# On Bounds of Eigenvalues of Complex Sturm-Liouville Boundary Value Problems

Authors: Wenwen Jian; Huaqing Sun
Journal: Abstract and Applied Analysis
 (2014)

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## 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.


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## Body

## 1. Introduction

Consider the regular complex Sturm-Liouville problem(1)τy:=-y′′+qy=λwy
in Lw2[0,1] associated with the Dirichlet boundary conditions
(2)y(0)=y(1)=0,
where q is a complex-valued function and w is a real-valued function subjected to
(3)w(x)>0,a.e.x∈[0,1],q,w∈L1[0,1];λ is a spectral parameter, Lw2[0,1] is the weighted Hilbert space of all Lebesgue measurable complex-valued functions f on [0,1] satisfying ∫01‍w|f|2<∞ with the inner product 〈f,g〉=∫01‍wfg¯ and the norm ∥f∥=〈f,f〉1/2, and L1[0,1] is the set of all Lebesgue measurable complex-valued functions f on [0,1] for which ∫01‍|f|<∞.Equation (1) is formally self-adjoint, if and only if q is a real-valued function. Hence, (1) is formally non-self-adjoint, when the imaginary part of q 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 q 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 q is positive and w≡1. 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∥·∥C the maximum norm of C[0,1], which is the set of the continuous functions on [0,1], by ∥·∥1 the norm of the space L1[0,1] and by ∥·∥2 the norm of the space L2[0,1]. Let q1 and q2 be the real and imaginary parts of q, respectively; that is, q1=Req and q2=Im⁡q. Then, q=q1+iq2, where i=-1. For convenience, we set
(4)qk-:=-min⁡{qk,0},qk+:=max⁡{qk,0},k=1,2.
In addition, we denote
(5)S1(ɛ):={x∈[0,1]:w(x)<ɛ},m1(ɛ):=mesS1(ɛ),(6)S2(ɛ):={x∈[0,1]:w2(x)<ɛ},m2(ɛ):=mesS2(ɛ).
Then the result below is one of the main results of the paper.Theorem 1.
Ifλ is an eigenvalue of (1) and (2), then
(7)Reλ≥-8ε1(∥q1+∥1+2∥q1-∥1)∥q1-∥1;
if λ is an eigenvalue of (1) and (2), with Reλ≤0, then
(8)|Im⁡λ|≤8ε1∥q2∥1∥q1-∥1,
where ε1>0 satisfies 8∥q1-∥12m1(ε1)<1; and if λ is an eigenvalue of (1) and (2), with Reλ>0, then
(9)|Im⁡λ|≤8ε~1∥q2∥1(∥q1-∥1+Reλ∥w∥1),
where ε~1>0 satisfies 8(∥q1-∥1+Reλ∥w∥1)2m1(ε~1)<1.Proof.
We first consider the case whereλ is an eigenvalue of (1) and (2), with Reλ≤0. Let λ be such an eigenvalue and let ϕ(x) be the corresponding eigenfunction with ∥ϕ∥2=1. Then, ϕ(0)=ϕ(1)=0 and
(10)-ϕ′′+qϕ=λwϕ.
By multiplying both sides of (10) by ϕ¯ and integrating over the interval [0,1], we have
(11)∫01‍|ϕ′|2+∫01‍q|ϕ|2=λ∫01‍w|ϕ|2.
By separating the real and imaginary parts of both sides of (11), we get that
(12)Reλ∫01‍w|ϕ|2=∫01‍|ϕ′|2+∫01‍q1|ϕ|2,(13)Im⁡λ∫01‍w|ϕ|2=∫01‍q2|ϕ|2.
From Reλ≤0 and (12), it follows that
(14)∫01‍|ϕ′|2≤∫01‍q1-|ϕ|2.
Now, let Q1(x)=∫0x‍q1-(t)dt; then we have max⁡x∈[0,1]|Q1(x)|≤∥q1-∥1 and
(15)∫01‍q1-|ϕ|2=∫01‍Q1′|ϕ|2=-∫01‍Q1(ϕ′ϕ¯+ϕϕ′¯)=-2Re∫01‍Q1ϕ′ϕ¯≤2∥q1-∥1∫01‍|ϕ′||ϕ¯|≤2∥q1-∥1∥ϕ′∥2≤2∥q1-∥12+12∥ϕ′∥22
by the Cauchy-Schwarz inequality. It can be obtained from (14) and (15) that
(16)∥ϕ′∥22≤2∥q1-∥12+12∥ϕ′∥22,
and, hence,
(17)∥ϕ′∥22≤4∥q1-∥12,∫01‍q1-|ϕ|2≤4∥q1-∥12.
Similar to (15), we can get that
(18)∫01‍q1+|ϕ|2≤2∥q1+∥1∥ϕ′∥2,
which, together with the first relation of (17), gives that
(19)∫01‍q1+|ϕ|2≤4∥q1+∥1∥q1-∥1.
In addition, from ϕ(x)=∫0x‍ϕ′(t)dt and the Cauchy-Schwarz inequality, we have
