Dependence of eigenvalues of regular Sturm-Liouville operators on the boundary condition

Abstract

In this paper, we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem. The work not only provides a new and elementary proof of the above results, but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters. Furthermore, we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.

Graphical abstract

Keywords

Regular Sturm-Liouville operator / eigenvalue / implicit function theorem

1 Introduction

Many problems in mathematical physics can be reduced to certain eigenvalue problems of differential operators. Therefore, the spectral theory of differential operators has attracted great attention. Recently, a lot of progress has been made in the continuous dependence of eigenvalue of Sturm-Liouville (S-L) operator with respect to an endpoint, a boundary condition (BC), a coefficient, see, for example, [6-11] and references therein. They considered the eigenvalue of the regular S-L operator as a one-variable function of one of these parameters, and showed that the function is continuously differential and its derivative is given. As the reader may already be aware, these results have great significance in S-L problems, see [1, 2, 4, 13, 15]. By use of the continuous dependence of eigenvalues on the endpoints, one can get the result that the eigenvalue approximates the continuous spectrum, e.g., [1, 2, 4, 15]. By use of the continuous dependence of eigenvalues on the BC, one can establish the inequalities among eigenvalues between indefinite and definite S-L problems, see [13].

By using of implicit function theorem, we study the continuous dependence of eigenvalues of S-L operators on the BC. We not only provide a new and elementary proof of the above results, but also explicitly present the expressions for derivatives of the n-th eigenvalue with respect to BC. Furthermore, we obtain the new results that the position and number of the generated double eigenvalues under the real coupled BC. Our method is based on the eigenvalue inequalities in [5, 14], that is, an eigenvalue corresponding to any separated BC is between two Dirichlet eigenvalues and an eigenvalue corresponding to any coupled BC is between two eigenvalues corresponding to separated BCs. These inequalities provide a platform for the implicit function theorem. In fact, it is an effective and concise way to deal with the continuous dependence of eigenvalues on the endpoints and coefficients as well. On the other hand, it is important to identify the position of the double eigenvalues because the eigenvalues generated by the real coupled BCs may be double. The method developed by us remains valid for this problem.

The paper is organized as follows. In Section 2, the basic concepts of the S-L operator and two lemmas needed subsequently are given. In Section 3, the continuous dependence of the n-th eigenvalue for separated BC on the BC is considered. In Section 4, we study the same problem associated with coupled BC, and show the position and number of the generated double eigenvalues under the real coupled BC.

2 Preliminaries

Consider the regular S-L differential operator [3] defined on [0,1]

(2.1)\begin{array}{rl}& Ly=:\frac{1}{r(x)}\left[-{\left(p(x)y^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}+q(x)y\right],y\in \mathfrak{D}(L),\\ & \mathfrak{D}(L)=\left\{y\in L_r^2[0,1]|y,y^{[1]}\in AC[0,1],Ly\in L_r^2[0,1],y\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{s}(2.3)\mathrm{o}\mathrm{r}(2.5)\right\},\end{array}

where y^{[1]}=(p{y}^{\prime })(x) is the quasi-derivative of y, the coefficients p,q,r are real-valued functions satisfying

(2.2)\frac{1}{p},q,r\in L^1[0,1],p(x),r(x)>0,\mathrm{a}.\mathrm{e}.x\in [0,1].

Here, L_r^2[0,1] denotes the set of real-valued functions satisfying , while AC[0,1] stands for the set of real-valued functions which are absolutely continuous on [0,1]. In this paper, we concerned with two self-adjoint BCs, that is, separated and coupled BCs, see below (2.3) and (2.5).

(i)Separated boundary condition

(2.3)\begin{array}{l}\{\begin{array}{l}{y}^{[1]}(0)-h_{-}y(0)=0,\\ y^{[1]}(1)+h_{+}y(1)=0,\end{array}\end{array}

where h_{\pm }=\mathrm{\infty } represents the Dirichlet BC. It is well known [13] that the operator L subject to (2.3) has an infinite but countable numbers of eigenvalues, these are all real, simple, bounded from below and can be ordered to satisfy

(2.4)

Moreover, the eigenfunction u(x,{\lambda }_n) of {\lambda }_n has exactly n zeros in (0,1), see [15, p.73].

(ii) Coupled boundary condition

(2.5)\begin{array}{l}\left(\begin{array}{c}y(1)\\ y^{[1]}(1)\end{array}\right)={\mathrm{e}}^{\mathrm{i}\theta }K\left(\begin{array}{c}y(0)\\ y^{[1]}(0)\end{array}\right),\end{array}

where K={\left(k_{ij}\right)}_{2\times 2},k_{ij}\in \mathbb{R},|K|=1 and \theta \in (-\pi ,\pi ). The operator L subject to (2.5) has an infinite but countable numbers of eigenvalues, these are all real, bounded from below and can be ordered to satisfy

Each eigenvalue may be simple or double but there cannot be two consecutive equalities above. In particular, if \theta \ne 0, then the eigenvalue is simple, see [15, p.72].

