Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 62, pp. 1–30. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF FOURTH-ORDER DIFFERENTIAL OPERATORS WITH SPECTRAL PARAMETER IN THE BOUNDARY CONDITIONS DMITRY M. POLYAKOV Abstract. We consider a spectral problem for a fourth-order differential equa- tion with spectral parameter dependent boundary conditions. We determine the high energy eigenvalue behavior for this operator. Moreover, if the co- efficient of differential equation is sufficiently smooth, we can obtain sharp eigenvalue asymptotic behavior. This behavior exhibits a non-standard high- frequency effect generated by the spectral parameter in the boundary condi- tions. 1. Introduction and main results We consider the eigenvalue problem y(4)(x)− (p(x)y′(x))′ = λy(x), y(0) = y′(0) = y′′(1) = 0, (aλ+ b)y(1)− (cλ+ d)Ty(1) = 0, (1.1) where λ ∈ C is a spectral parameter, p is a real absolutely continuous function on [0, 1], Ty = y′′′ − py′, a, b, c, d ∈ R, and σ = bc− ad > 0. This problem arises in the boundary value problem describing the free bending vibrations of a homogeneous beam when the left end of this beam is fixed, and at the right end there is concentrated a load (see [24]). If b = c = 0 and d = 1, then this situation corresponds to the right end of beam having a load hanging, the mass of which is equal to −a (see [13]). Boundary value problems for a fourth-order differential equation with a spectral parameter in the boundary conditions have been studied quite actively in previous years. A general theory of boundary value problems for ordinary differential op- erators with a spectral parameter in the boundary conditions was constructed in [25]. There the main results are devoted to completeness, minimality, and basis property of system of eigenfunctions and associated functions for these operators. The spectral properties of boundary eigenvalue problems for differential equations of the form Ny = λPy investigated in [26]. There N and P are regular differential operators of order n and p, with n > p ≥ 0, and the boundary conditions depend on 2020 Mathematics Subject Classification. 34L20, 34B08, 34B09. Key words and phrases. Eigenvalue; asymptotic behavior; fourth-order eigenvalue problem; spectral parameter in boundary conditions; fourth-order differential operator. ©2024. This work is licensed under a CC BY 4.0 license. Submitted July 23, 2024. Published October 16, 2024. 1 2 D. M. POLYAKOV EJDE-2024/62 a spectral parameter polynomially. The main results there describe the complete- ness, minimality, and the Riesz basis properties of the corresponding eigenfunctions and associated eigenfunctions. Basis properties for fourth-order differential operators (1.1) with a spectral pa- rameter in the boundary conditions of different statements are investigated in [1, 2, 6, 14, 15]. General characteristic of the location of the eigenvalues on the real axis and oscillation properties of eigenfunctions for fourth-order differential operators with spectral parameter in three boundary conditions are obtained in [3]. Convergence of eigenfunction expansions for this operators are studied in [4, 5]. The asymptotic behavior of spectrum and a trace formula for fourth-order dif- ferential operator with unbounded operator coefficient and spectral parameter in boundary condition are given in [7, 8]. Sharp eigenvalue asymptotic behavior and a trace formula for this type operator without operator coefficient are obtained in [20, 21]. Fourth-order differential operators with squared spectral parameter in bound- ary conditions were considered in [12]. In this article we provide the simplicity and interlacing properties of the eigenvalues and the oscillation properties of the corresponding eigenfunctions. The location of the spectrum and asymptotic behavior of the eigenvalues for fourth-order differential operators with spectral parameter dependent boundary conditions of different type were investigated in [16, 17, 18, 27]. The main goal of this article is to determine the asymptotic behavior of the eigenvalues of the spectral problem (1.1). This property of problem (1.1) was studied in [14]. But in that work only the main term of the asymptotics was established. In our manuscript we obtain sharp asymptotics of the eigenvalues for an absolutely continuous coefficient and for a smoother one. Moreover, we compare this asymptotic formula with the spectral asymptotics of the same operator with similar boundary conditions, but without a spectral parameter in the boundary condition and describe the non-standard effects that appear here. Problem (1.1) can be reduced to the spectral problem for some linear operator H. In the Hilbert space H = L2(0, 1)⊕ C with inner product (y,v) = ∫ 1 0 y(x)v(x) dx+ σ−1ab, y = ( y(x) a ) ∈ H, v = ( v(x) b ) ∈ H, we define the operator Hy = H ( y(x) cTy(1)− ay(1) ) = ( (Ty(x))′ by(1)− dTy(1) ) on the domain Dom(H) = { y = ( y(x) cTy(1)− ay(1) ) ∈ H, y ∈W 4,1(0, 1), (Ty)′ ∈ L2(0, 1), y(0) = y′(0) = y′′(1) = 0 } , where W 4,1(0, 1) is the standard Sobolev space. This domain is dense in H [25, Lemma 1.5]. The operator is well-defined in H. Therefore, we conclude that prob- lem (1.1) is equivalent to the spectral problem Hy = λy, y ∈ Dom(H), EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 3 i.e. the eigenvalues λn of the problem (1.1) and those of the operatorH coincide (see also [25, Lemma 1.4]). Moreover, the problem (1.1) is regular in the sense of [25] and, in particular, it has a discrete spectrum. Aliev and Kerimov [14, Lemma 4.2] proved that the eigenvalues λn of (1.1) form a countable set without finite limit points. Furthermore, all eigenvalues are simple. We consider the differential equation y(4) − (py′)′ = λy, λ ∈ C. (1.2) Now we define the fundamental solutions φj , j = 1, 2, 3, 4, of this equation. These solutions satisfy the conditions φ (k−1) j (0, λ) = δjk, k = 1, 2, 3, (φ′′′ j − pφ′ j)(0, λ) = δj4, where δjk is the Kronecker symbol. Note that φj(x, ·), x ∈ [0, 1], is an entire function. Moreover, the spectrum is σ(H) = {λ ∈ C : D(λ) = 0}, where D is an entire function defined by D(λ) = det ( φ′′ 3(1, λ) φ′′ 4(1, λ) (Gφ3)(1, λ) (Gφ4)(1, λ) ) , λ ∈ C. (1.3) Here and below we denote by G the function (Gφj)(1, λ) = (aλ+ b)φj(1, z)− (cλ+ d)Tφj(1, z), j = 1, 2, 3, 4. (1.4) Now we consider the case p = 0. The function D = D0 has the form D0(λ) = −cz 4 + d 2 (1 + cos z cos iz)− az4 + b 2z3 ( sin z cos iz + i cos z sin iz ) , (1.5) where z = λ1/4, arg z ∈ ( − π 4 , π 4 ] , as arg λ ∈ (−π, π]. The eigenvalues of (1.1) are the zeros of D0 and have the following asymptotic behavior (see [14, Theorem 6.1]): λ0n = {( − 3π/4 + πn )4 +O(n2), if c = 0,( − 3π/2 + πn )4 +O(n2), if c ̸= 0, n ∈ N. (1.6) Recall that the eigenvalues λ0n are simple [14, Theorem 2.1]. We denote the eigenvalues of the operator H by λn, n ∈ N. We introduce the coefficients f0 = ∫ 1 0 f(x) dx, f̂cn(ε) = ∫ 1 0 f(x) cosπ(2n− ε)x dx, f̂sn(ε) = ∫ 1 0 f(x) sinπ(2n− ε)x dx, for n ∈ Z, where ε is a positive constant. Our main result is devoted to the high energy asymptotic behavior of λn. Theorem 1.1. (i) Suppose that p ∈ W 1,1(0, 1). The eigenvalues λn are real, simple, and have the following asymptotic behaviour for c ̸= 0: λn = ( − 3π 2 + πn )4 + ( − 3π 2 + πn )2 p0 +O(n), (1.7) as n→ +∞. If c = 0, then the asymptotic behavior of the eigenvalue has the form λn = ( − 3π 4 + πn )4 + ( − 3π 4 + πn )2( p0 − 2d a ) +O(n), (1.8) 4 D. M. POLYAKOV EJDE-2024/62 as n→ +∞. (ii) Suppose that p ∈W 3,1(0, 1) and p(0) = p(1). If c ̸= 0, then λn = ( − 3π 2 + πn )4 + ( − 3π 2 + πn )2 p0 + 3p(1)π(2n− 3) + p20 − ∥p∥2 8 − 2p′(1) + 4a c − ρ1,n + p̂′′′cn(3) 4π(2n− 3) +O(n−2), (1.9) as n→ +∞, where ρ1,n = ∫ 1 0 e−π(2n−3)s ( p′′′(1− s)− p′′′(s) ) ds − 8 ∫ 1 0 e−π(n−3/2)s ( p′′′(s) + p′′′(1− s) ) sin ( − 3π 2 + πn ) s ds. (1.10) If c = 0, then the eigenvalues λn have the asymptotic behavior λn = ( − 3π 4 + πn )4 + ( − 3π 4 + πn )2( p0 − 2d a ) − ( − 3π 4 + πn )( p(1) + d2 a2 ) + p20 − ∥p∥2 8 − p′(1) 2 − p0d 2a + d2 2a2 − d3 3a3 − 7dp(1) 2a + ρ2,n − p̂′′′cn(3/2) 2π(4n− 3) +O(n−2), (1.11) as n→ +∞, with ρ2,n = ∫ 1 0 e−π(2n−3/2)sp′′′(s) ds+8 ∫ 1 0 e−π(n−3/4)sp′′′(s) sinπ(n−3/4)s ds. (1.12) Remark 1.2. In this manuscript, we do not discuss the numbering of