On Basis Property of Root Functions For a Class Second Order Differential Operator

We consider the boundary value problem for the differential equation on the interval [−1, 1] (1) $𝓁 (u) : \equiv - u^{''} + q (x) u = λ w (x) u,$ \ell (u): \equiv - {u^{''}} + q(x)u = \lambda w(x)u, with the boundary conditions (2) $L_{1} (u) : = u (- 1) + h u^{'} (- 1) = 0,$ {L_1}(u): = u( - 1) + h{u^'}( - 1) = 0,(3) $L_{2} (u) : = (β_{1} u (1) - β_{2} u^{'} (1)) + λ ({\tilde{β}}_{1} u (1) - {\tilde{β}}_{2} u^{'} (1)) = 0,$ {L_2}(u): = ({\beta _1}u(1) - {\beta _2}{u^'}(1)) + \lambda \left( {{{\widetilde \beta }_1}u(1) - {{\widetilde \beta }_2}{u^'}(1)} \right) = 0, and the transmission conditions (4) $L_{3} (y) : = γ_{1} u (- 0) - δ_{1} u (+ 0) = 0,$ {L_3}(y): = {\gamma _1}u( - 0) - {\delta _1}u( + 0) = 0,(5) $L_{4} (y) : = γ_{2} u^{'} (- 0) - δ_{2} u^{'} (+ 0) = 0,$ {L_4}(y): = {\gamma _2}{u^'}( - 0) - {\delta _2}{u^'}( + 0) = 0, where $w (x) = {\begin{matrix} w_{1}^{2}, & - 1 \leq x < 0, \\ w_{2}^{2}, & 0 < x \leq 1, \end{matrix}$ w(x) = \left\{ {\begin{array}{*{20}{c}} {w_1^2,}&{ - 1 \leqslant x < 0,} \\ {w_2^2,}&{0 < x \leqslant 1,} \end{array}} \right.w₁ ≠ w₂, the real valued function q(x) ∈ C ([−1, 0) ∪ (0, 1] ) and has finite limits $q (\pm 0) = lim_{x \to \pm 0} q (x), λ$ q( \pm 0) = \mathop {\lim }\limits_{x \to \pm 0} q(x),\lambda , λ is complex parameter and we assume that h, w_i, β_i, ${\tilde{β}}_{i}$ {\widetilde \beta _i}, γ_i, δ_i (i = 1, 2 ) are real numbers, γ_i, δ_i are positive coefficients, |β₁| + |β₂| ≠ 0, $| {\tilde{β}}_{1} | + | {\tilde{β}}_{2} | \neq 0$ \left| {{{\widetilde \beta }_1}} \right| + \left| {{{\widetilde \beta }_2}} \right| \ne 0 and $ρ : = β_{1} {\tilde{β}}_{2} - {\tilde{β}}_{1} β_{2}$ \rho : = {\beta _1}{\widetilde \beta _2} - {\widetilde \beta _1}{\beta _2}.

Spectral problems for the discontinuous Sturm-Liouville equations with eigenparameter dependent boundary conditions have been growing interest with physical applications and examined in [1, 2, 3, 4]. For example, boundary value problems with the form (1)–(7) is encountered in vibrating string problems investigated in [19]. This happens whenever one applies the method of seperation of variables to solve the corresponding partial differential equation which contain a directional derivative in boundary conditions. On eigenvalues problems for second order equation with spectral parameter in the boundary conditions are considered in [5, 6, 7, 8, 9, 10, 11, 12, 13, 20, 21, 22]. Some self adjoint problems on eigenvalues for second order equation with spectral parameter in the boundary conditions are considered in [5, 6, 7, 8, 9, 10, 11]. The corresponding problems led to the eigenvalue problem for a linear operator acting on the space L₂ ⊕ ℂ^N, where ℂ^N is N− dimensional Euclidean space of complex numbers. In [4] for distinct cases, it is shown that the eigenfunctions of the spectral problem formed a defect basis in L₂ (0, 1). In [14] Rayleigh-Ritz formula is developed for eigenvalues.