Let {\left\{{\lambda }_n\right\}}_{n=0}^{\mathrm{\infty }} be the eigenvalues of the S-L operator consisting of (2.1) and (2.3). For separated BCs, the n-th eigenvalue {\lambda }_n can be regarded as a two-variable function of h_{-},h_{+}. For each h_{-}\in \mathbb{R}, {\lambda }_n is a real-valued function of h_{+}\in \mathbb{R}. Our main purpose of the paper is to study the continuous differentiability of function {\lambda }_n={\lambda }_n(h_{+}). Analogously, we can also study the case of function {\lambda }_n={\lambda }_n(h_{-}). For the sake of simplicity, {\lambda }_n^D\left(h_{\pm }\right) represents the Dirichlet eigenvalue of h_{\mp } for each h_{\pm }. Moreover, it follows from [14] that {\lambda }_n={\lambda }_n(h_{-},h_{+}) satisfies the following inequalities.

Lemma 2.1　 Let h_{-},h_{+}\in \mathbb{R}. Then we have the inequalities

For coupled BCs, the n-th eigenvalue {\lambda }_n can be regarded as a function of {\mathrm{e}}^{\mathrm{i}\theta }K, denoted by {\lambda }_n={\lambda }_n\left({\mathrm{e}}^{\mathrm{i}\theta }K\right). According to a well-known classical result (see [5, Theorem 3.2]), we have the following inequalities.

Lemma 2.2　 Let {\left\{{\mu }_n\right\}}_{n=0}^{\mathrm{\infty }} and {\left\{{\nu }_n\right\}}_{n=0}^{\mathrm{\infty }} denote the eigenvalues of L subject to the separated BCs,

y(0)=0,k_{22}y(1)-k_{12}{y}^{[1]}(1)=0,

and

y^{[1]}(0)=0,k_{21}y(1)-k_{11}{y}^{[1]}(1)=0,

respectively.

(1) If k_{11}>0 and k_{12}⩽0, for any \theta \in (-\pi ,0)\cup (0,\pi ), we have

(2) If k_{11}⩽0 and , for any \theta \in (-\pi ,0)\cup (0,\pi ), we have

(3) If neither case (1) nor case (2) applies to K, then either case (1) or case (2) applies to -K.

The eigenvalues are multi-valued functions of all the parameters of the self-adjoint BCs. With the help of inequalities among eigenvalues in Lemma 2.1 and Lemma 2.2, we divide \mathbb{R} into several intervals to realize our purpose. We are devoted to study the continuous differentiability of the n-th eigenvalues with respect to each variable h_{+},h_{-} for the separated BC and \theta ,K for the coupled BC in the divided interval, which provides an essential prerequisite for implicit function theorem. By using of implicit function theorem, we will prove the continuous dependence of the n-th eigenvalues on the separated and coupled BCs and explicitly present the expressions for its derivatives.

3 The case of the separated boundary condition

This section is devoted to study the continuous differentiability of the n-th eigenvalue of the S-L operator consisting of (2.1) and (2.3) with regard to h_{-} or h_{+}. For h_{-},h_{+}\in \mathbb{R}, let \phi (x,\lambda ),\psi (x,\lambda ) denote the solutions of the S-L equation

(3.1)\begin{array}{l}Ly=\lambda y,\end{array}

satisfying the initial conditions

\phi (0,\lambda )=1,{\phi }^{[1]}(0,\lambda )=h_{-};\psi (1,\lambda )=-1,{\psi }^{[1]}(1,\lambda )=h_{+},

respectively. For each x\in [0,1], \phi (x,\lambda ),\psi (x,\lambda ) are entire functions of \lambda. According to Liouville's formula, the Wronskian of \phi ,\psi defined by

w(x,\lambda )=W(\phi ,\psi )=\left|\begin{array}{cc}\phi (x,\lambda )& \psi (x,\lambda )\\ {\phi }^{[1]}(x,\lambda )& {\psi }^{[1]}(x,\lambda )\end{array}\right|

is independent of x, and then we rewrite w(x,\lambda ) as w(\lambda ). Substituting x=0,1 into the Wronskian of \phi ,\psi then yields

w(\lambda )={\psi }^{[1]}(0,\lambda )-h_{-}\psi (0,\lambda )={\phi }^{[1]}(1,\lambda )+h_{+}\phi (1,\lambda ).

It is easy to see a number {\lambda }_0 is an eigenvalue of the S-L operator consisting of (2.1) and (2.3) if and only if entire function w(\lambda ) vanishes at {\lambda }_0.

