence of the Hadamard factorization theorem (see, for example, Theorem 14.2.6 on page 199 of [15]). So we will show that and have orders strictly less than one.
From the equations (10) and (11), we see that except for , small , both and are bounded by for large . So to show that they have the orders strictly less than one, it suffices to show that for , small , they are bounded by for some and .
From (14) with , one can see that
In this inequality, let lie in the region . Then since and are not in , we can use the asymptotic expression (10) to get that for all large ,
We know that depends only on by Lemma 5. Also the equations (29.4) and (29.7) imply that for fixed ,
Thus we see that for each ,
Hence the order of with respect to the variable is less than or equal to . Hence by combining this with (10), we conclude that the order of is , which is strictly less than one.
3. Proof of Theorem 2
When , the equation (1) is a translation of the harmonic oscillator. So there is nothing new here. We mention that since are . , the eigenvalues for the potential
Suppose and suppose that is an eigenvalue of the eigenproblem (1), then by Lemma 6 we have . Then from (19) and (20) along with (22), we have
Since the nonconstant function solves a linear second order ordinary differential equation, both and cannot be zero at the same time; otherwise, .
Suppose that . Then from Lemma 8 we have
Then by equating the absolute values of the two sides of the equation (and using ), we have
(25) 
Likewise, when , we get the following.
(26) 
We mention that and lie in the open lower halfplane for some . From Lemma 8 we know that and have infinitely many zeros . And (10) and (11) imply that the zeros near infinity lie near the negative real axis. Thus certainly and for some .
Below we will show that the hypotheses on the signs of the coefficients of force all the and to lie in the closed lower halfplane, which implies either
(27)  
since and are reflections of each other with respect to the real axis. If (25) holds, then (27) implies for all . If (26) holds, then (27) implies for all . Since