The goal of this work is to investigate the problem of completeness, minimality and basis property of the eigenfunctions of the boundary value problem (1)–(5). In this study, we introduce a special inner product in a special Hilbert space and construct a linear operator A in it so that the problem (1)–(5) can be interpreted as the eigenvalue problem for A. Let us give the main definitions and theorem which will be used in our main results:

Definition 1

[23] A sequence {f_j}_j_≥1 of vectors of a Hilbert space H is called a basis of this space if every vector f ∈ H can be expanded in a unique way in a series $f = \underset{j = 1}{\sum^{\infty}} c_{j} f_{j}$ f = \mathop {\mathop {\sum }\limits^\infty }\limits_{j = 1} {c_j}{f_j}, which converges in the norm of the space H.

Definition 2

[23] A basis {f_j}_j_≥1 of H is called a Riesz basis if it is obtained from an orthonormal basis by means of a bounded linear invertible operator.

Theorem 1

[24] Let A be selfadjoint compact operator or selfadjoint operator which has discrete spectrums. Then the eigenfunctions of the operator A form orthonormal basis in the Hilbert space H.

For example; $\begin{array}{l} u^{''} (x) + λ u (x) = 0, x \in (0, 1) \\ u (0) = 0, u^{'} (1) = d λ u (1), d > 0, \end{array}$ \begin{gathered} {u^{''}}(x) + \lambda u(x) = 0,\;\;\;x \in \left( {0,1} \right) \hfill \\ u(0) = 0,{u^'}(1) = d\lambda u(1),\;d > 0, \hfill \\ \end{gathered} this boundary value problem has only the eigenfunctions $u_{n} (x) = \sqrt{2} sin \sqrt{λ_{n}} x$ {u_n}(x) = \sqrt 2 \sin \sqrt {{\lambda _n}} x, n = 0, 1, 2.., with positive eigenvalues found from the equation $cot \sqrt{λ} = d \sqrt{λ}$ \cot \sqrt \lambda = d\sqrt \lambda . By deleting an arbitrary eigenfunction, one obtains a basis in the space L_p(0, 1) p > 1, even a Riesz bases in the case p = 2. [4]

The Operator Formulation of the Problem and Main Results

It is convenient to represent the spectral problem (1)–(5) as an eigenvalue problem for a linear problem in a Hilbert space. We denote by H = L₂[−1, 1] ⊕ ℂ the special Hilbert space of all elements $\tilde{u} = (\begin{matrix} u (x) \\ u_{1} \end{matrix}), \tilde{v} = (\begin{matrix} v (x) \\ v_{1} \end{matrix}) \in H,$ \widetilde u = \left( {\begin{array}{*{20}{c}} {u(x)} \\ {{u_1}} \end{array}} \right),\widetilde v = \left( {\begin{array}{*{20}{c}} {v(x)} \\ {{v_1}} \end{array}} \right) \in H, with the inner product (6) $(\tilde{u}, \tilde{v}) = w_{1}^{2} γ_{1} γ_{2} \int_{- 1}^{0} u (x) \bar{v (x)} dx + w_{2}^{2} δ_{1} δ_{2} \int_{0}^{1} u (x) \bar{v (x)} dx + \frac{δ_{1} δ_{2}}{ρ} u_{1} \bar{v_{1}} .$ \left( {\widetilde u,\widetilde v} \right) = w_1^2{\gamma _1}{\gamma _2}\int\limits_{ - 1}^0 u(x)\overline {v(x)} dx + w_2^2{\delta _1}{\delta _2}\int\limits_0^1 u(x)\overline {v(x)} dx + \frac{{{\delta _1}{\delta _2}}}{\rho }{u_1}\overline {{v_1}} . In the space we define the operator $A \tilde{u} = (\begin{matrix} \frac{1}{w (x)} [- u^{''} + q (x) u] \\ β_{1} u (1) - β_{2} u^{'} (1) \end{matrix}),$ A\widetilde u = \left( {\begin{array}{*{20}{c}} {\frac{1}{{w(x)}}[ - {u^{''}} + q(x)u]} \\ {{\beta _1}u(1) - {\beta _2}{u^'}(1)} \end{array}} \right), on the domain (7) $D (A) = {\begin{matrix} \tilde{u} | \tilde{u} = (u (x), u_{1}) \in H, u (x), u^{'} (x) \in AC ([- 1, 0) \cup (0, 1]), \\ u^{'} (\pm 0) = lim_{x \to \pm 0} u^{'} (x), 𝓁 (u) \in L_{2} [- 1, 1], \\ L_{1} u = L_{3} u = L_{4} u = 0, u_{1} = {\tilde{β}}_{1} u (1) - {\tilde{β}}_{2} u^{'} (1)), \end{matrix}}$ D(A) = \left\{ {\begin{array}{*{20}{c}} {\widetilde u|\widetilde u = \left( {u(x),{u_1}} \right) \in H,u(x),{u^'}(x) \in AC\left( {[ - 1,0) \cup \left( {0,1} \right]} \right),} \\ {{u^'}( \pm 0) = \mathop {\lim }\limits_{x \to \pm 0} {u^'}(x),\ell (u) \in {L_2}\left[ { - 1,1} \right],} \\ {{L_1}u = {L_3}u = {L_4}u = 0,{u_1} = {{\widetilde \beta }_1}u(1) - {{\widetilde \beta }_2}{u^'}(1)),} \end{array}} \right\} where AC ([−1, 1]) is the space of all absolutely continuous functions on the interval [−1, 1]. Obviously, the operator A is well defined in H. It is clear that the spectral problem (1)–(5) is equivalent to operator equation (8) $A \tilde{u} = λ \tilde{u},$ A\widetilde u = \lambda \widetilde u, and the eigenvalues of A coincide with the eigenvalues of the problem (1)–(5) (see Lemma 1.4. in [5]). Also, there exists a correspondence between eigenfunctions ${\tilde{u}}_{k} (x) \leftrightarrow (\begin{matrix} u_{k} (x) \\ {\tilde{β}}_{1} u (1) - {\tilde{β}}_{2} u^{'} (1)) \end{matrix})$ {\widetilde u_k}(x) \leftrightarrow \left( {\begin{array}{*{20}{c}} {{u_k}(x)} \\ {{{\widetilde \beta }_1}u(1) - {{\widetilde \beta }_2}{u^'}(1))} \end{array}} \right)

