A Boundary Value Problem for the Dirac System with a Spectral Parameter in the Boundary Conditions

Differential Equations, Vol. 38, No. 2, 2002, pp. 164–174. Translated from Differentsial’nye Uravneniya, Vol. 38, No. 2, 2002, pp. 155–164. c 2002 by Kerimov. Original Russian Text Copyright

ORDINARY DIFFERENTIAL EQUATIONS

A Boundary Value Problem for the Dirac System with a Spectral Parameter in the Boundary Conditions N. B. Kerimov Baku State University, Baku, Azerbaijan Received May 15, 2001

Problems with a spectral parameter in equations and boundary conditions form an important part of spectral theory of linear differential operators. A bibliography of papers in which such problems were considered in connection with specific physical processes can be found in [1, 2]. Boundary value problems for ordinary differential operators with a spectral parameter in boundary conditions were considered in various settings in numerous papers [3–14]. The completeness and the basis property of eigenfunctions of boundary value problems with a spectral parameter in equations and boundary conditions were studied in more detail in [7, 8]. Consider the following boundary value problem for the Dirac system with the same spectral parameter in the equations and the boundary conditions: u0 − {λ + P (x)}w = 0, w0 + {λ + R(x)}u = 0, (λ cos α + a0 ) u(0) − (λ sin α + b0 ) w(0) = 0, (λ cos β + a1 ) u(1) − (λ sin β + b1 ) w(1) = 0;

(1) (2) (3)

here 0 ≤ x ≤ 1, λ is the spectral parameter, P (x) and R(x) are real-valued functions from the class C[0, 1], and ak , bk (k = 1, 2), α, and β are real constants; moreover, −π/2 ≤ α ≤ π/2 and −π/2 ≤ β ≤ π/2. We shall study the properties of the eigenvalues and (vector-valued) eigenfunctions of the boundary value problem (1)–(3). The main results are given in Theorem 3.1 on the oscillation property and Theorem 5.1 on asymptotic formulas for the eigenvalues. Throughout the following, we assume that σ0 = a0 sin α − b0 cos α > 0,

σ1 = a1 sin β − b1 cos β < 0.

(4)

Note that the boundary conditions (λα11 + β11 )u(0) − (λα12 + β12 )w(0) = 0,

(λα21 + β21 )u(1) − (λα22 + β22 )w(1) = 0

can readily be reduced to the form (2), (3) provided that α12 β11 −α11 β12 > 0 and α22 β21 −α21 β22 < 0; moreover, condition (4) is satisfied in this case. 1. SOME PROPERTIES OF THE EIGENVALUES OF THE BOUNDARY VALUE PROBLEM (1)–(3) Lemma 1.1. The eigenvalues of the boundary value problem (1)–(3) are real. Proof. Let λ0 be a nonreal eigenvalue of the boundary value problem (1)–(3), and let (u(x), w(x)) be the corresponding eigenfunction. By (1), we have n o
 (d/dx) u(x)w(x) − u(x)w(x) = λ0 − λ0 |u(x)|2 + |w(x)|2 c 2002 MAIK “Nauka/Interperiodica” 0012-2661/02/3802-0164$27.00

A BOUNDARY VALUE PROBLEM FOR THE DIRAC SYSTEM

165

for 0 ≤ x ≤ 1. By integrating this identity from 0 to 1, we obtain n

Z1 o n o
 u(1)w(1) − u(1)w(1) − u(0)w(0) − u(0)w(0) = λ0 − λ0 |u(x)|2 + |w(x)|2 dx.

(1.1)

0

We rewrite conditions (2) and (3) in the form u(0) = [(λ0 sin α + b0 ) / (λ0 cos α + a0 )] w(0), u(1) = [(λ0 sin β + b1 ) / (λ0 cos β + a1 )] w(1). Taking account of these relations, we obtain
2 u(1)w(1) − u(1)w(1) = λ0 − λ0 σ1 |w(1)|2 / |λ0 cos β + a1 | ,
2 u(0)w(0) − u(0)w(0) = λ0 − λ0 σ0 |w(0)|2 / |λ0 cos α + a0 | , where σ0 and σ1 are the constants defined in (4). Since λ0 6= λ0 , it follows from the last two relations and (1.1) that σ1 |w(1)|2 σ0 |w(0)|2 − 2 2 = |λ0 cos β + a1 | |λ0 cos α + a0 |

Z1



|u(x)|2 + |w(x)|2 dx.

0

R1 This contradicts the conditions 0 {|u(x)|2 + |w(x)|2 } dx > 0, σ0 > 0, and σ1 < 0. Consequently, λ0 must be real. The proof of Lemma 1.1 is complete. One can readily show that there exists a unique solution of system (1) satisfying the initial condition u(0, λ) = λ sin α + b0 , w(0, λ) = λ cos α + a0 ; (1.2) moreover, for each given x ∈ [0, 1], the functions u(x, λ) and w(x, λ) are entire functions of the argument λ. The proof of this assertion reproduces that of Theorem 1.1 in [15, p. 14] with obvious modifications. Note that the eigenvalues of the boundary value problem (1)–(3) are the roots of the equation (λ cos β + a1 ) u(1, λ) − (λ sin β + b1 ) w(1, λ) = 0.

