## A Class of Differential Operators with Eigenparameter Dependent Boundary Conditions

Sun Kang, Gao Yunlan,

College of Sciences, Inner Mongolia University of Technology, Hohhot 010051

 基金资助: 国家自然科学基金.  11661059内蒙古自然科学基金.  2017MS(LH)0103

 Fund supported: the NSFC.  11661059the NSF of Inner Mongolia.  2017MS(LH)0103

Abstract

In this paper, A class of third-order differential operators with transition conditions and two boundary conditions containing spectral parameters is studied, and the analytical method is used to do two aspects of work. First, by constructing a new space and a new operator, the eigenvalues of the problem and the operator are connected so that the eigenvalues of the original problem are consistent with the eigenvalues of the new operator. Second, the properties of the eigenvalues of the original problem are studied, and the conclusion that the spectrum of the original problem has only point spectrum is given.

Keywords： Differential operator ; Boundary condition ; Transition condition ; Eigenvalues

Sun Kang, Gao Yunlan. A Class of Differential Operators with Eigenparameter Dependent Boundary Conditions. Acta Mathematica Scientia[J], 2022, 42(3): 661-670 doi:

## 1 引言

$$$ly = \frac{1}{w} (-{\rm i} y^{(3)}+q y) = \lambda y, x \in J,$$$

$$$[f(x), g(x)]_{1} = \int_{a}^{c} f_{1}(x) \bar{g}_{1}(x) w_{1}(x){\rm d}x+\int_{c}^{b} f_{2}(x) \bar{g}_{2}(x) w_{2}(x){\rm d}x,$$$

$H = H_{1} \oplus {\Bbb C}_{\rho_{1}} \oplus {\Bbb C}_{\rho_{2}}$, 对于任意的$F = (f(x), h, r)^{T}, G = (g(x), k, s)^{T}, h, r, k, s$均为复数, 在$H$内定义内积如下

$$$[F, G] = [f(x), g(x)]_{1}+\frac{1}{\rho_{1}} h \bar{k}+\frac{1}{\rho_{2}} r \bar{s},$$$

假设$D(A) $$H 中按内积(1.9)–(1.10)是不稠密的, 那么一定存在一个非零元素 F \in H , 对于 \forall G \in D(A) , 有 [F, G] = 0 . 其中 F = (f(x), h, r)^{T}, G = (g(x), k, s)^{T} . C_{0}^{\infty} 表示如下函数集合 其中 \varphi_{1}(x) \in C_{0}^{\infty}[a, c), \varphi_{2}(x)\in C_{0}^{\infty}(c, b] , 显然 C_{0}^{\infty}\oplus \{0\}\oplus \{0\}\subseteq D(A) . U = (u(x), 0, 0)^{T}\in C_{0}^{\infty}\oplus \{0\}\oplus \{0\} , 则 F\perp U . 根据内积定义有下列式子成立 又知 \bar{C}_{0}^{\infty} = H_{1} , 故 f_1(x) = 0, f_2(x) = 0 , 即 f(x) = 0 . G = (g(x), k, s)^{T}\in D(A) , 有 由于 k, s 是任意选取的, 故 h = 0, r = 0 . 于是 F = (f(x), h, r)^{T} = (0, 0, 0)^{T} , 这与最初的假设矛盾, 综上 D(A)$$ H$中是稠密的.

对于$F, G\in D(A)$,

$A$的定义, 知

$\begin{eqnarray} [AF, G]& = &[lf(x), g(x)]_{1}+\frac{1}{\rho_{1}}\left(\beta_{2} f^{[2]}(a)-\beta_{1} f(a)\right)\left(\alpha_{1} \bar{g}(a)-\alpha_{2} \bar{g}^{[2]}(a)\right) \\ &&+\frac{1}{\rho_{2}}\left(\beta_{4}f^{[2]}(b)-\beta_{3}f(b)\right)\left(\alpha_{3} \bar{g}(b)-\alpha_{4}\bar{g}^{[2]}(b)\right), \end{eqnarray}$

$\begin{eqnarray} [F, AG]& = &[f(x), lg(x)]_{1}+\frac{1}{\rho_{1}}\left(\alpha_{1}f(a)-\alpha_{2}f^{[2]}(a)\right)\left(\beta_{2}\bar{g}^{[2]}(a)-\beta_{1} \bar{g}(a)\right) \\ &&+\frac{1}{\rho_{2}}\left(\alpha_{3}f(b)-\alpha_{4}f^{[2]}(b)\right)\left(\beta_{4}\bar{g}^{[2]}(b)-\beta_{3}\bar{g}(b)\right), \end{eqnarray}$

由于$A$是对称的, 要证$A $$H 中是自伴的, 只需要证明: 若对于任意的 F = \left(f(x), \alpha_{1}f(a)-\alpha_{2}f^{[2]}(a), \alpha_{3}f(b)-\alpha_{4}f^{[2]}(b)\right)^{T}\in D(A) , 有若: [AF, V] = [F, U] 成立, 则 V\in D(A)$$ AV = U$ (其中$V = (v(x), m_{1}, m_{2})^{T}, U = (u(x), n_{1}, n_{2})^{T}$).