Lemma 2

The domain D(A) of the operator A is dense in the space H.

Proof

The proof is similar using the same method in [15]. Suppose that $\tilde{f} \in H$ \widetilde f \in H is orthogonal to all $\tilde{g} \in D (A)$ \widetilde g \in D(A) with respect to the inner product (6), where $\tilde{f} = (f (x), f_{1})$ \widetilde f = (f(x),{f_1}), $\tilde{g} = (g (x), g_{1})$ \widetilde g = (g(x),{g_1}). Let ${\tilde{C}}_{0}^{\infty}$ \widetilde C_0^\infty denote the set of functions $Φ (x) = {\begin{matrix} φ_{1} (x), x \in [- 1, 0), \\ φ_{2} (x), x \in (0, 1], \end{matrix}$ \;\;\;\Phi (x) = \left\{ {\begin{array}{*{20}{c}} {{\varphi _1}(x),x \in \left[ { - 1,0} \right),} \\ {{\varphi _2}(x),x \in \left( {0,1} \right],} \end{array}} \right. where $φ_{1} (x) \in C_{0}^{\infty} [- 1, 0)$ {\varphi _1}(x) \in C_0^\infty \left[ { - 1,0} \right) and $φ_{2} (x) \in C_{0}^{\infty} (0, 1]$ {\varphi _2}(x) \in C_0^\infty \left( {0,1} \right]. Since ${\tilde{C}}_{0}^{\infty} \oplus 0 \subset D (A)$ \widetilde C_0^\infty \oplus 0 \subset D(A) (0 ∈ ℂ), any $\tilde{u} = (u (x), 0) \in {\tilde{C}}_{0}^{\infty} \oplus 0$ \widetilde u = \left( {u(x),0} \right) \in \widetilde C_0^\infty \oplus 0 is orthogonal to $\tilde{f}$ \widetilde f , namely, $(\tilde{f}, \tilde{u}) = w_{1}^{2} γ_{1} γ_{2} \int_{- 1}^{0} f (x) \bar{u (x)} dx + w_{2}^{2} δ_{1} δ_{2} \int_{0}^{1} f (x) \bar{u (x)} dx = (f, u)_{1}$ (\widetilde f,\widetilde u) = w_1^2{\gamma _1}{\gamma _2}\int\limits_{ - 1}^0 f(x)\overline {u(x)} dx + w_2^2{\delta _1}{\delta _2}\int\limits_0^1 f(x)\overline {u(x)} dx = (f,u{)_1} where (,)₁ denotes inner product in L₂[−1, 1]. This implies that f(x) is orthogonal to ${\tilde{C}}_{0}^{\infty}$ \widetilde C_0^\infty and (f, u)₁ = 0. Hence, $(\tilde{f}, \tilde{g}) = \frac{δ_{1} δ_{2}}{ρ} f_{1} \bar{g_{1}} = 0 .$ (\widetilde f,\widetilde g) = \frac{{{\delta _1}{\delta _2}}}{\rho }{f_1}\overline {{g_1}} = 0. Thus f₁ = 0 since $g_{1} = {\tilde{β}}_{1} g (1) - {\tilde{β}}_{2} g^{'} (1)$ {g_1} = {\widetilde \beta _1}g(1) - {\widetilde \beta _2}{g^'}(1) can be chosen arbitrary. So $\tilde{f} = (0, 0) .$ \widetilde f = (0,0).. Therefore, D(A) is dense in H.