Lemma 3.1　 We introduce a shorthand notation {\phi }_n=\phi \left(x,{\lambda }_n\right),{\psi }_n=\psi \left(x,{\lambda }_n\right). Then we have

(3.2)\dot{w}\left({\lambda }_n\right)=-\frac{\left({\phi }_n,{\phi }_n\right)}{\phi \left(1,{\lambda }_n\right)}=\frac{\left({\psi }_n,{\psi }_n\right)}{\psi \left(0,{\lambda }_n\right)}\ne 0,

where \frac{\mathrm{d}w}{\mathrm{d}\lambda }=\dot{w}(\lambda ) and \left({\phi }_n,{\phi }_n\right)={\int }_0^1{\left|\phi \left(x,{\lambda }_n\right)\right|}^{2}r(x)\mathrm{d}x.

Proof　Differentiating both sides of L{\phi }_n=\lambda {\phi }_n with respect to \lambda, one infers that L\dot{{\phi }_n}=\lambda \dot{{\phi }_n}+{\phi }_n, where \frac{\mathrm{d}{\phi }_n}{\mathrm{d}\lambda }=:\dot{{\phi }_n}. Applying Green formula into \dot{{\phi }_n} and {\phi }_n on [0,1], one infers

\left(L{\dot{\phi }}_n,{\phi }_n\right)-\left({\dot{\phi }}_n,L{\phi }_n\right)={\left[{\dot{\phi }}_n{\overline{\phi }}_n\right]}_0^1.

Simple calculations show that

\left({\phi }_n,{\phi }_n\right)=-\dot{w}\left({\lambda }_n\right)\phi \left(1,{\lambda }_n\right).

By virtue of \phi (1,{\lambda }_n)\ne 0 (if not, we can derive \phi (x,{\lambda }_n)\equiv 0 by the uniqueness of initial value solutions, and this is a contradiction), the first identity holds. The proof of the latter identity is similar with above. The proof is complete.□

For each fixed h_{-}, we study the continuous differentiability of the n-th eigenvalue with regard to h_{+}.

Theorem 3.1　 For each fixed h_{-}\in \mathbb{R}, the n-th eigenvalue {\lambda }_n={\lambda }_n(h_{+}) is continuously differentiable on \mathbb{R}, and its derivative is given by

(3.3){\lambda }_n^{\prime }(h_{+})=\frac{|\phi (1,{\lambda }_n){|}^2}{({\phi }_n,{\phi }_n)}.

Proof　Fix h_{-}\in \mathbb{R}. According to Lemma 2.1, we have

where {\lambda }_{-1}^D(h_{-})=-\mathrm{\infty }. Consider the two-variable real-valued function F defined on \mathbb{R}\times \left({\lambda }_{n-1}^D\left(h_{-}\right),{\lambda }_n^D\left(h_{-}\right)\right)

F\left(h_{+},\lambda \right)={\phi }^{[1]}(1,\lambda )+h_{+}\phi (1,\lambda ).

It is easy to verify F\left(h_{+},\lambda \right) satisfies the following conditions:

(1) For any h_{+}^0\in \mathbb{R}, there exists an eigenvalue of L corresponding to h_{+}^0 in the divided interval \left({\lambda }_{n-1}^D(h_{-}),{\lambda }_n^D(h_{-})\right), denoted by {\lambda }_n^0. F is continuous on a neighborhood U(h_{+}^0,{\lambda }_n^0)\subset \mathbb{R}\times \left({\lambda }_{n-1}^D\left(h_{-}\right),{\lambda }_n^D\left(h_{-}\right)\right).

(2) F\left(h_{+}^0,{\lambda }_n^0\right)=0.

(3) \dot{F}\left(h_{+},\lambda \right)={\dot{\phi }}^{[1]}(1,\lambda )+h_{+}\dot{\phi }(1,\lambda ) is continuous.

(4) \dot{F}\left(h_{+}^0,{\lambda }_n^0\right)\ne 0.

By implicit function theorem [12], there exist a neighborhood U(h_{+}^0;\delta ) and a unique continuous function {\lambda }_n={\lambda }_n\left(h_{+}\right) such that

\{\begin{array}{l}F\left(h_{+},{\lambda }_n\left(h_{+}\right)\right)\equiv 0,h_{+}\in U(h_{+}^0;\delta ),\\ {\lambda }_n\left(h_{+}^0\right)={\lambda }_n^0.\end{array}

Applying implicit function theorem many times, we can derive a continuous function {\lambda }_n={\lambda }_n\left(h_{+}\right) defined on \mathbb{R} since the arbitrary of the initial point h_{+}^0\in \mathbb{R}.