eigenval- ues in a circle of large radius. The asymptotic behavior of the eigenvalues from Theorem 1.1 will be true only starting from some sufficiently large n. To obtain the information about the number of zeros of operator H, we need to obtain the number of zeros of the function D0 in a circle of large radius and the estimates for the difference D −D0. However, it is not trivial for higher-order operators. Now we discuss the main features of the considered operator and the main ad- vantages of our primary results. Note that the formulas (1.7) and (1.8) improve well-known asymptotics (1.6) from [14, Theorem 6.1]. Namely, we establish the sec- ond and the third terms of the eigenvalue asymptotics. If the coefficient p is smooth, then we obtain a more detailed asymptotic behavior of the eigenvalues (1.9) and (1.11). We discuss a structure of these asymptotics. It contains the term ρ1,n or ρ2,n of the form (1.10) and (1.12), respectively. These terms correlate with the corresponding terms from well-known spectral asymptotics for an operator of the form (1.1) with boundary conditions y(0) = y(1) = y′(0) = y′(1) = 0 (see details in [22, Theorem 1]), i.e. with identical fixation at the left end. At the same time, in comparison with [22, Theorem 1], additional constant terms appear in (1.9) and (1.11). These terms are nontrivial, but they are typical specifically for problems with a spectral parameter in the boundary condition. This effect also previously appeared when studying the properties of fourth-order operators with a free term [20, 21]. It follows from (1.8) that in some problems this nontrivial effect is already observed for the case of non-smooth coefficient p. However, it does not appear in problems for a fourth-order operator without a spectral parameter in the boundary conditions (see [9, 22, 23], and the references therein). EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 5 Consider the shifted operator Ht(pt) defined by (1.1), where pt = p(· + t), t ∈ T = R/Z. We denote by µn(t) the eigenvalues of the operator Ht. The asymptotics (1.9) and (1.11) show that the series ∞∑ n=1 ( λn(t)− λn(0)− 3π(2n− 3) ( p(t)− p(0) ) + 2 ( p′(t)− p′(0) )) , for c ̸= 0, and ∞∑ n=1 ( λn(t)− λn(0) + ( − 3π 4 + πn )( p(t)− p(0) ) + p′(t)− p′(0) 2 + 7d ( p(t)− p(0) ) 2a ) , for c = 0, converges. Thus, the regularized trace formula is correctly defined for this class of operators. However, at the moment it is an open question: to obtain exact trace formulas for both cases. We only determine the asymptotics of the characteristic function D of the form (1.3) (see Lemma 4.1). To establish the main results we use the Birkhoff method [19, Ch. 2]. In this work, we apply a combination of the matrix version of this method for higher-order operators (see [10, 23]) and the semiclassical method from [11]. This combination allows us to determine the second and the third term in the asymptotic behavior of the eigenvalues, as well as to obtain a precisely controlled remainder term. The plan of this article is as follows. In Section 2 we study the properties of the characteristic function D and the fundamental matrix of equation (1.2). Moreover, in this section we provide the representation of this fundamental matrix. In Section 3 we obtain the asymptotical formulas for the eigenvalues λn in the case p ∈W 1,1(0, 1). The case p ∈W 3,1(0, 1) is considered in Section 4. 2. Properties of the fundamental solutions In this section we introduce other fundamental solutions ϕj , j = 1, 2, 3, 4. These solutions are different from the solutions φj , j = 1, 2, 3, 4, but the asymptotic be- havior of ϕj , j = 1, 2, 3, 4, can be well controlled. We transform the equation (1.2). Using this transformation, we can determine the representation of the fundamental matrix and hold asymptotic analysis of the fundamental matrix. Recall that z = λ1/4, z ∈ Z, λ ∈ C, where Z = { z ∈ C : arg z ∈ ( − π 4 , π 4 ]} , Z = { z ∈ C : arg z ∈ ( − π 4 , π 4 )} . If λ ∈ C+, then z ∈ Z+, where Z+ = { z ∈ C : arg z ∈ ( 0, π 4 )} . Define the numbers ω1 = −ω4 = i, ω2 = −ω3 = 1. Therefore, Re(iω1z) ⩽ Re(iω2z) ⩽ Re(iω3z) ⩽ Re(iω4z), z ∈ Z+. It is easy to see that equation (1.2) with p = 0 has the fundamental solutions ϕ0j (x, z) = eizxωj , j = 1, 2, 3, 4. (2.1) Consider the perturbed equation (1.2). Let r > 0 be large enough and let z ∈ Z+(r), where Z+(r) = {z ∈ Z+ : |z| > r}, r > 0. Then equation (1.2) has the fundamental solutions ϕj(x, z), j = 1, 2, 3, 4, x ∈ [0, 1], z ∈ Z+(r), satisfying the asymptotics ϕj(x, z) = ϕ0j (x, z)(1 +O(z−1)), ϕ′j(x, z) = (ϕ0j ) ′(x, z)(1 +O(z−1)), ϕ′′j (x, z) = (ϕ0j ) ′′(x, z)(1 +O(z−1)), ϕ′′′j (x, z) = (ϕ0j ) ′′′(x, z)(1 +O(z−1)), (2.2) 6 D. M. POLYAKOV EJDE-2024/62 as |z| → ∞, uniformly in x ∈ [0, 1] (see [19]). Now we define the fundamental matrix A(x, z), x ∈ [0, 1], z ∈ Z+(r), of equa- tion (1.2) by A =  ϕ1 ϕ2 ϕ3 ϕ4 ϕ′1 ϕ′2 ϕ′3 ϕ′4 ϕ′′1 ϕ′′2 ϕ′′3 ϕ′′4 ϕ′′′1 − pϕ′1 ϕ′′′2 − pϕ′2 ϕ′′′3 − pϕ′1 ϕ′′′4 − pϕ′4  . (2.3) This matrix-valued function satisfies the equation A′ = PA, where P =  0 1 0 0 0 0 1 0 0 p 0 1 λ 0 0 0  . (2.4) We rewrite the function D, given by (1.3), in terms of the fundamental solu- tions ϕj , where the matrix-valued function ϕ has the form ϕ(z) =  ϕ1(0, z) ϕ2(0, z) ϕ3(0, z) ϕ4(0, z) ϕ′1(0, z) ϕ′2(0, z) ϕ′3(0, z) ϕ′4(0, z) ϕ′′1(1, z) ϕ′′2(1, z) ϕ′′3(1, z) ϕ′′4(1, z) (Gϕ1)(1, z) (Gϕ2)(1, z) (Gϕ3)(1, z) (Gϕ4)(1, z)  , (2.5) where G has the form (1.4). Repeating the arguments from [10, Lemma 3.2], we derive that the determinant D can be represented in the form D(λ) = detϕ(z) detA(0, z) , λ = z4. (2.6) The function detϕ is analytic in Z+(r). Hence the function D is entire and the identity (2.6) can be extended analytically from Z+(r) onto the whole complex plane. Note that the function detϕ is not an entire function of the variable λ, but its asymptotics at high energy is well controlled. Consider the unperturbed case p = 0. Let z ∈ Z+. Therefore, the fundamental matrix A = A0 satisfies A0 =  ϕ01 ϕ02 ϕ03 ϕ04 (ϕ01) ′ (ϕ02) ′ (ϕ03) ′ (ϕ04) ′ (ϕ01) ′′ (ϕ02) ′′ (ϕ03) ′′ (ϕ04) ′′ (ϕ01) ′′′ (ϕ02) ′′′ (ϕ03) ′′′ (ϕ04) ′′′  = ΩY0, Y0 = diag(ϕ01, ϕ 0 2, ϕ 0 3, ϕ 0 4), where ϕ0j , j = 1, 2, 3, 4, are defined by (2.1) and Ω =  1 1 1 1 −z iz −iz z z2 −z2 −z2 z2 −z3 −iz3 iz3 z3  . (2.7) Using the identity ϕ01ϕ 0 2ϕ 0 3ϕ 0 4 = 1, we obtain detA0(0, z) = detΩ = −16iz6. (2.8) EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 7 The matrix-valued function ϕ = ϕ0 has the form ϕ0(z) =  ϕ01(0, z) ϕ02(0, z) ϕ03(0, z) ϕ04(0, z) (ϕ01) ′(0, z) (ϕ02) ′(0, z) (ϕ03) ′(0, z) (ϕ04) ′(0, z) (ϕ01) ′′(1, z) (ϕ02) ′′(1, z) (ϕ03) ′′(1, z) (ϕ04) ′′(1, z) (Gϕ01)(1, z) (Gϕ02)(1, z) (Gϕ03)(1, z) (Gϕ04)(1, z)  =  1 1 1 1 −z iz −iz z z2e−z −z2eiz −z2e−iz z2ez (Gϕ01)(1, z) (Gϕ02)(1, z) (Gϕ03)(1, z) (Gϕ04)(1, z)  . Then detϕ0(z) = 8iz6(cz4+d)(1+cos z cos iz)+8iz3(az4+ b) ( sin z cos iz+ i cos z sin iz ) . The identities (2.6) and (2.8) show that the entire function D0 in this case satisfies (1.5). It follows from formula (2.6) that to obtain the asymptotics of D we need to analyse the asymptotics of the function detϕ. Now we derive the formula for the fourth-order determinant detϕ. More precisely we express detϕ in terms of linear combinations for product of second-order determinants. Lemma 2.1. Suppose that p ∈W 1,1(0, 1), and |z| → ∞. Then detϕ(z) = { e2Re zO(z10), for c ̸= 0, e2Re zO(z7), for c = 0, z ∈ Z+. (2.9) Moreover, detϕ(z) = { γ1(z)γ2(z) + γ3(z)γ4(z) +O(z10), for c ̸= 0, γ1(z)γ2(z) + γ3(z)γ4(z) +O(z7), for c = 0, z ∈ Z+, (2.10) where γ1(z) = det ( ϕ1(0, z) ϕ2(0, z) ϕ′1(0, z) ϕ′2(0, z) ) , (2.11) γ2(z) = det ( ϕ′′3(1, z) ϕ′′4(1, z) (Gϕ3)(1, z) (Gϕ4)(1, z) ) , (2.12) γ3(z) = det ( ϕ3(0, z) ϕ1(0, z) ϕ′3(0, z) ϕ′1(0, z) ) , (2.13) γ4(z) = det ( ϕ′′2(1, z) ϕ′′4(1, z) (Gϕ2)(1, z) (Gϕ4)(1, z) ) , (2.14) and Gϕj, j = 1, 2, 3, 4, have the form (1.4). 