Lemma 3

The operator A is selfadjoint.

Proof

In this case integrating by parts, we obtain that $(A \tilde{f}, \tilde{g})$ (A\widetilde f,\widetilde g) is real. Taking into account Lemma 2, we find that the operator A is symmetric in the space H. Boundary value problem (1)–(5) is solvable for every non-eigenvalue λ and has discrete spectrum. Therefore the operator A is symmetric and has discrete spectrum. Hence, the operator A is selfadjoint in H.

Theorem 4

The eigenfunctions of the operator A form an orthonormal basis in the space H = L₂[−1, 1] ⊕ ℂ.

Proof

The operator A has countable many eigenvalues ${λ_{n}}_{n = 1}^{\infty}$ \left\{ {{\lambda _n}} \right\}_{n = 1}^\infty which have the asymptotic form [16]: $λ_{n} = \frac{1}{w_{1} + w_{2}} π (n - 1) + O (\frac{1}{n}), n \to \infty .$ {\lambda _n} = \frac{1}{{{w_1} + {w_2}}}\pi (n - 1) + O(\frac{1}{n}),n \to \infty .

Then for any number λ which is not an eigenvalue and arbitary $\tilde{f} \in H$ \widetilde f \in H it can be found an element ũ ∈ D(A) satisfying the condition $(A - λ I) \tilde{u} = \tilde{f} .$ \left( {A - \lambda I} \right)\widetilde u = \widetilde f.. Thus the operator (A − λI) is invertible except for the isolated eigenvalues. Without loss of generality we assume that the point λ = 0 is not an eigenvalue. Then we obtain that the bounded inverse operator A⁻¹ is defined in H. Thus, the selfadjoint operator A⁻¹ has at most countable many eigenvalues and each one of them converges to zero at the infinity. So, the selfadjoint operator A⁻¹ is compact. Applying the Hilbert-Schmidt theorem to this operator we obtain that the eigenfunctions of the operator A form an orthonormal basis in H. Theorem is proved.

Now we consider the case ρ < 0. We assume that the operator A is defined by formula (7) on the domain D(A). In the space H = L₂ ⊕ ℂ for ũ, ṽ ∈ H the scalar product is defined by formula (9) $(\tilde{u}, \tilde{v}) = w_{1}^{2} γ_{1} γ_{2} \int_{- 1}^{0} u (x) \bar{v (x)} dx + w_{2}^{2} δ_{1} δ_{2} \int_{0}^{1} u (x) \bar{v (x)} dx - \frac{δ_{1} δ_{2}}{ρ} u_{1} \bar{v_{1}} .$ \left( {\widetilde u,\widetilde v} \right) = w_1^2{\gamma _1}{\gamma _2}\int\limits_{ - 1}^0 u(x)\overline {v(x)} dx + w_2^2{\delta _1}{\delta _2}\int\limits_0^1 u(x)\overline {v(x)} dx - \frac{{{\delta _1}{\delta _2}}}{\rho }{u_1}\overline {{v_1}} . In this case the operator A is not selfadjoint in the space H. Therefore we introduce the operator J is defined by $J = (\begin{matrix} I & 0 \\ 0 & - I \end{matrix})$ J = \left( {\begin{array}{*{20}{c}} I&0 \\ 0&{ - I} \end{array}} \right) where I is the identity operator in H. Operator J is selfadjoint and invertible.