(1.3)

Lemma 1.2. The eigenvalues of the boundary value problem (1)–(3) form an at most countable set without finite limit points. All eigenvalues of the boundary value problem (1)–(3) are simple. Proof. The eigenvalues are the zeros of the entire function occurring on the left-hand side in Eq. (1.3). We have shown (see Lemma 1.1) that this function does not vanish for nonreal λ. In particular, it does not vanish identically. Therefore, its zeros form an at most countable set without finite limit points. By (1), we have (d/dx){u(x, λ)w(x, µ) − u(x, µ)w(x, λ)} = (λ − µ){u(x, λ)u(x, µ) + w(x, λ)w(x, µ)}. By integrating this identity from 0 to 1 and by taking account of relation (1.2), we obtain u(1, λ)w(1, µ) − u(1, µ)w(1, λ) − σ0 (λ − µ) Z1 = (λ − µ) {u(x, λ)u(x, µ) + w(x, λ)w(x, µ)}dx. 0

DIFFERENTIAL EQUATIONS

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(1.4)

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We shall use the relation ∂u(1, λ) ∂w(1, λ) w(1, λ) − u(1, λ) = σ0 + ∂λ ∂λ

Z1



 {u(x, λ)}2 + {w(x, λ)}2 dx,

(1.5)

0

which can be obtained from (1.4) by dividing both sides by (λ − µ) with the subsequent passage to the limit as µ → λ. Let us show that Eq. (1.3) has only simple roots. Indeed, if λ = λ∗ is a multiple root of Eq. (1.3), then (λ∗ cos β + a1 ) u (1, λ∗ ) − (λ∗ sin β + b1 ) w (1, λ∗ ) = 0, u (1, λ∗ ) cos β + (λ∗ cos β + a1 ) ∂u (1, λ∗ ) /∂λ (1.6) ∗ ∗ ∗ − w (1, λ ) sin β − (λ sin β + b1 ) ∂w (1, λ ) /∂λ = 0. 2

2

Since σ1 6= 0, we have (λ∗ cos β + a1 ) + (λ∗ sin β + b1 ) 6= 0. Let λ∗ cos β + a1 6= 0. From (1.6), we obtain the relations u (1, λ∗ ) = [(λ∗ sin β + b1 ) / (λ∗ cos β + a1 )] w (1, λ∗ ) , ∂u (1, λ∗ ) λ∗ sin β + b1 ∂w (1, λ∗ ) σ1 w (1, λ∗ ) + = . 2 ∂λ λ∗ cos β + a1 ∂λ (λ∗ cos β + a1 ) Considering (1.5) with λ = λ∗ and taking account of the last relations, we obtain ∗

2

∗

2

σ1 {w (1, λ )} /(λ cos β + a1 ) = σ0 +

Z1 h

{u (x, λ∗ )} + {w (x, λ∗ )} 2

2

i dx,

0

which contradicts condition (4). The case in which λ∗ sin β + b1 6= 0 can be treated in a similar way. The proof of Lemma 1.2 is complete. 2. SOME AUXILIARY RESULTS I. Let Pν (x) and Rν (x) (ν = 1, 2) be functions of class C[0, 1] such that P2 (x) > P1 (x) > 0 and R2 (x) > R1 (x) > 0 for 0 ≤ x ≤ 1. Consider two systems of equations ϕ0ν − Pν (x)ψν = 0,

ψν0 + Rν (x)ϕν = 0,

ν = 1, 2.

(2.1ν )

Let (ϕ1 (x), ψ1 (x)) and (ϕ2 (x), ψ2 (x)) be arbitrary nontrivial solutions of the first and the second system, respectively. The following two assertions are a straightforward consequence of the comparison theorems in [16, pp. 152–153 of the Russian translation]. Assertion 2.1. If either ϕ1 (0) = 0 or ϕ1 (0) 6= 0, ϕ2 (0) 6= 0, and ψ1 (0)/ϕ1 (0) ≥ ψ2 (0)/ϕ2 (0), then the function ϕ2 (x) has at least as many zeros as ϕ1 (x) on the half-open interval 0 < x ≤ 1; moreover, the kth zero of ϕ2 (x) is less than the kth zero of ϕ1 (x). If either ψ1 (0) = 0 or ψ1 (0) 6= 0, ψ2 (0) 6= 0, and ϕ1 (0)/ψ1 (0) ≤ ϕ2 (0)/ψ2 (0), then the function ψ2 (x) has at least as many zeros as ψ1 (x) on the half-open interval 0 < x ≤ 1; moreover, the kth zero of ψ2 (x) is less than the kth zero of ψ1 (x). Assertion 2.2. If ϕ1 (x) and ϕ2 (x) have equally many zeros on the interval 0 < x < 1, ϕ1 (1) 6= 0, and ϕ2 (1) 6= 0, then ψ1 (1)/ϕ1 (1) > ψ2 (1)/ϕ2 (1). If ψ1 (x) and ψ2 (x) have equally many zeros on the interval 0 < x < 1, ψ1 (1) 6= 0, and ψ2 (1) 6= 0, then ϕ1 (1)/ψ1 (1) < ϕ2 (1)/ψ2 (1). DIFFERENTIAL EQUATIONS