Moreover, F(h_{+},\lambda ) has a derivative with respect to h_{+}:

F_{h_{+}}\left(h_{+},\lambda \right)=\phi (1,\lambda ),

and F_{h_{+}}\left(h_{+},\lambda \right) is continuous on U(h_{+}^0,{\lambda }_n^0). Then, we have {\lambda }_n={\lambda }_n\left(h_{+}\right) is continuously differentiable on \mathbb{R}

{\lambda }_n^{\mathrm{\prime }}\left(h_{+}\right)=-\frac{F_{h_{+}}\left(h_{+},{\lambda }_n\right)}{\dot{F}\left(h_{+},{\lambda }_n\right)}=-\frac{F_{h_{+}}\left(h_{+},{\lambda }_n\right)}{\dot{w}\left({\lambda }_n\right)}=\frac{{\left|\phi \left(1,{\lambda }_n\right)\right|}^2}{\left({\phi }_n,{\phi }_n\right)}.

The proof is complete.□

Analogously, for each fixed h_{+}, we can get the continuous differentiability of the n-th eigenvalue with regard to h_{-}.

Theorem 3.2　 For each fixed h_{+}\in \mathbb{R}, the n-th eigenvalue {\lambda }_n={\lambda }_n(h_{-}) is continuously differentiable on \mathbb{R}, and its derivative is given by

(3.4){\lambda }_n^{\mathrm{\prime }}\left(h_{-}\right)=\frac{{\left|\psi \left(0,{\lambda }_n\right)\right|}^2}{\left({\psi }_n,{\psi }_n\right)}.

Formulas (3.3) and (3.4) yield {\lambda }_n^{\mathrm{\prime }}\left(h_{+}\right) and {\lambda }_n^{\mathrm{\prime }}\left(h_{-}\right)>0, then {\lambda }_n is strictly monotonic increasing on \mathbb{R} with respect to h_{+} or h_{-}. From the above arguments, we have the n-th eigenvalue {\lambda }_n is continuously differentiable and strictly monotonic increasing on \mathbb{R} with respect to each variable h_{+}, h_{-} for the separated BC. Furthermore, it follows from implicit function theorem that its derivatives are also continuous.

4 The case of the coupled boundary condition

For the coupled BCs, by [4, Lemma 3.1], a number \lambda is an eigenvalue of the S-L operator consisting of (2.1) and (2.5) if and only if the characteristic

(4.1)\begin{array}{l}\mathrm{\Delta }(\lambda )=:(-k_{11}{u}_2^{[1]}+k_{12}{u}_1^{[1]}+k_{21}{u}_2-k_{22}{u}_1)(1,\lambda )+2\cos \theta =0,\end{array}

where u_1(x,\lambda ),u_2(x,\lambda ) are the fundamental solutions of the S-L equation (3.1) satisfying the initial conditions

(u_1(0,\lambda ),u_1^{[1]}(0,\lambda ))=(1,0),(u_2(0,\lambda ),u_2^{[1]}(0,\lambda ))=(0,1).

Lemma 4.1　 Let {\lambda }_n be the n-th eigenvalue of the S-L operator consisting of (2.1) and (2.5). Then we have

(4.2)\begin{array}{l}\dot{\mathrm{\Delta }}({\lambda }_n)=({\phi }_n,{\phi }_n)(-k_{12}{u}_2^{[1]}+k_{22}{u}_2)(1,{\lambda }_n),\end{array}

where \frac{\mathrm{d}\mathrm{\Delta }}{\mathrm{d}\lambda }=:\dot{\mathrm{\Delta }}(\lambda ),{\phi }_n=:\phi (x,{\lambda }_n)=u_1(x,{\lambda }_n)+d_n{u}_2\left(x,{\lambda }_n\right) is an eigenfunction of the n-th eigenvalue {\lambda }_n and

(4.3)d_n=\frac{{\mathrm{e}}^{\mathrm{i}\theta }{k}_{11}-u_1}{u_2-{\mathrm{e}}^{\mathrm{i}\theta }{k}_{12}}(1,{\lambda }_n).

Proof　It is easy to verify \frac{d{\phi }_n}{d\lambda }=:{\dot{\phi }}_n is the solution of the non-homogeneous linear equation (L-\lambda )y=\phi satisfying the initial conditions y(0,\lambda )=y^{[1]}(0,\lambda )=0. By means of a method of constant variation, one infers

\begin{array}{l}{\dot{u}}_1\left(1,{\lambda }_n\right)=\left(u_1,u_2\right)u_1\left(1,{\lambda }_n\right)-\left(u_1,u_1\right)u_2\left(1,{\lambda }_n\right),\\ {\dot{u}}_2\left(1,{\lambda }_n\right)=\left(u_2,u_2\right)u_1\left(1,{\lambda }_n\right)-\left(u_1,u_2\right)u_2\left(1,{\lambda }_n\right),\end{array}

where \frac{\mathrm{d}{u}_i}{\mathrm{d}\lambda }=:{\dot{u}}_i(1,\lambda ) and \frac{\mathrm{d}{u}_i^{[1]}}{\mathrm{d}\lambda }=:{\dot{u}}_i^{[1]}(1,\lambda ) for i=1,2. Inserting the above identities into \dot{\mathrm{\Delta }}({\lambda }_n) then yields