In this case, the boundary value problem (1)–(5) is equivalent to eigenvalue problem for the operator pencil (10) $(B - λ J) \tilde{u} = 0,$ (B - \lambda J)\widetilde u = 0, in the space H such that B = JA. We obtain that (8) is equivalent to (10).

Lemma 5

The operator A is J− selfadjoint in the Hilbert space H.

Proof

Analogously to Lemma 2, we can show that the domain D(A) is dense in space H. From (9) and (10) applying two times integration by parts, (Bũ, ũ) is real. Hence, the operator B is symmetric. Therefore, the operator A is J− symmetric in the space H. In this case it can be proved that the operator A has a discrete spectrum. Taking into consideration that the operator B is symmetric we have that the operator JA is selfadjoint.

Corollary 6

From the system ${u_{n}}_{0}^{\infty}$ \left\{ {{u_n}} \right\}_0^\infty one can eliminate one element so that the remaining elements will form a complete and minimal system in the space L₂[−1, 1].

Proof

By Theorem 4, the system of eigenfunctions ${\tilde{u}}_{n} (x) = {\begin{matrix} u_{n} (x) \\ u_{1} \end{matrix}},$ {\widetilde u_n}(x) = \left\{ {\begin{array}{*{20}{c}} {{u_n}(x)} \\ {{u_1}} \end{array}} \right\}, (u₁ ∈ ℂ ) of the operator A forms an orthonormal basis in H. Hence, the system of the eigenfunctions ${{\tilde{u}}_{n} (x)}_{1}^{\infty}$ \left\{ {{{\widetilde u}_n}(x)} \right\}_1^\infty is complete and minimal in the space H. Thus, of course, codimP = 1, then by Lemma 2.1 in [5], the system {Pũ_n(x)} = {u_n(x)} whose one element is omitted from forms a complete and minimal system in P(H) = L₂[−1, 1]. Hence, the eigenfunctions ${u_{n} (x)}_{0}^{\infty}$ \left\{ {{u_n}(x)} \right\}_0^\infty (n ≠ n₀, n₀ is an arbitrary nonnegative integer) of the boundary problem (1)–(5) are complete and minimal system in L₂[−1, 1].

Theorem 7

The eigenfunctions of the operator A form a Riesz basis in the Hilbert space H.

Proof

From the operator J is a bounded operator, using the idea in Theorem 4, it can be shown that the operator B = JA is invertible since the selfadjoint operator B⁻¹ is compact. Since the selfadjoint operator B⁻¹ has at most countable many eigenvalues which converge to zero at infinity. Hence the operator B⁻¹ is compact. Then applying Azizov-Iokhvidov theorem in section IV of [18] to operator B we obtain that the eigenfunctions of the J− selfadjoint operator A form a Riesz basis in the space H = L₂ ⊕ ℂ.