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Let Q = {µ : µ cos α + a0 = 0 or µ sin α + b0 = 0}, µ∗ = min Q, and µ∗ = max Q. Assertion 2.1 has the following corollary. Lemma 2.1. Let (ϕν (x), ψν (x)) be an arbitrary solution of system (2.1ν ) with the initial condition ϕν (0) = λ(ν) sin α + b0 , ψν (0) = λ(ν) cos α + a0 (ν = 1, 2). (2.2) Furthermore, suppose that either λ(1) < λ(2) < µ∗ or µ∗ < λ(1) < λ(2) < µ∗ or µ∗ < λ(1) < λ(2) . Then ϕ2 (x) [respectively, ψ2 (x)] has at least as many zeros as ϕ1 (x) [respectively, ψ1 (x)] on the half-open interval 0 < x ≤ 1; moreover, the kth zero of ϕ2 (x) [respectively, ψ2 (x)] is less than the kth zero of ϕ1 (x) [respectively, ψ1 (x)]. −1

−2

Proof. Let T0 (λ) = (λ cos α + a0 ) (λ sin α + b0 ) . We have T00 (λ) = −σ0 (λ sin α + b0 ) . Since σ0 > 0 by (4), it follows that the function T0 (λ) is
strictly decreasing on any interval where
λ sin α + b0 6= 0. Consequently, ψ1 (0)/ϕ1 (0) = T0 λ(1) > T0 λ(2) = ψ2 (0)/ϕ2 (0). This, together with Assertion 2.1, implies the part of Lemma 2.1 concerning ϕ1 (x) and ϕ2 (x). −1 −2 Let T1 (λ) = (λ sin α + b0 ) (λ cos α + a0 ) . We have T10 (λ) = σ0 (λ cos α + a0 ) . Consequently, T1 (λ) is strictly increasing on each
of the intervals (−∞, µ∗ ), (µ∗ , µ∗ ), and (µ∗ , +∞). Therefore,
(1) ϕ1 (0)/ψ1 (0) = T1 λ < T1 λ(2) = ϕ2 (0)/ψ2 (0). This, together with Assertion 2.1, implies the part of Lemma 2.1 dealing with the functions ψ1 (x) and ψ2 (x). The proof of Lemma 2.1 is complete. The following assertion can be proved in a similar way. Lemma 2.2. Let (ϕν (x), ψν (x)) be a solution of the system of equations ϕ0ν + Pν (x)ψν = 0, − Rν (x)ϕν = 0 with the initial condition (2.2). Furthermore, suppose that either λ(2) < λ(1) < µ∗ or µ∗ < λ(2) < λ(1) < µ∗ or µ∗ < λ(2) < λ(1) . Then the assertion of Lemma 2.1 is valid. ψν0

II. Let (ϕ(x), ψ(x)) be an arbitrary nontrivial solution of the system of equations ϕ0 −p(x)ψ = 0, ψ + r(x)ϕ = 0, where p(x) ∈ C[0, 1], r(x) ∈ C[0, 1], and p(x), r(x) > 0 for 0 ≤ x ≤ 1. By x(0) (f ) [respectively, x(1) (f )] we denote the minimum (respectively, maximum) zero of a function f (x) ∈ C[0, 1] on the open interval (0, 1). 0

Lemma 2.3. There is a unique zero of ψ(x) [respectively, ϕ(x)] between any two consecutive zeros of ϕ(x) [respectively, ψ(x)]. Moreover, if x(0) (ϕ) [respectively, x(0) (ψ)] exists and the condition ϕ(0)ψ(0) > 0 [respectively, ϕ(0)ψ(0) < 0] is satisfied, then x(0) (ψ) [respectively, x(0) (ϕ)] also exists and x(0) (ψ) < x(0) (ϕ) [respectively, x(0) (ϕ) < x(0) (ψ)]. If x(1) (ψ) [respectively, x(1) (ϕ)] exists and the inequality ϕ(1)ψ(1) > 0 [respectively, ϕ(1)ψ(1) < 0] is valid, then x(1) (ϕ) [respectively, x(1) (ψ)] also exists and x(1) (ψ) < x(1) (ϕ) [respectively, x(1) (ϕ) < x(1) (ψ)]. Proof. The first part of the lemma is trivial. Let us prove the second part. Suppose that x(0) (ϕ) exists and ϕ(0)ψ(0) > 0. Suppose that nevertheless, either x(0) (ψ) does not exist at all or x(0) (ψ) > x(0) (ϕ). In both cases, ϕ(x)ψ(x) > 0

(2.3)

for 0 ≤ x ≤ x(0) (ϕ). 0 By integrating the identity (ϕ2 (x)) = 2p(x)ϕ(x)ψ(x) from 0 to x(0) (ϕ), we obtain x(0) Z (ϕ)