\begin{array}{rl}\dot{\mathrm{\Delta }}({\lambda }_n)& =(-k_{11}{\dot{u}}_2^{[1]}+k_{12}{\dot{u}}_1^{[1]}+k_{21}{\dot{u}}_2-k_{22}{\dot{u}}_1)(1,{\lambda }_n)\\ & =(u_2,u_2)(-k_{11}{u}_1^{[1]}+k_{21}{u}_1)(1,{\lambda }_n)+(u_1,u_1)(-k_{12}{u}_2^{[1]}+k_{22}{u}_2)(1,{\lambda }_n)\\ & +(u_1,u_2)(k_{11}{u}_2^{[1]}+k_{12}{u}_1^{[1]}-k_{21}{u}_2-k_{22}{u}_1)(1,{\lambda }_n).\end{array}

Since {\phi }_n satisfies the BC (2.5), we derive the system of homogeneous linear equations

\{\begin{array}{l}\left(u_1+d_n{u}_2\right)\left(1,{\lambda }_n\right)={\mathrm{e}}^{\mathrm{i}\theta }\left(k_{11}+k_{12}{d}_n\right),\\ (u_1^{[1]}+d_n{u}_2^{[1]})\left(1,{\lambda }_n\right)={\mathrm{e}}^{\mathrm{i}\theta }\left(k_{21}+k_{22}{d}_n\right).\end{array}

This implies

d_n=\frac{{\mathrm{e}}^{\mathrm{i}\theta }{k}_{11}-u_1}{u_2-{\mathrm{e}}^{\mathrm{i}\theta }{k}_{12}}\left(1,{\lambda }_n\right)=\frac{{\mathrm{e}}^{\mathrm{i}\theta }{k}_{21}-u_1^{[1]}}{u_2^{[1]}-{\mathrm{e}}^{\mathrm{i}\theta }{k}_{22}}\left(1,{\lambda }_n\right).

Simple calculations show that

\begin{array}{rl}(-k_{11}{u}_1^{[1]}+k_{21}{u}_1)(1,{\lambda }_n)& ={\mathrm{e}}^{-\mathrm{i}\theta }[-({\mathrm{e}}^{\mathrm{i}\theta }{k}_{11}-u_1)u_1^{[1]}+({\mathrm{e}}^{\mathrm{i}\theta }{k}_{21}-u_1^{[1]})u_1](1,{\lambda }_n)\\ & ={\mathrm{e}}^{-\mathrm{i}\theta }{d}_n[-(u_2-{\mathrm{e}}^{\mathrm{i}\theta }{k}_{12})u_1^{[1]}+(u_2^{[1]}-{\mathrm{e}}^{\mathrm{i}\theta }{k}_{22})u_1](1,{\lambda }_n)\\ & =d_n{\mathrm{e}}^{\mathrm{i}\theta }[({\mathrm{e}}^{-\mathrm{i}\theta }{k}_{11}-u_1)u_2^{[1]}-({\mathrm{e}}^{-\mathrm{i}\theta }{k}_{21}-u_1^{[1]})u_2](1,{\lambda }_n)\\ & ={\left|d_n\right|}^2{\mathrm{e}}^{\mathrm{i}\theta }[(u_2-{\mathrm{e}}^{-\mathrm{i}\theta }{k}_{12})u_2^{[1]}-(u_2^{[1]}-{\mathrm{e}}^{-\mathrm{i}\theta }{k}_{22})u_2](1,{\lambda }_n)\\ & ={\left|d_n\right|}^2(-k_{12}{u}_2^{[1]}+k_{22}{u}_2)(1,{\lambda }_n).\end{array}

Similarly, we have

(k_{11}{u}_2^{[1]}+k_{12}{u}_1^{[1]}-k_{21}{u}_2-k_{22}{u}_1)(1,{\lambda }_n)=2\mathrm{Re}{d}_n(-k_{12}{u}_2^{[1]}+k_{22}{u}_2)(1,{\lambda }_n),

lemma is proved by replacing \dot{\mathrm{\Delta }}({\lambda }_n) by the above results.□

Basing on Lemma 4.1, we study the continuous dependence of the n-th eigenvalue on the coupled BC by means of implicit function theorem similar with the case of separated BC. Since the eigenvalues are all simple for the complex coupled BC and the eigenvalues may be simple or double for the real coupled BC, we will discuss the two cases, respectively.

4.1 The case of the complex coupled boundary condition

To begin with, we study the continuous differentiability of the n-th eigenvalue with respect to \theta.

Theorem 4.1　 For each fixed K, the n-th eigenvalue {\lambda }_n={\lambda }_n(\theta ) is continuously differentiable on (-\pi ,0)\cup (0,\pi ), and its derivative is given by

(4.4){\lambda }_n^{\mathrm{\prime }}(\theta )=\frac{-2\mathrm{Im}\left(\phi {\overline{\phi }}^{[1]}\right)\left(1,{\lambda }_n\right)}{\left({\phi }_n,{\phi }_n\right)}.