Let us consider another boundary value problem for the Sturm-Liouville equation (11) $𝓁 u : = - p (x) u^{''} + q (x) u = λ u,$ \ell u: = - p(x){u^{''}} + q(x)u = \lambda u, on [−1, 0) ∪ (0, 1] with boundary condition, (12) $L_{1} (u) : = α_{1} u (- 1) + α_{2} u^{'} (- 1) = 0$ {L_1}(u): = {\alpha _1}u( - 1) + {\alpha _2}{u^'}( - 1) = 0(13) $L_{2} (λ) u : = λ ({\tilde{β}}_{1} u (1) - {\tilde{β}}_{2} u^{'} (1)) + (β_{1} u (1) - β_{2} u^{'} (1)) = 0,$ {L_2}(\lambda )u: = \lambda ({\widetilde \beta _1}u(1) - {\widetilde \beta _2}{u^'}(1)) + ({\beta _1}u(1) - {\beta _2}{u^'}(1)) = 0, with transmission conditions at the point of discontinuity x = 0, (14) $L_{3} (u) : = γ_{1} u (- 0) - δ_{1} u (+ 0) = 0,$ {L_3}(u): = {\gamma _1}u( - 0) - {\delta _1}u( + 0) = 0,(15) $L_{4} (u) : = γ_{2} u^{'} (- 0) - δ_{2} u^{'} (+ 0) = 0,$ {L_4}(u): = {\gamma _2}{u^'}( - 0) - {\delta _2}{u^'}( + 0) = 0, here $p (x) = {\begin{matrix} p_{1}^{2}, & - 1 \leq x < 0, \\ p_{2}^{2}, & 0 < x \leq 1, \end{matrix}$ p(x) = \left\{ {\begin{array}{*{20}{c}} {p_1^2,}&{ - 1 \leqslant x < 0,} \\ {p_2^2,}&{0 < x \leqslant 1,} \end{array}} \right.p₁ ≠ p₂, the real valued function q(x) is continuous in [−1, 0) ∪ (0, 1] and has finite limits $q (\pm 0) = lim_{x \to \pm 0} q (x)$ q( \pm 0) = \mathop {\lim }\limits_{x \to \pm 0} q(x), λ is complex parameter, we assume that h, p_i, β_i, ${\tilde{β}}_{i}$ {\widetilde \beta _i}, γ_i, δ_i (i = 1, 2 ) are real numbers, γ_i, δ_i are positive coefficients, |β₁|+ |β₂| ≠ 0, $| {\tilde{β}}_{1} | + | {\tilde{β}}_{2} | \neq 0$ \left| {{{\widetilde \beta }_1}} \right| + \left| {{{\widetilde \beta }_2}} \right| \ne 0 and $ρ : = β_{1} {\tilde{β}}_{2} - {\tilde{β}}_{1} β_{2} > 0$ \rho : = {\beta _1}{\widetilde \beta _2} - {\widetilde \beta _1}{\beta _2} > 0.

It is convenient to represent the spectral problem (11)–(15) as an eigenvalue problem for a linear problem in a special Hilbert space. We denote by H = L₂ (−1, 1) ⊕ ℂ the special Hilbert space of all elements $f = (\begin{matrix} f (x) \\ f_{1} \end{matrix}), g = (\begin{matrix} g (x) \\ g_{1} \end{matrix}) \in H,$ f = \left( {\begin{array}{*{20}{c}} {f(x)} \\ {{f_1}} \end{array}} \right),g = \left( {\begin{array}{*{20}{c}} {g(x)} \\ {{g_1}} \end{array}} \right) \in H, with the inner product (16) $(f, g) = \frac{1}{p_{1}^{2}} γ_{1} γ_{2} \int_{- 1}^{0} f (x) \bar{g (x)} dx + \frac{1}{p_{2}^{2}} δ_{1} δ_{2} \int_{0}^{1} f (x) \bar{g (x)} dx + \frac{δ_{1} δ_{2}}{ρ} f_{1} \bar{g_{1}}$ \left( {f,g} \right) = \frac{1}{{p_1^2}}{\gamma _1}{\gamma _2}\int\limits_{ - 1}^0 f(x)\overline {g(x)} dx + \frac{1}{{p_2^2}}{\delta _1}{\delta _2}\int\limits_0^1 f(x)\overline {g(x)} dx + \frac{{{\delta _1}{\delta _2}}}{\rho }{f_1}\overline {{g_1}} In the space we define the operator $Lu = (\begin{matrix} - p (x) u^{''} + q (x) u \\ β_{1} u (1) - β_{2} u^{'} (1) \end{matrix}),$ Lu = \left( {\begin{array}{*{20}{c}} { - p(x){u^{''}} + q(x)u} \\ {{\beta _1}u(1) - {\beta _2}{u^'}(1)} \end{array}} \right), on the domain (17) $D (L) = {\begin{matrix} \tilde{u} | \tilde{u} = (u (x), u_{1}) \in H, u (x), u^{'} (x) \in AC ([- 1, 0) \cup (0, 1]), \\ u^{'} (\pm 0) = lim_{x \to \pm 0} u^{'} (x), 𝓁 (u) \in L_{2} [- 1, 1], \\ L_{1} u = L_{3} u = L_{4} u = 0, u_{1} = {\tilde{β}}_{1} u (1) - {\tilde{β}}_{2} u^{'} (1)), \end{matrix}}$ D(L) = \left\{ {\begin{array}{*{20}{c}} {\widetilde u|\widetilde u = \left( {u(x),{u_1}} \right) \in H,u(x),{u^'}(x) \in AC\left( {[ - 1,0) \cup \left( {0,1} \right]} \right),} \\ {{u^'}( \pm 0) = \mathop {\lim }\limits_{x \to \pm 0} {u^'}(x),\ell (u) \in {L_2}\left[ { - 1,1} \right],} \\ {{L_1}u = {L_3}u = {L_4}u = 0,{u_1} = {{\widetilde \beta }_1}u(1) - {{\widetilde \beta }_2}{u^'}(1)),} \end{array}} \right\}u(±0) has finite limits. Now we can rewrite the problem (11)–(15) in the operator form as (18) $Lu = λ u .$ Lu = \lambda u. The eigenvalues of the operator L coincide with the eigenvalues of the spectral problem (11)–(15).