(20)|ϕ(x)|2=|∫0x‍ϕ′(t)dt|2≤x∫0x‍|ϕ′(t)|2dt≤∫01‍|ϕ′|2=∥ϕ′∥22,vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvx∈[0,1],
by which, together with (17), the definition of m1(ε1) in (5), and 8∥q1-∥12m1(ε1)≤1, we get that
(21)∫01‍w|ϕ|2≥∫[0,1]∖S1(ε1)‍w|ϕ|2≥ε1∫[0,1]∖S1(ε1)‍|ϕ|2=ε1(1-∫S1(ε1)‍|ϕ|2)≥ε1(1-∥ϕ′∥22m1(ε1))≥ε12.
Using (12), (17), (19), and (21), we can easily conclude that, for every eigenvalue λ with Reλ≤0,
(22)|Reλ|ε12≤4(∥q1+∥1+2∥q1-∥1)∥q1-∥1,
which implies that (7) holds for all the eigenvalues of (1) and (2).
LetQ2(x)=∫0x‍q2(t)dt. Then, from (17),
(23)∫01‍q2|ϕ|2≤2∥q2∥1∥ϕ′∥2≤4∥q2∥1∥q1-∥1,
which, together with (13) and (21), implies that (8) holds for every eigenvalue λ with Reλ≤0.
Now, letλ0 be an eigenvalue of (1) and (2), with Reλ0>0. Then, we consider the problem
(24)-y′′+(q-Reλ0w)y=λwy
with the Dirichlet boundary conditions (2). It can be easily verified that λ0-Reλ0 is an eigenvalue of (24) and (2). Clearly, Re(λ0-Reλ0)=0. Hence, by (8), there exists ε~1 satisfying 8(∥q1-∥1+Reλ0∥w∥1)2m1(ε~1)<1 such that
(25)|Im⁡λ0|=|Im⁡(λ0-Reλ0)|≤8ε~1∥q2∥1(∥q1-∥1+Reλ0∥w∥1).
This completes the proof.The following corollary is a direct consequence of Theorem1.Corollary 2.
Ifq1≥0, then all eigenvalues λ of (1) and (2) satisfy that Reλ≥0. Consequently, if there exists q0∈R such that q1≥q0w on [0,1], then all eigenvalues λ of (1) and (2) satisfy that Reλ≥q0.Ifw∈AC[0,1] and w′∈L2[0,1], where AC[0,1] denotes the set of functions which are locally absolutely continuous on [0,1], then we have the following result.Theorem 3.
Assume thatw∈AC[0,1] and w′∈L2[0,1]. If λ is an eigenvalue of (1) and (2), then
(26)Reλ≥-8ε2∥q1-∥12[∥w∥C(1+∥q1∥1)+∥w′∥2];
if λ is an eigenvalue of (1) and (2), with Reλ≤0, then
(27)|Im⁡λ|≤8ε2∥q1-∥12(∥w∥C∥q2∥1+∥w′∥2),
where ε2>0 satisfies 8∥q1-∥12m2(ε2)<1; and if λ is an eigenvalue of (1) and (2), with Reλ>0, then
(28)|Im⁡λ|≤8ε~2(∥q1-∥1+Reλ∥w∥1)2(∥w∥C∥q2∥1+∥w′∥2),
where ε~2>0 satisfies 8(∥q1-∥1+Reλ∥w∥1)2m2(ε~2)<1.Proof .
Letλ be an eigenvalue of (1) and (2) with Reλ≤0 and let ϕ(x) be the corresponding eigenfunction with ∥ϕ∥2=1. Then, ϕ(0)=ϕ(1)=0 and (10) holds. By multiplying both sides of (10) by wϕ¯ and integrating over the interval [0,1], we have
(29)∫01‍w|ϕ′|2+∫01‍w′ϕ′ϕ¯+∫01‍wq|ϕ|2=λ∫01‍w2|ϕ|2.
By separating the real and imaginary parts of both sides of (29), we get that
(30)Reλ∫01‍w2|ϕ|2=∫01‍w|ϕ′|2+Re(∫01‍w′ϕ′ϕ¯)+∫01‍wq1|ϕ|2,(31)Im⁡λ∫01‍w2|ϕ|2=Im⁡(∫01‍w′ϕ′ϕ¯)+∫01‍wq2|ϕ|2.
On the other hand, it follows from (17) and (20) that
(32)∫01‍w|ϕ′|2≤∥w∥C∥ϕ′∥22≤4∥w∥C∥q1-∥12,|∫01‍wq1|ϕ|2|≤∥w∥C∥ϕ′∥22∥q1∥1≤4∥w∥C∥q1-∥12∥q1∥1,|∫01‍w′ϕ′ϕ¯|≤∥w′∥2∥ϕ′∥22≤4∥w′∥2∥q1-∥12,|∫01‍wq2|ϕ|2|≤∥w∥C∥q2∥1∥ϕ′∥22≤4∥w∥C∥q1-∥12∥q2∥1.
In addition, by (17), the definition of m2(ε2) in (6), and 8∥q1-∥12m2(ε2)<1, it can be obtained that
(33)∫01‍w2|ϕ|2≥∫[0,1]∖S2(ε2)‍w2|ϕ|2≥ε2∫[0,1]∖S2(ε2)‍|ϕ|2=ε2(1-∫S2(ε2)‍|ϕ|2)≥ε2(1-4∥q1-∥12m2(ε2))≥ε22,
which, together with (30) and (32), implies that (26) holds for every eigenvalue λ of (1) and (2), with Reλ≤0, and, hence, (26) holds for all eigenvalues of (1) and (2). Furthermore, if λ is an eigenvalue of (1) and (2), with Reλ≤0, 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.



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*Source: 362340-2014-05-12.xml*