(1) $v_{1}(x), v_{1}^{[1]}(x), v_{1}^{[2]}(x)\in AC[a, c), v_{2}(x), v_{2}^{[1]}(x), v_{2}^{[2]}(x)\in AC(c, b], lv(x)\in H_{1}$;

(2) $m_{1} = \alpha_{1}v(a)-\alpha_{2}v^{[2]}(a)$;

(3) $m_{2} = \alpha_{3}v(b)-\alpha_{4}v^{[2]}(b)$;

(4) $l_{2}v = l_{4}v = l_{5}v = l_{6}v = 0$;

(5) $u(x) = lv(x)$;

(6) $n_{1} = \beta_{2}u^{[2]}(a)-\beta_{1}u(a)$;

(7) $n_{2} = \beta_{4}u^{[2]}(b)-\beta_{3}u(b)$;

$[lf(x), v(x)]_{1} = [f(x), u(x)]_{1},$由标准的Sturm-Liouville理论可知$v(x)\in D(l)$, 故(1)成立.

$\begin{eqnarray} [lf(x), v(x)]_{1}-[f(x), lv(x)]_{1} & = &\frac{1}{\rho_{1}}[\left(\alpha_{1}f(a)-\alpha_{2}f^{[2]}(a)\right)\bar{n}_{1}-\left(\beta_{2} f^{[2]}(a)-\beta_{1}f(a)\right)\bar{m}_{1}] \\ &&+\frac{1}{\rho_{2}}[\left(\alpha_{3}f(b)-\alpha_{4}f^{[2]}(b)\right)\bar{n}_{2}-\left(\beta_{4}f^{[2]}(b)-\beta_{3}f(b)\right)\bar{m}_{2}], {\qquad} \end{eqnarray}$

$\begin{eqnarray} [lf(x), v(x)]_{1}-[f(x), lv(x)]_{1} = f^{[2]}(a)\bar{v}(a)-f(a)\bar{v}^{[2]}(a)+f(b)\bar{v}^{[2]}(b) -f^{[2]}(b)\bar{v}(b), \end{eqnarray}$

$(2.3) = (2.4),$根据Naimark Patching Lemma, 存在这样的$F = (f(x), \alpha_{1}f(a)-\alpha_{2}f^{[2]}(a), \alpha_{3}f(b)-\alpha_{4}f^{[2]}(b))^{T}\in D(A)$, 使得

$\begin{eqnarray} &&[lf(x), v(x)]_{1}-[f(x), lv(x)]_{1} \\ & = &f(b)\bar{v}^{[2]}(b) -f^{[2]}(b)\bar{v}(b)+{\rm i} f^{[1]}(b)\bar{v}^{[1]}(b)+f^{[2]}(a)\bar{v}(a)-f(a)\bar{v}^{[2]}(a)\\ &&-{\rm i} f^{[1]}(a)\bar{v}^{[1]}(a)+f(c-)\bar{v}^{[2]}(c-) -f^{[2]}(c-)\bar{v}(c-)+{\rm i} f^{[1]}(c-)\bar{v}^{[1]}(c-)\\ &&+f^{[2]}(c+)\bar{v}(c+)-f(c+)\bar{v}^{[2]}(c+)-{\rm i} f^{[1]}(c+)\bar{v}^{[1]}(c+), \end{eqnarray}$

$\begin{eqnarray} &&{\rm i} f^{[1]}(b)\bar{v}^{[1]}(b)-{\rm i} f^{[1]}(a)\bar{v}^{[1]}(a) +f(c-)\bar{v}^{[2]}(c-) -f^{[2]}(c-)\bar{v}(c-)\\ && +{\rm i} f^{[1]}(c-)\bar{v}^{[1]}(c-)+f^{[2]}(c+)\bar{v}(c+) -f(c+)\bar{v}^{[2]}(c+)-{\rm i} f^{[1]}(c+)\bar{v}^{[1]}(c+) = 0, \end{eqnarray}$

由于原问题的特征值与算子$A$的特征值是一致的, 且自共轭算子$A$的特征值是实的, 故该推论成立.

## 3 特征值的性质

在常微分方程中由解的存在唯一性, 我们可得到该结论.

$\varphi_{1}(x, \lambda), \chi_{1}(x, \lambda), \psi_{1}(x, \lambda)$分别是方程

$\varphi_{2}(x, \lambda), \chi_{2}(x, \lambda), \psi_{2}(x, \lambda)$是方程

$y(x, \lambda)$带入边界条件$A_{\lambda}Y(a)+B_{\lambda}Y(b) = 0$, 可得

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