Theorem 8

The eigenfunctions of the operator L form an orthonormal basis in the space H = L₂[−1, 1] ⊕ ℂ.

Proof

The operator L has countable many eigenvalues ${λ_{n}}_{n = 1}^{\infty}$ \left\{ {{\lambda _n}} \right\}_{n = 1}^\infty which have the asymptotic form [17]: $λ_{n} = \frac{α_{1} α_{2}}{α_{1} + α_{2}} π (n - 1) + O (\frac{1}{n}), n \to \infty .$ {\lambda _n} = \frac{{{\alpha _1}{\alpha _2}}}{{{\alpha _1} + {\alpha _2}}}\pi (n - 1) + O(\frac{1}{n}),n \to \infty . Then using the idea of the proof of Theorem 4, it can be proved analogusly.

Now for the boundary value problem (11)–(15) we consider the case ρ < 0. In the space H = L₂ ∈ ℂ for f, g ∈ H the scalar product is defined by formula $(f, g) = \frac{1}{p_{1}^{2}} γ_{1} γ_{2} \int_{- 1}^{0} f (x) \bar{g (x)} dx + \frac{1}{p_{2}^{2}} δ_{1} δ_{2} \int_{0}^{1} f (x) \bar{g (x)} dx - \frac{δ_{1} δ_{2}}{ρ} f_{1} \bar{g_{1}} .$ \left( {f,g} \right) = \frac{1}{{p_1^2}}{\gamma _1}{\gamma _2}\int\limits_{ - 1}^0 f(x)\overline {g(x)} dx + \frac{1}{{p_2^2}}{\delta _1}{\delta _2}\int\limits_0^1 f(x)\overline {g(x)} dx - \frac{{{\delta _1}{\delta _2}}}{\rho }{f_1}\overline {{g_1}} . Since ρ < 0, the operator A is not selfadjoint in the space H. Therefore we introduce the operator J is defined by $J = (\begin{matrix} I & 0 \\ 0 & - I \end{matrix})$ J = \left( {\begin{array}{*{20}{c}} I&0 \\ 0&{ - I} \end{array}} \right) where I is the identity operator in H. Operator J is selfadjoint and invertible in H.

In this case, the boundary value problem (11)–(15) is equivalent to eigenvalue problem for the operator pencil (19) $(B - λ J) \tilde{u} = 0,$ (B - \lambda J)\widetilde u = 0, where B = JL is symmetric and L is J− symmetric in the space H.. Similary to Lemma 5, The operator L is J− selfadjoint in the Hilbert space H and we obtain the following results:

Theorem 9

The eigenfunctions of the operator L form a Riesz basis in the Hilbert space H.

Corollary 10

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On Basis Property of Root Functions For a Class Second Order Differential Operator

Published Online: Mar 31, 2020

Page range: 361 - 368

Received: Aug 01, 2019

Accepted: Sep 30, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00034

KeywordsBasis Property, Sturm-Liouville, Eigenvalues

© 2019 Volkan Ala et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Basis Property, Sturm-Liouville, Eigenvalues