−ϕ2 (0) = 2

p(x)ϕ(x)ψ(x)dx. 0

By (2.3), the right-hand side of the last formula is positive, while the left-hand side is negative. Consequently, x(0) (ψ) exists and satisfies x(0) (ψ) < x(0) (ϕ). DIFFERENTIAL EQUATIONS

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The remaining cases can be treated in a similar way. The proof of the lemma is complete. We denote the number of zeros of a function f (x) ∈ C[0, 1] on the interval (0, 1) by N (f ) and the zeros themselves by tj (f ) [j = 1, . . . , N (f )] : 0 < t1 (f ) < · · · < tN (f ) (f ) < 1. Lemma 2.3 implies the following assertion. Corollary 2.1. If ϕ(0)ψ(0) > 0 and ϕ(1)ψ(1) > 0, then N (ϕ) = N (ψ),

0 < t1 (ψ) < t1 (ϕ) < · · · < tN (ψ) (ψ) < tN (ϕ) (ϕ) < 1.

If ϕ(0)ψ(0) < 0 and ϕ(1)ψ(1) < 0, then N (ϕ) = N (ψ),

0 < t1 (ϕ) < t1 (ψ) < · · · < tN (ϕ) (ϕ) < tN (ψ) (ψ) < 1.

If ϕ(0)ψ(0) > 0 and ϕ(1)ψ(1) < 0, then N (ψ) = N (ϕ) + 1,

0 < t1 (ψ) < t1 (ϕ) < · · · < tN (ϕ) (ϕ) < tN (ψ) (ψ) < 1.

If ϕ(0)ψ(0) < 0 and ϕ(1)ψ(1) > 0, then N (ϕ) = N (ψ) + 1,

0 < t1 (ϕ) < t1 (ψ) < · · · < tN (ψ) (ψ) < tN (ϕ) (ϕ) < 1.

3. OSCILLATION PROPERTIES OF THE EIGENFUNCTIONS OF THE BOUNDARY VALUE PROBLEM (1)–(3) Let (u(x, λ), w(x, λ)) be the solution of system (1) with the initial condition (1.2). We write G = {µ : µ cos β + a1 = 0 or µ sin β + b1 = 0}, µ ˜∗ = min(Q ∪ G), µ ˜∗ = max(Q ∪ G), ∗ ˜ ∗ = min {˜ ˜ = max {˜ M = max {max0≤x≤1 |P (x)|, max0≤x≤1 |R(x)|}, λ µ∗ , −M }, and λ µ∗ , M }, where Q is defined in Section 2. Lemmas 2.1 and 2.2 imply the following assertion. ˜ ∗ or λ00 < λ0 < λ ˜ ∗ , then u (x, λ00 ) [respectively, w (x, λ00 )] has Corollary 3.1. If λ00 > λ0 > λ 0 at least as many zeros as u (x, λ ) [respectively, w (x, λ0 )] on the half-open interval 0 < x ≤ 1; moreover, the kth zero of u (x, λ00 ) [respectively, w (x, λ00 )] is less than the kth zero of u (x, λ0 ) [respectively, w (x, λ0 )]. Consider the equation u(x, λ) = 0 [or w(x, λ) = 0], where 0 ≤ x ≤ 1. Obviously, the roots of this equation depend on λ. ˜∗, λ ˜ ∗ ]. In the following two assertions, we assume that λ ∈ [λ Lemma 3.1. If x0 (0 < x0 ≤ 1) is a zero of the function u (x, λ0 ) [respectively, w (x, λ0 )], then for an arbitrary sufficiently small ε > 0, there exists a δ > 0 such that u(x, λ) [respectively, w(x, λ)] has exactly one zero on the interval |x − x0 | < ε whenever |λ − λ0 | < δ. Proof. The proof of this fact reproduces that of Lemma 3.1 in [15, p. 30] word for word. This lemma has the following corollary. Corollary 3.2. As λ varies, the function u(x, λ) [or w(x, λ)] can lose or gain a zero only if the zero enters or leaves the interval through one of the endpoints 0 and 1. The existence of countably many eigenvalues of the boundary value problem (1)–(3) is justified in the following assertion. ∞

Theorem 3.1. There exists an unboundedly decreasing sequence {λ−n }n=1 of negative eigenval∞ ues and an unboundedly increasing sequence {λn }n=1 of nonnegative eigenvalues of the boundary value problem (1)–(3) : · · · < λ−n < λ−n+1 < · · · < λ−1 < λ1 < · · · < λn−1 < λn < · · · DIFFERENTIAL EQUATIONS

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Moreover, there exist numbers n∗, n∗ , k∗, k∗ ∈ N ∪{0} such that the eigenfunctions (un (x), wn (x)), n > n∗ , and (u−n (x), w−n (x)) , n > n∗ , corresponding to the eigenvalues λn (n > n∗ ) and λ−n (n > n∗ ), respectively, have the following properties : (a1 ) if α ∈ {−π/2} ∪ (0, π/2] and β ∈ [0, π/2), then N (wn ) = N (un ) = n + k∗ − n∗ ,

x(0) (wn ) < x(0) (un ) ;