8 D. M. POLYAKOV EJDE-2024/62 Proof. Let z ∈ Z+ and |z| → ∞. Then 0 ⩽ Im z ⩽ Re z. Consider the definition (2.5). Direct calculations imply detϕ(z) = det ( ϕ1(0, z) ϕ2(0, z) ϕ′1(0, z) ϕ′2(0, z) ) det ( ϕ′′3(1, z) ϕ′′4(1, z) (Gϕ3)(1, z) (Gϕ4)(1, z) ) + det ( ϕ3(0, z) ϕ1(0, z) ϕ′3(0, z) ϕ′1(0, z) ) det ( ϕ′′2(1, z) ϕ′′4(1, z) (Gϕ2)(1, z) (Gϕ4)(1, z) ) + det ( ϕ1(0, z) ϕ4(0, z) ϕ′1(0, z) ϕ′4(0, z) ) det ( ϕ′′2(1, z) ϕ′′3(1, z) (Gϕ2)(1, z) (Gϕ3)(1, z) ) + det ( ϕ2(0, z) ϕ3(0, z) ϕ′2(0, z) ϕ′3(0, z) ) det ( ϕ′′1(1, z) ϕ′′4(1, z) (Gϕ1)(1, z) (Gϕ4)(1, z) ) + det ( ϕ4(0, z) ϕ2(0, z) ϕ′4(0, z) ϕ′2(0, z) ) det ( ϕ′′1(1, z) ϕ′′3(1, z) (Gϕ1)(1, z) (Gϕ3)(1, z) ) + det ( ϕ3(0, z) ϕ4(0, z) ϕ′3(0, z) ϕ′4(0, z) ) det ( ϕ′′1(1, z) ϕ′′2(1, z) (Gϕ1)(1, z) (Gϕ2)(1, z) ) . (2.15) Let c ̸= 0. Here and below we denote 1 + O(z−1) by [1]−1. Consider the term (Gϕ1)(1, z). Using (1.4) and (2.2), we have (Gϕ1)(1, z) = (az4 + b)ϕ1(1, z)− (cz4 + d) ( ϕ′′′1 (1, z)− p(1)ϕ′1(1, z) ) = (az4 + b)e−z[1]−1 − (cz4 + d) ( − z3e−z[1]−1 + p(1)ze−z[1]−1 ) = e−z ( (az4 + b)[1]−1 + (cz4 + d)z3[1]−1 ) = cz7e−z[1]−1. (2.16) Similar arguments give (Gϕ2)(1, z) = ciz7eiz[1]−1, (Gϕ3)(1, z) = −ciz7e−iz[1]−1, (Gϕ4)(1, z) = −cz7ez[1]−1. (2.17) Then the estimates 0 ⩽ Im z ⩽ Re z and the asymptotics (2.2), (2.16), and (2.17) imply det ( ϕ3(0, z) ϕ4(0, z) ϕ′3(0, z) ϕ′4(0, z) ) det ( ϕ′′1(1, z) ϕ′′2(1, z) (Gϕ1)(1, z) (Gϕ2)(1, z) ) = det ( [1]−1 [1]−1 −iz[1]−1 z[1]−1 ) det ( z2e−z[1]−1 −z2eiz[1]−1 cz7e−z[1]−1 ciz7eiz[1]−1 ) = e−z+izO(z10) = e−Re z−Im zO(z10) = e−2Re zO(z10) and det ( ϕ4(0, z) ϕ2(0, z) ϕ′4(0, z) ϕ′2(0, z) ) det ( ϕ′′1(1, z) ϕ′′3(1, z) (Gϕ1)(1, z) (Gϕ3)(1, z) ) = det ( [1]−1 [1]−1 z[1]−1 iz[1]−1 ) det ( z2e−z[1]−1 −z2e−iz[1]−1 cz7e−z[1]−1 −ciz7e−iz[1]−1 ) = e−z−izO(z10) = e−Re z+Im zO(z10) = O(z10). Moreover, det ( ϕ2(0, z) ϕ3(0, z) ϕ′2(0, z) ϕ′3(0, z) ) det ( ϕ′′1(1, z) ϕ′′4(1, z) (Gϕ1)(1, z) (Gϕ4)(1, z) ) EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 9 = det ( [1]−1 [1]−1 iz[1]−1 −iz[1]−1 ) det ( z2e−z[1]−1 z2ez[1]−1 cz7e−z[1]−1 −cz7ez[1]−1 ) = O(z10) and det ( ϕ1(0, z) ϕ4(0, z) ϕ′1(0, z) ϕ′4(0, z) ) det ( ϕ′′2(1, z) ϕ′′3(1, z) (Gϕ2)(1, z) (Gϕ3)(1, z) ) = det ( [1]−1 [1]−1 −z[1]−1 z[1]−1 ) det ( −z2eiz[1]−1 −z2e−iz[1]−1 ciz7eiz[1]−1 −ciz7e−iz[1]−1 ) = O(z10). Substituting these asymptotics into (2.15), we get the first formula in (2.10). Using (2.2), (2.16), (2.17), and the estimates 0 ⩽ Im z ⩽ Re z again, we obtain γ1(z)γ2(z) + γ3(z)γ4(z) = det ( [1]−1 [1]−1 −z[1]−1 iz[1]−1 ) det ( −z2e−iz[1]−1 z2ez[1]−1 −ciz7e−iz[1]−1 −cz7ez[1]−1 ) + det ( [1]−1 [1]−1 −iz[1]−1 −z[1]−1 ) det ( −z2eiz[1]−1 z2ez[1]−1 ciz7eiz[1]−1 −cz7ez[1]−1 ) = ez−izO(z10) + ez+izO(z10) = ez−izO(z10) = eRe z+Im zO(z10) = e2Re zO(z10). This yields the first asymptotics in (2.9). Let c = 0. We first compute the elements (Gϕj)(1, z), j = 1, 2, 3, 4. The formulas (1.4) and (2.2) give (Gϕ1)(1, z) = (az4 + b)e−z[1]−1 − d ( − z3e−z[1]−1 + p(1)ze−z[1]−1 ) = az4e−z[1]−1. Using similar arguments, we get (Gϕ2)(1, z) = az4eiz[1]−1, (Gϕ3)(1, z) = az4e−iz[1]−1, (Gϕ4)(1, z) = az4ez[1]−1. Repeating the above procedure as in the case c ̸= 0, we obtain the second asymp- totics in (2.9) and (2.10). □ Now we investigate the properties of the fundamental matrix A of equation (2.4). Since the contribution of bounded and decreasing elements of the matrix A completely disappears against the background of the contribution of increasing ones, the asymptotic analysis of the matrix A is a rather difficult problem. It is clearly seen that the matrix P in equation (2.4) contains growing elements. We transform this equation into a first-order differential equation in such a way that the increasing terms are separated into a separate term, and the right side of the equation is decreasing as z−1. Further, we apply the Birkhoff method [19, Chapter 2] to this first-order equation. In this manuscript we use the matrix version of this scheme from [10, § 2] (see also [23]). Applying this modification, we reduce this equation to the equivalent Fredholm integral equation with a “small kernel”. Using the method of simple iterations, we find a solution of the Fredholm integral equation. This solution allows us to obtain the representation (2.35) of the fundamental matrix A. The main feature of (2.35) is that all growing elements are separated into a separate diagonal matrix. This makes it possible to control their contribution to the asymptotic behavior of the eigenvalues. If the coefficient p has the necessary additional smoothness, then we can transform (2.4) into a first-order differential equation of the same type as before, but with a better rate of decrease of the right- hand side. In order to do this we use semiclassical method [11, Chapter V.1.3]. 