Proof　Fix K. Without loss of generality, we assume k_{11}>0 and k_{12}⩽0. According to Lemma 2.2, we have

where {\mu }_{-1}={\nu }_0. Consider the two-variable real-valued function F defined on (-\pi ,0)\cup (0,\pi )\times \left(max\left\{{\mu }_{n-1},{\nu }_n\right\},min\left\{{\mu }_n,{\nu }_{n+1}\right\}\right)

F(\theta ,\lambda )=(-k_{11}{u}_2^{[1]}+k_{12}{u}_1^{[1]}+k_{21}{u}_2-k_{22}{u}_1)(1,\lambda )+2\cos \theta .

The proof is similar with the proof of Theorem 3.1 and hence is omitted.□

Theorems 3.1, 3.2 and 4.1 provide a new proof of the continuous dependence of the n-th eigenvalue on the BC in [7, Theorem 4.2]. The methods employed is simpler, and we prove that the dependence of the n-th eigenvalue on the BC. In addition, the derivatives derived by us are continuous.

We next study the continuous differentiability of the n-th eigenvalue with respect to K=(k_{ij}{)}_{2\times 2}. In particular, we only consider the case for k_{11}=\frac{1}{k_{22}}=k(k>0),k_{12}=k_{21}=0.

Theorem 4.2　 For each fixed \theta \in (-\pi ,0)\cup (0,\pi ), the n-th eigenvalue {\lambda }_n={\lambda }_n(k) is continuously differentiable on {\mathbb{R}}_{+}, and its derivative is given by

(4.5){\lambda }_n^{\mathrm{\prime }}(k)=\frac{2d_n}{k\left({\phi }_n,{\phi }_n\right)},

where d_n is defined by (4.3).

Proof　Fix \theta \in (-\pi ,0)\cup (0,\pi ). Consider the two-variable function F defined on {\mathbb{R}}_{+}\times \left(max\left\{{\mu }_{n-1},{\nu }_n\right\},min\left\{{\mu }_n,{\nu }_{n+1}\right\}\right)

F(k,\lambda )=2\cos \theta -(k{u}_2^{[1]}+u_1/k)(1,\lambda ).

The proof is similar with the proof of Theorem 3.1 and hence is omitted.□

In fact, the method of implicit function theorem should work for the general case K=(k_{ij}{)}_{2\times 2}.

4.2 The case of the real coupled boundary condition

Different from the previous cases, the eigenvalues may be simple or double for the real coupled BC. For convenience, we study the continuous dependence of the n-th eigenvalue of L subject to the quasi-periodic BC

(4.6)\begin{array}{l}\left(\begin{array}{c}y(1)\\ y^{[1]}(1)\end{array}\right)=\left(\begin{array}{cc}k& 0\\ 0& {\displaystyle \frac{1}{k}}\end{array}\right)\left(\begin{array}{c}y(0)\\ y^{[1]}(0)\end{array}\right).\end{array}

Without loss of generality, we assume k>0. The n-th eigenvalue of the S-L operator consisting of (2.1) and (4.6) can be regarded as a one-variable function of k.

We will prove the n-th eigenvalue {\lambda }_n={\lambda }_n(k) is continuously differentiable by using of implicit function theorem. Basing on Lemma 3.2, we obtain the new results that the position and number of the generated double eigenvalues under the real coupled BC.

Lemma 4.2 [5]　 An eigenvalue \lambda of the S-L operator consisting of (2.1) and (4.6) is double if and only if there exist n,m\in {\mathbb{N}}_0 such that \lambda ={\mu }_n={\nu }_m, where {\mu }_n and {\nu }_m defined in front of Lemma 2.2.

Without loss of generality, we assume n is odd. According to Lemma 2.2, we have

where {\left\{{\mu }_n\right\}}_{n=0}^{\mathrm{\infty }} and {\left\{{\nu }_n\right\}}_{n=0}^{\mathrm{\infty }} denote the Dirichlet eigenvalues and Neumann eigenvalues, respectively. By Lemma 3.2, the n-th eigenvalue {\lambda }_n={\lambda }_n(k) is double only if {\mu }_n={\nu }_{n+1}.

Lemma 4.3　 If {\mu }_n={\nu }_{n+1}, then the following statements are equivalent.

(1) u_1\left(1,{\lambda }_n(k)\right)=k;

(2) u_2\left(1,{\lambda }_n(k)\right)=0;

(3) {\lambda }_n={\lambda }_n(k) is double.