(b1 ) if α ∈ {−π/2} ∪ (0, π/2] and β ∈ [−π/2, 0) ∪ {π/2}, then N (wn ) = N (un ) + 1 = n + k∗ − n∗ + 1,

x(0) (wn ) < x(0) (un ) ;

(c1 ) if α ∈ (−π/2, 0] and β ∈ [0, π/2), then N (un ) = N (wn ) + 1 = n + k∗ − n∗ ,

x(0) (un ) < x(0) (wn ) ;

(d1 ) if α ∈ (−π/2, 0] and β ∈ [−π/2, 0) ∪ {π/2}, then N (un ) = N (wn ) = n + k∗ − n∗ ,

x(0) (un ) < x(0) (wn ) ;

(a2 ) if α ∈ [0, π/2) and β ∈ {−π/2} ∪ (0, π/2], then N (w−n ) = N (u−n ) = n + k∗ − n∗ ,

x(0) (w−n ) < x(0) (u−n ) ;

(b2 ) if α ∈ [0, π/2) and β ∈ (−π/2, 0], then N (w−n ) = N (u−n ) + 1 = n + k∗ − n∗ + 1,

x(0) (w−n ) < x(0) (u−n ) ;

(c2 ) if α ∈ [−π/2, 0) ∪ {π/2} and β ∈ {−π/2} ∪ (0, π/2], then N (u−n ) = N (w−n ) + 1 = n + k∗ − n∗ ,

x(0) (u−n ) < x(0) (w−n ) ;

(d2 ) if α ∈ [−π/2, 0) ∪ {π/2} and β ∈ (−π/2, 0], then N (u−n ) = N (w−n ) = n + k∗ − n∗ ,

x(0) (u−n ) < x(0) (w−n ) .

Proof. We prove only the existence and properties of the sequence of nonnegative eigenvalues of the boundary value problem (1)–(3). The assertions pertaining to negative eigenvalues can be proved in a similar way. ˜ ∗ + 1, where λ ˜ ∗ is the number defined at the beginning of this section and Let λ ≥ λ∗ = λ (u(x, λ), w(x, λ)) is the solution of system (1) with the initial condition (1.2). By Corollary 3.1, the number of zeros of u(x, λ) is a nondecreasing function of λ. Let (ϕ(x, λ), ψ(x, λ)) be the solution of the system of equations ϕ0 − (λ − M − 1/2)ψ = 0,

ψ 0 + (λ − M − 1/2)ϕ = 0

(3.1)

ψ(0, λ) = λ cos α + a0 .

(3.2)

with the initial condition ϕ(0, λ) = λ sin α + b0 , One can readily see that ϕ(x, λ) = (λ cos α + a0 ) sin(λ − M − 1/2)x + (λ sin α + b0 ) cos(λ − M − 1/2)x. As λ → ∞, the number of zeros of this function on (0, 1) grows unboundedly. Let us compare the boundary value problem (1), (1.2) with the boundary value problem (3.1), (3.2). It follows from Lemma 2.1 that the number of zeros of u(x, λ) on (0, 1) also grows unboundedly as λ → ∞. DIFFERENTIAL EQUATIONS

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Consider the equation u(x, λ) = 0 for λ ≥ λ∗ . By Lemma 3.1, the roots of this equation continuously depend on λ. On the other hand, by Corollary 3.2, as λ grows, every zero of u moves to the left but cannot pass through 0, since the number of zeros does not decrease. By Corollary 3.2, ˜ 0 be the first value of the parameter λ ≥ λ∗ such that zeros enter through the point 1. Let λ u(1, λ) = 0. Obviously, this value exists.
˜ 0 has k∗ zeros on (0, 1). Suppose that the function u x, λ ˜ 1 (λ ˜1 > λ ˜ 0 ) be the second value of the parameter λ such that u(x, λ) = 0 and so on. Let λ
˜ 1 has exactly k∗ + 1 zeros on (0, 1). The sequence λ ˜0, λ ˜1 , λ ˜ 2 , . . . has Obviously, the function u x, λ ˜ k−1 < λ ≤ λ ˜ k has exactly k + k∗ zeros on (0, 1); the following property: the function u(x, λ) with λ
˜ ˜ k (k = 0, 1, . . .) are not eigenvalues moreover, u 1, λk = 0. One can readily see that the numbers λ of the boundary value problem (1)–(3).
˜ k−1 , λ ˜ k . Since By Assertion 2.2, the function w(1, λ)/u(1, λ) is strictly decreasing on λ     ˜k−1 = u 1, λ ˜k = 0, u 1, λ
˜ k−1 , λ ˜k . it follows that w(1, λ)/u(1, λ) strictly decreases from +∞ to −∞ on λ −1

