A Dirac System with Transmission Condition and Eigenparameter in Boundary Condition

Lemma 1. The dom (A) is dense in ℍ.

Proof. It is easily seen that there is no nonzero vector $$ such that for every $$, 〈F,U〉_ℍ = 0. This implies dom (A) ^⊥ = {Θ}, where Θ = (0,0, 0). Therefore, dom (A) is dense in ℍ.

Theorem 2. The operator A is symmetric.

Proof. For each U, V ∈ dom (A) from the inner product (7) and the integration by parts, we have

Since U and V satisfy the same boundary condition (4) at x = a, From transmission condition (6), it follows that Furthermore, Now substituting (13), (14), and (15) in (12), we obtain

Corollary 3. All the eigenvalues of the system (1)–(6) are real and to every eigenvalue λ_n, there corresponds a vector-valued eigenfunction $$,u_2n(x, λ_n)). Moreover, vector-valued eigenfunctions belonging to different eigenvalues are orthogonal in the sense of

Remark 4. The vector-valued eigenfunctions stated in Corollary 3 are not orthogonal in the usual sense in the Hilbert space L₂[a, b].

Theorem 5. The Dirac system (1) has a solution Φ(x, λ) on [a, b] satisfying boundary condition (4) and transmission condition (6). For each x, Φ(x, λ) is a vector-valued entire function of λ.

Proof. From the classical theory of differential equations (see [10]), since the Dirac system

with the initial conditions is continuous on the interval [a, c), this system has a unique solution Φ₁(x, λ) = (Φ₁₁(x, λ), Φ₂₁(x, λ)) ^T which is an entire function of λ on [a, c).

Now consider the Dirac system of differential equations

and nonstandard initial conditions contain eigenparameter

Let us denote solutions of (20) by u₀(x, λ) = (u₁₀(x, λ), u₂₀(x, λ)) ^T in the case p₁(x) = p₂(x) ≡ 0. It is clear that the vector-valued function u₀(x, λ) is written as

From the initial conditions (21), we obtain constants c₁ and c₂. Then, inserting these values into (22) and using some basic trigonometric identities, we arrive at

By applying the method of variation of the constants as in [11, page 243], we find the following system of integral equations:

In what follows, we use the method of successive approximations, which is helpful in constructing a solution of the integral equation system (24). This method requires a sequence of functions {u_n(x, λ)} for n = 1,2, … defined as

where u₁₀(x, λ) and u₂₀(x, λ) are defined in (23). It is obvious that each of u_n(x, λ) is an entire function of λ for every x ∈ (c, b].

Set

where $$, and let M₁ = max _x∈(c,b]|p₁(x)|, M₂ = max _x∈(c,b]|p₂(x)|, M = max (M₁, M₂), N₁(λ) = max _x∈(c,b]|u₁₀(x, λ)|, N₂(λ) = max _x∈(c,b]|u₂₀(x, λ)|. Then, where the norm ∥·∥ can be any convenient norm in ℍ, but for the sake of presentation, we used 1 − norm . Furthermore, let N₁ = max _|λ|≤RN₁(λ), N₂ = max _|λ|≤RN₂(λ), and N_R = max (N₁, N₂) in closed contour {λ ∈ ℂ : |λ| ≤ R}; then Similarly, and so generally, Now, consider the infinite series The nth partial sum of this series is u_n(x, λ); that is, Therefore, the sequence {u_n(x, λ)} converges if and only if series (31) does so. In view of (30), it follows that series (31) is uniformly convergent with respect to x on (c, b] and λ in the closed contour {λ ∈ ℂ : |λ| ≤ R}. Let the sum of series (31) be Φ₂(x, λ) = (Φ₁₂(x, λ), Φ₂₂(x, λ)) ^T; that is, and so, (32) gives

Finally, we will show next that the limit function Φ₂(x, λ) satisfies (20). For this, we need to find $$. From (33),

For the first term on the right-hand side of (35), if we take n = 1 in (25), then now from (25) and the fact that (u₁₀, u₂₀) ^T is a solution of the homogeneous system, we have For the second term on the right-hand side of (35), it follows from (25) and (26) that and its derivative is In this equation By using (39) and (40), the second term on the right-hand side of (35) becomes Substituting (37) and (41) into (35) gives so that Φ₂(x, λ) satisfies (20) on (c, b]. It also clearly satisfies the boundary conditions (21). As a result, the vector-valued function Φ(x, λ) defined by satisfies the Dirac system (1), (4), and (6).

Theorem 6. For any λ ∈ ℂ, the Dirac system

has a solution on [a, c) ∪ (c, b] satisfying the boundary condition (5) and transmission condition (6). For each x ∈ [a, c) ∪ (c, b], Ψ(x, λ) is a vector-valued entire function of λ.

Proof. The proof of this theorem is similar to that of Theorem 5 and hence is omitted.

Theorem 7. The eigenvalues of the problem (1)–(6) are the zeros of the function ω(λ).

Proof. Let ω(λ_n) = 0 for any λ = λ_n. Then, it follows from (51) that the Wronskian of Φ₂(x, λ_n) and Ψ₂(x, λ_n) is zero, so that Ψ₂(x, λ_n) is a constant multiple of Φ₂(x, λ_n), say

It follows that Ψ(x, λ_n) also fulfils the boundary condition (5) and, therefore, is a vector-valued eigenfunction of the problem (1)–(6) for eigenvalue λ_n.

Conversely, let u_n(x, λ_n) be a vector-valued eigenfunction corresponding to eigenvalue λ_n, but ω(λ_n) ≠ 0. Then, from (51), at least one of the pair of the functions $$ and $$ would be linearly independent. Therefore, u_n(x, λ_n) can be expressed as

where at least one of the constants C₁, C₂, D₁, D₂ is not zero. Since u_n(x, λ_n) is a vector-valued eigenfunction corresponding to eigenvalue λ_n by substitution in conditions (4)–(6), we obtain a system of linear, homogeneous equations and the determinant of this system is zero. This means that W(λ_n) = 0, and from (51), ω(λ_n) = 0 which yields a contradiction to the assumption that ω(λ_n) ≠ 0. This completes the proof.

Theorem 8. The Dirac system (1)–(6) has at most denumerably many eigenvalues, and these eigenvalues have no finite limit point.