Proof　(1)\Rightarrow(2): Suppose that u_1\left(1,{\lambda }_n(k)\right)=k. By virtue of u_1\left(1,{\lambda }_n(k)\right)+d_n{u}_2\left(1,{\lambda }_n(k)\right)=\phi \left(1,{\lambda }_n(k)\right)=k, we obtain d_n{u}_2\left(1,{\lambda }_n(k)\right)=0. If u_2\left(1,{\lambda }_n(k)\right)=0, then (2) holds. If d_n=0, then the eigenfunction {\phi }_n=u_1. Hence, u_1 satisfies the BC (4.6) then yields u_1^{[1]}\left(1,{\lambda }_n(k)\right)=0. This implies {\lambda }_n(k)={\nu }_{n+1}={\mu }_n and so u_2\left(1,{\lambda }_n(k)\right)=u_2\left(1,{\mu }_n\right)=0.

(2)\Rightarrow(3): If u_2\left(1,{\lambda }_n(k)\right)=0, then we have {\lambda }_n(k)={\mu }_n={\nu }_{n+1}. Hence {\lambda }_n={\lambda }_n(k) is double.

(3)\Rightarrow(1): If {\lambda }_n={\lambda }_n(k) is double, then we have {\lambda }_n(k)={\mu }_n={\nu }_{n+1} and so

u_2\left(1,{\lambda }_n(k)\right)=u_2\left(1,{\mu }_n\right)=0,u_1^{[1]}\left(1,{\lambda }_n(k)\right)=u_1^{[1]}\left(1,{\nu }_{n+1}\right)=0.

It is obvious that F\left(k,{\lambda }_n(k)\right)=0,\dot{F}\left(k,{\lambda }_n(k)\right)=0. This implies

\{\begin{array}{l}k{u}_2^{[1]}\left(1,{\lambda }_n(k)\right)+{\displaystyle \frac{u_1\left(1,{\lambda }_n(k)\right)}{k}}=2,\\ k{u}_2^{[1]}\left(1,{\lambda }_n(k)\right)-{\displaystyle \frac{u_1\left(1,{\lambda }_n(k)\right)}{k}}=0.\end{array}

Solving the system of equations, we obtain u_1\left(1,{\lambda }_n(k)\right)=k,u_2^{[1]}\left(1,{\lambda }_n(k)\right)=\frac{1}{k}.□

Theorem 4.3　 If {\mu }_n={\nu }_{n+1}, then there exists a unique k_0=u_1\left(1,{\nu }_{n+1}\right) such that the n-th eigenvalue {\lambda }_n={\lambda }_n(k_0) is double.

Proof　If the eigenvalue {\lambda }_n^0={\lambda }_n\left(k_0\right) is double, by Lemma 3.3, then

k_0=u_1\left(1,{\lambda }_n\left(k_0\right)\right)=u_1\left(1,{\nu }_{n+1}\right).

According to the oscillatory of eigenfunction, u_1\left(1,{\nu }_{n+1}\right) has exactly n+1 (even) zeros in (0,1). Since u_1\left(0,{\nu }_{n+1}\right)=1>0, one infers u_1\left(1,{\nu }_{n+1}\right)⩾0. If u_1\left(1,{\nu }_{n+1}\right)=0, by the uniqueness of initial value solutions, then u_1\left(x,{\nu }_{n+1}\right)\equiv 0, which is a contradiction. Hence u_1\left(1,{\nu }_{n+1}\right)>0. The proof is complete.□

It is well known that if {\lambda }_n(k_0) is a simple eigenvalue of the S-L operator consisting of (2.1) and (4.6), then there exists a neighborhood U of k_0 such that {\lambda }_n(k) is simple for any k in U. This guarantees the application of implicit function theorem. The continuous differentiability of the n-th eigenvalue {\lambda }_n={\lambda }_n(k) for k\in {\mathbb{R}}_{+} is shown below.

Theorem 4.4　 The n-th eigenvalue {\lambda }_n={\lambda }_n(k) of the S-L operator consisting of (2.1) and (4.6) is continuously differentiable on {\mathbb{R}}_{+}, and its derivative is given by

(4.7){\lambda }_n^{\mathrm{\prime }}(k)=\frac{2{\hat{d}}_n(k)}{k\left({\phi }_n,{\phi }_n\right)},

where {\hat{d}}_n(k) is a one-variable function of k, defined by

{\hat{d}}_n(k)=\{\begin{array}{l}{\displaystyle \frac{k-u_1(1,{\lambda }_n)}{u_2\left(1,{\lambda }_n\right)}},k\ne k_0,\\ 0,k=k_0.\end{array}

Remark 4.1　If {\mu }_n={\nu }_{n+1}, then {\lambda }_n^{\mathrm{\prime }}(k_0)=0 but {\lambda }_n^{\mathrm{\prime }}(k)\ne 0 for k\in {\mathbb{R}}_{+}\mathrm{\setminus }{k}_0, that is, k_0 is the unique extreme point of {\lambda }_n(k). Since the maximum of {\lambda }_n(k) is reached at k_0, the n-th eigenvalue {\lambda }_n(k) is monotone increasing in (0,k_0) and monotone decreasing in (k_0,+\mathrm{\infty }). Similarly, we can also obtain the continuous differentiability of the (n+1)-th eigenvalue {\lambda }_{n+1}={\lambda }_{n+1}(k). Then the relation of eigenvalues {\lambda }_n={\lambda }_n(k) and {\lambda }_{n+1}={\lambda }_{n+1}(k) if {\mu }_n={\nu }_{n+1} is shown as Fig.1.