−2

Let T (λ) = (λ cos β + a1 ) (λ sin β + b1 ) . We have T 0 (λ) = −σ1 (λ sin β + b1 ) . Since σ1 < 0 by assumption (4), it follows that the function T (λ) is strictly decreasing on an arbitrary interval, where λ sin β + b1 6= 0. ˜0 ]. Suppose that n∗ is the number of eigenvalues of the boundary value problem (1)–(3) on [0, λ
˜ ˜ Since the function w(1, λ)/u(1, λ) strictly decreases from +∞ to −∞ on λk−1 , λk , it follows that
˜ k−1 , λ ˜ k such that w (1, λn∗ +k ) /u (1, λn∗ +k ) = T (λn∗ +k ), there exists a unique value λn∗ +k ∈ λ i.e., condition (3) is satisfied. Consequently, λn∗ +k is an eigenvalue of the boundary value problem (1)–(3), and the first component of the corresponding eigenfunction (u (x, λn∗ +k ) , w (x, λn∗ +k ))
˜ has as many zeros as u x, λk (i.e., k + k∗ ) on (0, 1). Let us prove assertion (a1 ). By the definition of λ∗ , the roots of the linear functions µ cos α + a0 , µ sin α + b0 , µ cos β + a1 , and µ sin β + b1 lie on (−∞, λ∗ ). Consequently, if n > n∗ , then  if α ∈ (0, π/2)  sgn(sin 2α) sgn (un (0)wn (0)) = sgn ((λn cos α + a0 ) (λn sin α + b0 )) = sgn (a0 sin α) if α = ±π/2  sgn (b0 cos α) if α = 0. This, together with (4), implies that un (0)wn (0) > 0 for α ∈ {−π/2} ∪ (0, π/2] and n > n∗ . In a similar way, we can show that un (1)wn (1) > 0 for β ∈ [0, π/2) and n > n∗ . To complete the proof of assertion (a1 ), it remains to use Corollary 2.1. Assertions (b1 )–(d2 ) can be proved in a similar way. The proof of Theorem 3.1 is complete. 4. DEFINITION AND PROPERTIES OF THE FUNCTION θn (x). SOME AUXILIARY RESULTS Throughout this section, we assume that n > n∗ , where n∗ is the number defined in the proof of Theorem 3.1. Let (un (x), wn (x)) be the eigenfunction of the boundary value problem (1)–(3) corresponding to the eigenvalue λn . We introduce the angular variable θn (x) = Arc tan (un (x)/wn (x)), or, more precisely, θn (x) = arg {wn (x) + iun (x)} .

(4.1)

Taking account of relation (2), we define the initial value as λn sin α + b0 1 − l0 θn (0) = arctan + π, λn cos α + a0 2  1 for α ∈ {−π/2} ∪ (0, π/2] l0 = −1 for α ∈ (−π/2, 0]. DIFFERENTIAL EQUATIONS

(4.2) (4.3) Vol. 38

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For the remaining values of x, the function θn (x) is given by (4.1) modulo 2π, since the functions un (x) and wn (x) cannot vanish simultaneously. We fix the choice by the condition that θn (x) must satisfy condition (4.2) and be continuous with respect to x. This determines θn (x) uniquely. Lemma 4.1. The function θn (x) satisfies the differential equation θn0 (x) = λn + P (x) cos2 θn (x) + R(x) sin2 θn (x)

(4.4)

and is strictly increasing on the closed interval [0, 1]. Proof. Equation (4.4) readily follows from the definition of θn (x). Since λn > λ∗ > M for n > n∗ , where M is the number defined in Section 3, we have θn0 (x) ≥ λn − M cos2 θn (x) − M sin2 θn (x) = λn − M > 0

if

0 ≤ x ≤ 1.

This implies that θn (x) is strictly increasing on [0, 1]. The proof of Lemma 4.1 is complete. It follows from (4.1) that the zeros of un (x) are just the points at which θn (x) is a multiple of π. Considering the function un (x) as x increases from 0 to 1, we find that it has a zero at a point x ∈ (0, 1) if and only if θn (x) passes through a value that is a multiple of π at this point. Since 0 < θn (0) < π, it follows that θn (x) successively takes finitely many values π, 2π, . . . as x varies from 0 to 1. Since θn (x) cannot decrease to a multiple of π, we see that it takes values that are multiples of π in increasing order. By xn,k (k = 1, . . . , kn ) we denote the zeros of un (x) on (0, 1) : 0 < xn,1 < · · · < xn,kn < 1. It follows from the oscillation theorem 3.1 that kn = n + k∗ − n∗ for n > n∗ . We can readily see that θn (xn,k ) = πk

(k = 1, . . . , kn ) , λn sin β + b1 1 − l1 θn (1) = arctan + π + πkn , λn cos β + a1 2  1 for β ∈ [0, π/2) l1 = −1 for β ∈ [−π/2, 0) ∪ {π/2}.

(4.5) (4.6) (4.7)

Lemma 4.2. The asymptotic formulas λn = πn + O(1),

∆xn,k = xn,k+1 − xn,k = O n−1




(k = 0, . . . , kn )

(4.8)

are valid, where xn,0 = 0 and xn,kn +1 = 1. Proof. By integrating identity (4.4) from 0 to 1, we obtain θn (1) − θn (0) = λn +

Z1 

P (x) cos2 θn (x) + R(x) sin2 θn (x) dx.

0

Since the integral occurring in the last relation is O(1), we have λn = θn (1) − θn (0) + O(1).

