Abstract
The paper is concerned with eigenvalues of complex Sturm-Liouville boundary value problems. Lower bounds on the real parts of all eigenvalues are given in terms of the coefficients of the corresponding equation and the bound on the imaginary part of each eigenvalue is obtained in terms of the coefficients of this equation and the real part of the eigenvalue.
1. Introduction
Consider the regular complex Sturm-Liouville problem in associated with the Dirichlet boundary conditions where is a complex-valued function and is a real-valued function subjected to is a spectral parameter, is the weighted Hilbert space of all Lebesgue measurable complex-valued functions on satisfying with the inner product and the norm , and is the set of all Lebesgue measurable complex-valued functions on for which .
Equation (1) is formally self-adjoint, if and only if is a real-valued function. Hence, (1) is formally non-self-adjoint, when the imaginary part of is nonzero. The boundary value problems associated with (1) with real coefficients have been deeply studied (cf., e.g., [1–5] and their references). Similar to regular self-adjoint boundary value problems, one can prove that (1) and (2) also have only countable eigenvalues without finite accumulation points using the spectral theory of compact operators in Hilbert spaces when (1) is formally non-self-adjoint. Unlike self-adjoint boundary value problems, (1) and (2) may have infinitely many nonreal eigenvalues, when the imaginary part of is nonzero (see [6, Theorem 1.1]). The asymptotic behavior of eigenvalues of boundary value problems associated with (1) has been studied (cf. [7–9]). Sufficient conditions were given in [10] for all eigenvalues of (1) with the periodic, antiperiodic, Dirichlet, or Neumann boundary conditions to be simple. For more related results for non-self-adjoint differential expressions, the reader is referred to [11–14] and the references cited therein.
In this paper, we are interested in the bounds on the eigenvalues of (1) and (2). In the case where (1) is formally self-adjoint, the bounds on eigenvalues of (1) and (2) were constructed in terms of the coefficients of (1), the coefficients of a comparing equation, and the eigenvalues of the comparing eigenvalue problem using the comparison theorem in [4]. The Rayleigh-Ritz method was used in [3] to obtain the bounds of eigenvalues of (1) and (2), when is positive and . It is noted that the comparison theorem and the Rlayleigh-Ritz method are not applicable to boundary value problems (1) and (2), for the case where (1) is formally non-self-adjoint. Here, we obtain lower bounds on the real parts of all eigenvalues of (1) and (2) in terms of the coefficients of (1) and get the bound on the imaginary part of each eigenvalue in terms of the coefficients of (1) and the real part of the eigenvalue.
In the next section, we will present the main results of this paper.
2. The Bounds on the Eigenvalues
We denote by the maximum norm of , which is the set of the continuous functions on , by the norm of the space and by the norm of the space . Let and be the real and imaginary parts of , respectively; that is, and . Then, , where . For convenience, we set In addition, we denote Then the result below is one of the main results of the paper.
Theorem 1. If is an eigenvalue of (1) and (2), then if is an eigenvalue of (1) and (2), with , then where satisfies ; and if is an eigenvalue of (1) and (2), with , then where satisfies .
Proof. We first consider the case where is an eigenvalue of (1) and (2), with . Let be such an eigenvalue and let be the corresponding eigenfunction with . Then, and
By multiplying both sides of (10) by and integrating over the interval , we have
By separating the real and imaginary parts of both sides of (11), we get that
From and (12), it follows that
Now, let ; then we have and
by the Cauchy-Schwarz inequality. It can be obtained from (14) and (15) that
and, hence,
Similar to (15), we can get that
which, together with the first relation of (17), gives that
In addition, from and the Cauchy-Schwarz inequality, we have
by which, together with (17), the definition of in (5), and , we get that
Using (12), (17), (19), and (21), we can easily conclude that, for every eigenvalue with ,
which implies that (7) holds for all the eigenvalues of (1) and (2).
Let . Then, from (17),
which, together with (13) and (21), implies that (8) holds for every eigenvalue with .
Now, let be an eigenvalue of (1) and (2), with . Then, we consider the problem
with the Dirichlet boundary conditions (2). It can be easily verified that is an eigenvalue of (24) and (2). Clearly, . Hence, by (8), there exists satisfying such that
This completes the proof.
The following corollary is a direct consequence of Theorem 1.
Corollary 2. If , then all eigenvalues of (1) and (2) satisfy that . Consequently, if there exists such that on , then all eigenvalues of (1) and (2) satisfy that .
If and , where denotes the set of functions which are locally absolutely continuous on , then we have the following result.
Theorem 3. Assume that and . If is an eigenvalue of (1) and (2), then if is an eigenvalue of (1) and (2), with , then where satisfies ; and if is an eigenvalue of (1) and (2), with , then where satisfies .
Proof . Let be an eigenvalue of (1) and (2) with and let be the corresponding eigenfunction with . Then, and (10) holds. By multiplying both sides of (10) by and integrating over the interval , we have By separating the real and imaginary parts of both sides of (29), we get that On the other hand, it follows from (17) and (20) that In addition, by (17), the definition of in (6), and , it can be obtained that which, together with (30) and (32), implies that (26) holds for every eigenvalue of (1) and (2), with , and, hence, (26) holds for all eigenvalues of (1) and (2). Furthermore, if is an eigenvalue of (1) and (2), with , then (27) follows from (31)–(33). With a similar argument to that in the proof of Theorem 1, (28) can be proved. This completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the NNSF of China (Grant 11101241), the NNSFs of Shandong Province (Grants ZR2011AQ002 and ZR2012AM002), and the Special Fund for Postdoctoral Innovative Programs of Shandong Province (Grant 201301010).