The Proof of Theorem 4.4　Let F be the two-variable real-valued function of k,\lambda defined on {\mathbb{R}}_{+}\mathrm{\setminus }\left\{k_0\right\}\times \left(max\left\{{\mu }_{n-1},v_n\right\},min\left\{{\mu }_n,v_{n+1}\right\}\right)

F(k,\lambda )=2-\left(k{u}_2^{[1]}+\frac{1}{k}{u}_1\right)(1,\lambda ).

Similar with the proof of Theorem 3.1, there exists a unique continuously differentiable function {\lambda }_n={\lambda }_n(k) defined on {\mathbb{R}}_{+}\mathrm{\setminus }\left\{k_0\right\}, and its derivative is given by

{\lambda }_n^{\mathrm{\prime }}(k)=\frac{2{\hat{d}}_n(k)}{k\left({\phi }_n,{\phi }_n\right)}.

We have verified the continuous differentiability of the n-th eigenvalue {\lambda }_n={\lambda }_n(k) on {\mathbb{R}}_{+}\mathrm{\setminus }\left\{k_0\right\} by implicit function theorem. The rest of proof is divided into two steps: the continuity and smoothness of the function {\lambda }_n={\lambda }_n(k) at k_0.

Step I　The continuity. If {\mu }_n={\nu }_{n+1}, by Lemma 3.3 and (4.7), then {\lambda }_n^{\mathrm{\prime }}(k)\ne 0 for k\in {\mathbb{R}}_{+}\mathrm{\setminus }\left\{k_0\right\}, that is, k_0 is not an extreme point of {\lambda }_n(k). Since the maximum of {\lambda }_n(k) is reached at k_0, the n-th eigenvalue {\lambda }_n(k) is monotone increasing in (0,k_0) and monotone decreasing in (k_0,+\mathrm{\infty }). The fact that {\lambda }_n(k) is monotone and bounded on a neighborhood of k_0 then yields the limit of {\lambda }_n(k) at k_0 exists, denoted by {\lambda }_n^0. By the continuity of F\left(k,{\lambda }_n(k)\right), we obtain

0=\underset{k\to k_0}{lim}F\left(k,{\lambda }_n(k)\right)=F\left(k_0,{\lambda }_n^0\right).

Note that the n-th eigenvalue of L subject to the same BC is unique, by F\left(k_0,{\lambda }_n\left(k_0\right)\right)=0, we have

{\lambda }_n\left(k_0\right)={\lambda }_n^0.

Therefore, {\lambda }_n(k) is continuous at k_0.

Step II　The smoothness. The inner product of {\phi }_n=u_1+{\hat{d}}_n{u}_2 and u_2 is \left({\phi }_n,u_2\right)=\left(u_1,u_2\right)+{\hat{d}}_n\left(u_2,u_2\right). We can divide by (u_2,u_2)\ne 0 to conclude

{\hat{d}}_n(k)=\frac{\left({\phi }_n-u_1,u_2\right)}{\left(u_2,u_2\right)}.

Then \underset{k\to k_0}{lim}{\hat{d}}_n(k) exists and so \underset{k\to k_0}{lim}{\lambda }_n^{\mathrm{\prime }}(k) exists. Since {\lambda }_n={\lambda }_n(k) is continuous on [k_0-\delta ,k_0+\delta ] and {\lambda }_n^{\mathrm{\prime }}(k) is continuous in the deleted neighborhood U^{\circ }\left(k_0;\delta \right), one infers {\lambda }_n^{\mathrm{\prime }}\left(k_0\right)=\underset{k\to k_0}{lim}{\lambda }_n^{\mathrm{\prime }}(k) exists and thus {\lambda }_n(k) is differentiable at k_0. By virtue of k_0 is the extreme point of {\lambda }_n(k), we have {\lambda }_n^{\mathrm{\prime }}(k_0)=0 and \underset{k\to k_0}{lim}{\hat{d}}_n(k)=0={\hat{d}}_n(k_0). The derivative of {\lambda }_n(k) at k_0 is continuous and can be represented as (4.7) as well. Therefore, {\lambda }_n^{\mathrm{\prime }}(k) is continuous at k_0.□

Basing on the above argument, under the hypothesis {\mu }_n={\nu }_{n+1}, we obtain the position and number of the generated double eigenvalues {\lambda }_n={\lambda }_n(k) and its analytic properties under the real coupled BC.

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