(4.9)

Taking account of (4.2) and (4.6), we obtain 0 < θn (0) < π and πkn < θn (1) < π (kn + 1), whence it follows that θn (1) − θn (0) = πkn + O(1). This relation, together with (4.9) and the formula kn = n + k∗ − n∗ , implies the first asymptotic formula in (4.8). Let us prove the second formula in (4.8). By integrating identity (4.4) from xn,k to xn,k+1 and by taking account of the fact that the integral in the resulting relation is O(1), we obtain λn ∆xn,k = θn (xn,k+1 ) − θn (xn,k ) + O(1). DIFFERENTIAL EQUATIONS

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(4.10)

172

KERIMOV

Since θn (x) is an increasing function, it follows from (4.2), (4.5), and (4.6) that ( 0 ≤ θn (xn,k+1 ) − θn (xn,k ) =

π − θn (0) π θn (1) − πkn

if k = 0 if 1 ≤ k ≤ kn − 1 if k = kn

) ≤ π.

This, together with (4.10), implies that λn ∆xn,k = O(1). Now the validity of the second formula in (4.8) follows with regard to the first relation in (4.8). The proof of Lemma 4.2 is complete. R1 Lemma 4.3. The relation 0 f (x) cos 2θn (x)dx = O (wf (n−1 )) is valid for an arbitrary function f (x) ∈ C[0, 1], where wf (δ) = δ +wf∗ (δ) and wf∗ (δ) is the modulus of continuity of f (x) on the closed interval [0, 1]. Proof. First, we show that xZ n,k+1

cos 2θn (x)dx = O n−2




1 ≤ k ≤ kn − 1.

if

xn,k

Since θn0 (x) = λn + O(1), it follows from Lemma 4.2 that xZ n,k+1

1 cos 2θn (x)dx = λn

xn,k

xZ n,k+1

1 θn0 (x) cos 2θn (x)dx + λn

xn,k

1 = λn

xZ n,k+1

O(1) cos 2θn (x)dx xn,k



(k+1)π Z

cos ϕ dϕ + O

∆xn,k λn



 =O

∆xn,k λn

 = O n−2




kπ

if 1 ≤ k ≤ kn − 1. Consequently, Z1

xZ xn,1 n,k+1 Z Z1 kX n −1 f (x) cos 2θn (x)dx = f (x) cos 2θn (x)dx + f (x) cos 2θn (x)dx + f (x) cos 2θn (x)dx

0

0

= O (∆xn,1 ) + O (∆xn,kn ) +

kX n −1

f (xk )

k=1



= O n−1 + O n−2 = O ωf n−1





kX n −1 k=1

k=1 x n,k

xn,kn xZ n,k+1

|f (xk )| +

cos 2θn (x)dx + xn,k

kX n −1

kX n −1

xZ n,k+1

(f (x) − f (xk )) cos 2θn (x)dx

k=1 x n,k





O ωf∗ (∆xn,k ) ∆xn,k = O n−1 + O ωf∗ n−1

k=1

.

The proof of Lemma 4.3 is complete. 5. ASYMPTOTIC FORMULAS FOR THE EIGENVALUES OF THE BOUNDARY VALUE PROBLEM (1)–(3) Throughout this section, we assume that n is an integer with sufficiently large absolute value. Let (Φn (x), Ψn (x)) be an eigenfunction of the boundary value problem (1)–(3); moreover, suppose that Φn (x) has |n| zeros on (0, 1). By µn we denote the eigenvalue corresponding to the eigenfunction (Φn (x), Ψn (x)). It follows from the oscillation theorem 3.1 that µn = λn−k∗ +n∗ for n > 0 and µn = λn+k∗ −n∗ for n < 0. DIFFERENTIAL EQUATIONS

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A BOUNDARY VALUE PROBLEM FOR THE DIRAC SYSTEM

173

Theorem 5.1. The asymptotic formulas π 1 µn = πn + β − α − (sgn β + sgn(−α)) − 2 2 +O ω n

(P (x) + R(x))dx (5.1)

0



−1

for

n > 0,

π 1 µn = πn + β − α + (sgn α + sgn(−β)) − 2 2 + O ω |n|−1

Z1

Z1 (P (x) + R(x))dx 0





for

(5.2)

n<0

are valid, where sgn t = 1 for t ≥ 0 and sgn t = −1 for t < 0, w(δ) = δ + w1 (δ), and w1 (δ) is the modulus of continuity of P (x) − R(x) on the closed interval [0, 1]. Proof. First, we prove (5.1). Consider the function θ˜n (x) = arg(Ψn (x)+iΦn (x)). By Lemma 4.1, it satisfies the differential equation θ˜n0 (x) = µn + 2−1 (P (x) + R(x)) + 2−1 (P (x) − R(x)) cos 2θ˜n (x)

(5.3)

and is strictly increasing on [0, 1]; moreover, θ˜n (0) and θ˜n (1) are given by (4.2) and (4.6), respectively, with λn and kn replaced by µn and n. A straightforward computation with regard to these formulas readily shows that
θ˜n (0) = α + π/2 + (π/2) sgn(−α) + O n−1 , (5.4)
θ˜n (1) = πn + β + π/2 − (π/2) sgn β + O n−1 . Integrating both sides of Eq. (5.3) from 0 to 1 and taking account of relation (5.4), we obtain
π (sgn β + sgn(−α)) + O n−1 2 Z1 Z1 1 1 = µn + (P (x) + R(x))dx + (P (x) − R(x)) cos 2θ˜n (x)dx. 2 2

πn + β − α −

0

0

R1 To complete the proof of (5.1), it suffices to note that 0 (P (x) − R(x)) cos 2θ˜n (x)dx = O (ω (n−1 )) by Lemma 4.2. Let us prove (5.2). Let n be a negative integer. Consider the boundary value problem 0

Φ∗ (x) − {λ − P (1 − x)}Ψ∗ (x) = 0, 0

Ψ∗ (x) + {λ − R(1 − x)}Φ∗ (x) = 0, ∗ ∗ (λ cos α + a0 ) Φ∗ (0) − (λ sin α∗ + b∗0 ) Ψ∗ (0) = 0, (λ cos β ∗ + a∗1 ) Φ∗ (1) − (λ sin β ∗ + b∗1 ) Ψ∗ (1) = 0, where α∗ = β, β ∗ = α, a∗0 = −a1 , b∗0 = −b1 , a∗1 = −a0 , and b∗1 = −b0 . Since σ0∗ = a∗0 sin α∗ − b∗0 cos α∗ = −a1 sin β + b1 cos β = −σ1 > 0, σ1∗ = a∗1 sin β ∗ − b∗1 cos β ∗ = −a0 sin α + b0 cos α = −σ0 < 0, it follows that Theorem 3.1 can be applied to the boundary value problem (5.5), (5.6). If
Φ∗|n| (x), Ψ∗|n| (x) DIFFERENTIAL EQUATIONS

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(5.5) (5.6)

174

KERIMOV

is the eigenfunction of the boundary value problem (5.5), (5.6) corresponding to the eigenvalue µ∗|n| and the function Φ∗|n| (x) has |n| zeros on (0, 1), then, by (5.1), we have µ∗|n|

π 1 = π|n| + β − α − (sgn β ∗ + sgn (−α∗ )) + 2 2 ∗

∗

Z1

(P (1 − x) + R(1 − x))dx + O ω |n|−1





,

0

or, which is the same, µ∗|n|

π 1 = π|n| + α − β − (sgn α + sgn(−β)) + 2 2

Z1

(P (x) + R(x))dx + O ω |n|−1





.

(5.7)

0


We can readily see that the vector function Φ∗|n| (1 − x), Ψ∗|n| (1 − x) is the eigenfunction of the
boundary value problem (1)–(3) corresponding to the eigenvalue − µ∗|n| ; moreover, the function Φ∗|n| (1 − x) has exactly |n| zeros on (0, 1). Consequently, µn = −µ∗|n| for n < 0. This, together with (5.7), implies the desired formula (5.2) and completes the proof of Theorem 5.1. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Walter, J., Math. Z., 1973, vol. 133, no. 4, pp. 301–312. Fulton, C.T., Proc. Roy. Soc. Edinburgh. Sect. A, 1977, vol. 77, pp. 293–308. Schneider, A., Math. Z., 1974, vol. 136, no. 2, pp. 163–167. Hinton, D.B., Quart. J. Math. Oxford. Ser. 2 , 1979, vol. 30, no. 2, pp. 33–42. Russakovskii, E.M., Funkts. Analiz i Ego Prilozheniya, 1975, vol. 9, no. 4, pp. 91–92. Shkalikov, A.A., Funkts. Analiz i Ego Prilozheniya, 1982, vol. 16, no. 4, pp. 92–93. Shkalikov, A.A., Tr. Sem. im. I.G. Petrovskogo, 1983, no. 9, pp. 190–229. Makhmudov, A.P., Dokl. Akad. Nauk SSSR, 1984, vol. 277, no. 6, pp. 1318–1323. Meleshko, S.V. and Pokornyi, Yu.V., Differents. Uravn., 1987, vol. 23, no. 8, pp. 1466–1467. Kerimov, N.B. and Allakhverdiev, T.I., Differents. Uravn., 1993, vol. 29, no. 1, pp. 54–60. Kerimov, N.B. and Allakhverdiev, T.I., Differents. Uravn., 1993, vol. 29, no. 6, pp. 952–960. Binding, P.A., Browne, P.J., and Seddighi, K., Proc. Edinburgh. Math. Soc. Ser. 2 , 1994, vol. 37, no. 1, pp. 57–72. Binding, P.A. and Browne, P.J., Proc. Roy. Soc. Edinburgh. Sect. A. 127 , 1997, no. 6, pp. 1123–1136. Kerimov, N.B. and Mamedov, Kh.R., Sib. Mat. Zh., 1999, vol. 40, no. 2, pp. 325–335. Levitan, B.M. and Sargsyan, I.S., Vvedenie v spektral’nuyu teoriyu (Introduction to Spectral Theory), Moscow, 1970. Kamke, E.W.H., Differentialgleichungen, Leipzig, 1959. Translated under the title Spravochnik po obyknovennym differentsial’nym uravneniyam, Moscow, 1981.

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