10 D. M. POLYAKOV EJDE-2024/62 Then we again apply the Birkhoff method and obtain a representation (2.37) of the fundamental matrix A. Now we transform equation (2.4). Introduce the matrix-valued function Y1(x, z) by A(x, z) = Ω(z)Y1(x, z), (x, z) ∈ [0, 1]×Z+(r), (2.18) and the matrix T = diag(ω1, ω2, ω3, ω4) = diag(i, 1,−1,−i). Here Ω is defined by (2.7). Lemma 2.2. Suppose that p ∈ W 1,1(0, 1) and z ∈ Z+(r), where r > 0 is large enough. Then the matrix-valued function Y1, defined by (2.18), satisfies the equation Y ′ 1 − izT1Y1 = 1 z Φ1Y1, (2.19) where T1 and Φ1 are the matrix-valued functions of the form T1 = T − p 4z2 T 3, Φ1 = F1 = p 4 (P + iT 3), (2.20) with P =  −1 i −i 1 1 −i i −1 1 −i i −1 −1 i −i 1  , Q =  −1 −1 −1 −1 i i i i −i −i −i −i 1 1 1 1  . The proof of the above lemma can be found in [23, Lemma 1]. Now we consider the case of the smooth coefficient. Let p ∈ W 3,1(0, 1). Using the method [11, Ch. V.1.3], we transform equation (2.19) so that the matrix coefficient on the right side decreases as z−4. We introduce a new unknown matrix-valued function Y4(x, z) by Y1(x, z) = ( I4 + W(x, z) z2 ) Y4(x, z), (x, z) ∈ [0, 1]×Z+(r), (2.21) where Y1 is the solution of equation (2.19) and W has the form W = pW1 + p′ z W2 + p′′ 32z2 Q1 − p2 64z2 Q2. (2.22) We choose the matrices W1, W2, Q1, and Q2 in such a way that the coefficient on the right side of equation (2.25) decreases as z−4. Thus, W1 = 1 8  0 1 + i 1− i 1 −1 + i 0 −1 −1− i −1− i −1 0 −1 + i 1 1− i 1 + i 0  , W2 = 1 16  0 −2 −2 −1 2i 0 i 2i −2i −i 0 −2i 1 2 2 0  , (2.23) and Q1 =  0 2− 2i 2 + 2i 1 2 + 2i 0 1 2− 2i 2− 2i 1 0 2 + 2i 1 2 + 2i 2− 2i 0  , Q2 =  −2 1− i 1 + i 0 1 + i 2i 0 1− i 1− i 0 −2i 1 + i 0 1 + i 1− i 2  . (2.24) We have the following result. EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 11 Lemma 2.3. Suppose that p ∈ W 3,1(0, 1) and z ∈ Z+(r), where r > 0 is large enough. Then the matrix-valued function Y4, given by (2.21), satisfies the equation Y ′ 4 − izT4Y4 = 1 z4 Φ4Y4, (2.25) where T4 = T − p 4z2 T 3 + p2 32z4 T + ipp′ 64z5 (−3I4 + 4iT ), (2.26) Φ4 = F4 +O(z−1), F4 = −p ′′′ 32 Q1 + pp′ 64 (Q3 + 3I4 − 4iT ), (2.27) as |z| → ∞, uniformly in x ∈ [0, 1]. The matrix Q1 has the form (2.24) and Q3 is a matrix. The proof of this lemma can be found in [23, Lemma 4]. Remark 2.4. The matrix Q3 has a specific form. It is not important for further calculations, since this term is O(z−1). To transform the differential equations (2.19) and (2.25) we use the Birkhoff method. Moreover, we obtain the representation of the fundamental matrix A. The formula (2.3) shows that the matrix A contains exponentially increasing, bounded, and decreasing entries at high energy. Using the Birkhoff method, we extract exponentially increasing elements into a separate diagonal matrix. It is known (see [10, Theorem 4.5] and [23, Lemma 5]) that the matrix-valued function Yσ(x, z) = X (x, z)eiz ∫ x 0 Tσ(s,z) ds, σ = 1, 4, satisfies the differential equations (2.19) and (2.25) if and only if X is a solution of the integral equations X = I4 + 1 zσ KX , σ = 1, 4, (2.28) where T1 and T4 have the form (2.20) and (2.26), respectively, and K is an integral operator in the space C[0, 1] of 4× 4 matrix-valued functions defined by (KX )lj(x, z) = ∫ 1 0 Klj(x, s, z)(ΦσX )lj(s, z) ds, l, j = 1, 2, 3, 4, σ = 1, 4, for all X ∈ C[0, 1], (x, z) ∈ [0, 1]×Z+(r), and Klj(x, s, z) = { eiz ∫ x s (Θl(u,z)−Θj(u,z)) du χ(x− s), l < j, −eiz ∫ x s (Θl(u,z)−Θj(u,z)) du χ(s− x), l ⩾ j, χ(s) = { 1, s ⩾ 0, 0, s < 0. (2.29) Here Φ1 and Φ4 satisfy (2.20) and (2.27), respectively. Note that the integral operator K is a contraction for z ∈ Z+(r), where r > 0 is large enough. The equations (2.28) have a unique solution X (·, z) ∈ C[0, 1]. Moreover, each matrix- valued function X (x, ·), x ∈ [0, 1], is analytic on Z+(r) and satisfies the asymptotics X (x, z) = I4 + 1 zσ (KI4)(x, z) +O(z−2σ), σ = 1, 4, (2.30) 12 D. M. POLYAKOV EJDE-2024/62 as |z| → ∞, z ∈ Z+, uniformly in x ∈ [0, 1], where (KI4)lj(x, z) = ∫ 1 0 Klj(x, s, z)Φσ,lj(s, z) ds, l, j = 1, 2, 3, 4, σ = 1, 4. Relations (2.20) and (2.27) show that in equations (2.19) and (2.25) the function KI4 has the asymptotics KI4 = Bσ + O(z−1), σ = 1, 4, where the matrix-valued functions Bσ(x, z), (x, z) ∈ [0, 1]×Z+, σ = 1, 4, are defined by Bσ,lj(x, z) = ∫ 1 0 Klj(x, s, z)Fσ,lj(s) ds. Moreover, the formula (2.29) yields Bσ,jj(x, z) = 0, j = 1, 2, 3, 4, (2.31) Bσ,lj(x, z) = − ∫ 1 x e−iz(s−x)(ωl−ωj)Fσ,lj(s) ds, 1 ⩽ j < l ⩽ 4, (2.32) Bσ,lj(x, z) = ∫ x 0 eiz(x−s)(ωl−ωj)Fσ,lj(s) ds, 1 ⩽ l < j ⩽ 4. (2.33) Recall that the functions Fσ, σ = 1, 4, are defined by (2.20) and (2.27). Therefore, the asymptotics (2.30) gives X (x, z) = I4 + Bσ(x, z) zσ +O(z−σ−1), σ = 1, 4. Remark 2.5. Note that the matrix-valued functions Bσ(x, ·), x ∈ [0, 1], σ = 1, 4, are analytic and bounded in Z+. Using these results, we represent the matrix A of equation (1.2) as a prod- uct of the bounded matrix X , the simple matrix Ω, and the diagonal matrix exp{iz ∫ x 0 Tσ(s, z) ds}. Note that all exponentially increasing terms contain into this diagonal matrix. To obtain sharp eigenvalue asymptotics we provide two fac- torizations of the fundamental matrix. It is more convenient for obtaining sharp eigenvalue asymptotics. For σ = 1, 4, l, j = 1, 2, 3, 4, we introduce the functions ζσ,lj(x, z) = Bσ,lj(x, z) zσ + Wlj(x, z) z2 . (2.34) Note that these functions are analytic and bounded in Z+. If p ∈ W 1,1(0, 1), then σ = 1 and W = 0 and if p ∈W 3,1(0, 1), then σ = 4 and W is defined by (2.22). Now we formulate the following lemma about factorization of the fundamental matrix A. Recall that the matrix Ω is given by (2.7), the diagonal 4 × 4 matrix- valued functions T1 and T4 have the form (2.20) and (2.26), respectively, and the matrix-valued functions Bσ, σ = 1, 4, are defined by (2.31)–(2.33). Lemma 2.6. Suppose that p ∈W 1,1(0, 1), x ∈ [0, 1], and z ∈ Z+(r) for some r > 0 large enough. Then (i) The fundamental matrix A of equation (1.2) satisfies the asymptotics A(x, z) = Ω(z) ( I4 + B1(x, z) z +O(z−2) ) eiz ∫ x 0 T1(s,z) ds, (2.35) uniformly in x ∈ [0, 1]. Moreover, the function detA(0, z) has the asymptotics detA(0, z) = −16iz6(1 +O(z−1)), (2.36) EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 13 as |z| → ∞. (ii) Suppose that p ∈W 3,1(0, 1). Then A(x, z) = Ω(z) ( I4 + W(x, z) z2 )( I4 + B4(x, z) z4 +O(z−5) ) eiz ∫ x 0 T4(s,z) ds. (2.37) (iii) The fundamental solutions ϕj, j = 1, 2, 3, 4, given by (2.2), have the asymp- totics ϕ1 ϕ2 ϕ3 ϕ4 ϕ′1 ϕ′2 ϕ′3 ϕ′4 ϕ′′1 ϕ′′2 ϕ′′3 ϕ′′4 ϕ′′′1 − pϕ′1 ϕ′′′2 − pϕ′2 ϕ′′′3 − pϕ′1 ϕ′′′4 − pϕ′4  (2.38) = Ω ( 1 + [ζσ,11]−σ−1 ) cσ,1 [ζσ,12]−σ−1cσ,2 [ζσ,13]−σ−1cσ,3 [ζσ,14]−σ−1cσ,4 [ζσ,21]−σ−1cσ,1 ( 1 + [ζσ,22]−σ−1 ) cσ,2 [ζσ,23]−σ−1cσ,3 [ζσ,24]−σ−1cσ,4 [ζσ,31]−σ−1cσ,1 [ζσ,32]−σ−1cσ,2 ( 1 + [ζσ,33]−σ−1 ) cσ,3 [ζσ,34]−σ−1cσ,4 [ζσ,41]−σ−1cσ,1 [ζσ,42]−σ−1cσ,2 [ζσ,43]−σ−1cσ,3 ( 1 + [ζσ,44]−σ−1 ) cσ,4  , as |z| → ∞, z ∈ Z+, uniformly in x ∈ [0, 1], where cσ,j(x, z) = eiz ∫ x 0 Tσ,j(s,z) ds, [ζσ,lj ]−σ−1 = ζσ,lj +O(z−σ−1), for l, j = 1, 2, 3, 4, σ = 1, 4, and ζσ,lj, σ = 1, 4, l, j = 1, 2, 3, 4, are given by (2.34). The proof of this lemma is similar to [23, Lemma 6], we omit it. 3. Eigenvalue asymptotics in the case p ∈W 1,1(0, 1) 3.1. Asymptotics of the functions γj, j = 1, 2, 3, 4. The main goal of this section is to obtain eigenvalue asymptotics of the operator H at high energy. But first we deduce more convenient form for the functions γj , j = 1, 2, 3, 4, defined by (2.12)–(2.14). The formulas (2.20) and (2.26) imply ασ(z) = ∫ 1 0 Tσ,2(s, z) ds = − ∫ 1 0 Tσ,3(s, z) ds, βσ(z) = ∫ 1 0 Tσ,1(s, z) ds = − ∫ 1 0 Tσ,4(s, z) ds, z ∈ Z+, (3.1) for σ = 1, 4, where Tσ,j are entries of the matrices Tσ = (Tσ,j) 4 j=1. Therefore, the functions ασ and βσ have the form α1(z) = 1− p0 4z2 , β1(z) = i+ ip0 4z2 , α4(z) = 1 + ∥p∥2 32z4 − p0 4z2 , β4(z) = i+ i∥p∥2 32z4 + ip0 4z2 . (3.2) In the following lemma, one assumes that the functions ζσ,lj , σ = 1, 4, l, j = 1, 2, 3, 4, satisfy (2.34). Now we introduce the functions κσ,1(z) = ( ζσ,22 − iζσ,32 + (1− i)ζσ,42 + ζσ,11 + (1 + i)ζσ,31 + iζσ,41 ) (0, z) + (1− i) 2 ( Mσ,1 + M̃σ,1 ) (0, z), (3.3) κσ,2(z) = ( ζσ,11 + (1− i)ζσ,21 − iζσ,41 + iζσ,23 + ζσ,33 + (1 + i)ζσ,43 ) (0, z) + (1 + i) 2 ( Mσ,2 + M̃σ,2 ) (0, z), (3.4) 14 D. M. POLYAKOV EJDE-2024/62 κσ,3(z) = ( ζσ,14 + ζσ,44 + ζσ,23 + ζσ,33 + (ζσ,23 + ζσ,33)(ζσ,14 + ζσ,44) − (ζσ,13 + ζσ,43)(ζσ,24 + ζσ,34) ) (1, z), (3.5) κσ,6(z) = ( ζσ,22 + ζσ,32 + ζσ,14 + ζσ,44 + (ξσ,22 + ξσ,32)(ξσ,14 + ξσ,44) − (ξσ,12 + ξσ,42)(ξσ,24 + ξσ,34) ) (1, z), (3.6) κσ,4(z) = ψ1(z) + 1− i 2 ( Mσ,3 + M̃σ,3 ) (1, z), κσ,5(z) = ψ1(z) +O(z−4), (3.7) κσ,7(z) = ψ2(z) + 1 + i 2 ( Mσ,5 + M̃σ,5 ) (1, z), κσ,8(z) = ψ2(z) +O(z−4), (3.8) where ψ1(z) = ( iζσ,14 + (−1− i)ζσ,24 + ζσ,44 + (−1 + i)ζσ,13 − iζσ,23 + ζσ,33 ) (1, z), ψ2(z) = ( (−1− i)ζσ,12 + ζσ,22 + iζσ,32 − iζσ,14 + (−1 + i)ξσ,34 + ξσ,44 ) (1, z), and Mσ,1(0, z) = (−ζσ,12 + iζσ,22 − iζσ,32 + ζσ,42)(ζσ,11 + ζσ,21 + ζσ,31 + ζσ,41)(0, z), M̃σ,1(0, z) = (ζσ,12 + ζσ,22 + ζσ,32 + ζσ,42)(ζσ,11 − iζσ,21 + iζσ,31 − ζσ,41)(0, z), (3.9) Mσ,2(0, z) = (ζσ,11 − iζσ,21 + iζσ,31 − ζσ,41)(ζσ,13 + ζσ,23 + ζσ,33 + ζσ,43)(0, z), M̃σ,2(0, z) = (ζσ,11 + ζσ,21 + ζσ,31 + ζσ,41)(−ζσ,13 + iζσ,23 − iζσ,33 + ζσ,43)(0, z), and Mσ,3(1, z) = (−ζσ,13 − iζσ,23 + iζσ,33 + ζσ,43)(ζσ,14 − ζσ,24 − ζσ,34 + ζσ,44)(1, z), M̃σ,3(1, z) = (−ζσ,13 + ζσ,23 + ζσ,33 − ζσ,43)(−ζσ,14 − iζσ,24 + iζσ,34 + ζσ,44)(1, z), (3.10) Mσ,4(1, z) = (−iζσ,14 + iζσ,24 + iζσ,34 − iζσ,44)(−iζσ,12 + ζσ,22 − ζσ,32 + iζσ,42)(1, z), M̃σ,4(1, z) = (−ζσ,12 + ζσ,22 + ζσ,32 − ζσ,42)(−ζσ,14 − iζσ,24 + iζσ,34 + ζσ,44)(1, z). Now we formulate the main result of this subsection. Lemma 3.1. Suppose that p ∈W 1,1(0, 1), |z| → ∞. Then the functions γ1 and γ3 have the form γ1(z) = (1 + i)z ( 1 + κσ,1(z) +O(z−σ−1) ) , γ3(z) = (−1 + i)z ( 1 + κσ,2(z) +O(z−σ−1) ) , (3.11) for z ∈ Z+, where κσ,1, κσ,2, σ = 1, 4, satisfy (3.3) and (3.4). If c ̸= 0, then the functions γ2 and γ4 are defined by γ2(z) = z9e−iασz−iβσz ( c(1 + i) ( 1 + κσ,4(z) ) + c(−1 + i)p(1) z2 ( 1 + κσ,5(z) ) − 2a z3 + d(1 + i) z4 +O(z−σ−1) ) , (3.12) γ4(z) = z9eiασz−iβσz ( c(1− i) ( 1 + κσ,7(z) ) + c(−1− i)p(1) z2 ( 1 + κσ,8(z) ) − 2a z3 + d(1− i) z4 +O(z−σ−1) ) , (3.13) EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 15 where κσ,j, j = 3, 4, 5, 6, 7, 8, are given by (3.5)–(3.8). If c = 0, then the functions γ2 and γ4 satisfy γ2(z) = z6e−iασz−iβσz ( − 2a ( 1 + κσ,3(z) ) + d(1 + i) z ( 1 + κσ,4(z) ) + d(−1 + i)p(1) z3 − 2b z4 +O(z−σ−1) ) , (3.14) γ4(z) = z6eiασz−iβσz ( − 2a ( 1 + κσ,6(z) ) + d(1− i) z ( 1 + κσ,7(z) ) + d(−1− i)p(1) z3 − 2b z4 +O(z−σ−1) ) . (3.15) Proof. Let z ∈ Z+, |z| → ∞. Substitute (2.38) into the first formula from (2.12). Then γ1(z) = iz ( 1 + (ζσ,11 + ζσ,21 + ζσ,31 + ζσ,41)(0, z) +O(z−σ−1) ) × ( 1 + (iζσ,12 + ζσ,22 − ζσ,32 − iζσ,42)(0, z) +O(z−σ−1) ) + z ( 1 + (ζσ,12 + ζσ,22 + ζσ,32 + ζσ,42)(0, z) +O(z−σ−1) ) × ( 1 + (ζσ,11 − iζσ,21 + iζσ,31 − ζσ,41)(0, z) +O(z−σ−1) ) . Now the first equation in (3.11) follows immediately. Consider the first definition from (2.14). Arguments similar to this provide the second equation in (3.11). Now we prove the first formula from (3.12). We define the functions ξ1(z) = ϕ′′3(1, z)ϕ4(1, z)− ϕ3(1, z)ϕ ′′ 4(1, z), ξ2(z) = ϕ′′′3 (1, z)ϕ′′4(1, z)− ϕ′′3(1, z)ϕ ′′′ 4 (1, z), ξ3(z) = ϕ′′3(1, z)ϕ ′ 4(1, z)− ϕ′3(1, z)ϕ ′′ 4(1, z). Equalities (1.4) and (2.12) imply γ2(z) = ϕ′′3(1, z) ( (az4 + b)ϕ4(1, z)− (cz4 + d) ( ϕ′′′4 (1, z)− p(1)ϕ′4(1, z) )) − ϕ′′4(1, z) ( (az4 + b)ϕ3(1, z)− (cz4 + d) ( ϕ′′′3 (1, z)− p(1)ϕ′3(1, z) )) = z4 ( aξ1(z) + cξ2(z) + cp(1)ξ3(z) + b z4 ξ1(z) + d z4 ξ2(z) + dp(1) z4 ξ3(z) ) . (3.16) 16 D. M. POLYAKOV EJDE-2024/62 We consider all the terms in the last equality separately. Using again (2.38), we obtain ξ1(z) = −z2e−iασz−iβσz ( 1 + (−ζσ,13 + ζσ,23 + ζσ,33 − ζσ,43)(1, z) +O(z−σ−1) ) × ( 1 + (ζσ,14 + ζσ,24 + ζσ,34 + ζσ,44)(1, z) +O(z−σ−1) ) − z2e−iασz−iβσz ( 1 + (ζσ,13 + ζσ,23 + ζσ,33 + ζσ,43)(1, z) +O(z−σ−1) ) × ( 1 + (ζσ,14 − ζσ,24 − ζσ,34 + ζσ,44)(1, z) +O(z−σ−1) ) = −2z2e−iασz−iβσz ( 1 + κσ,3(z) +O(z−σ−1) ) , (3.17) where κσ,3 has the form (3.5). Similarly, ξ2(z) = iz5e−iασz−iβσz ( 1 + (iζσ,13 − ζσ,23 + ζσ,33 − iζσ,43)(1, z) +O(z−σ−1) ) × ( 1 + (ζσ,14 − ζσ,24 − ζσ,34 + ζσ,44)(1, z) +O(z−σ−1) ) + z5e−iασz−iβσz ( 1 + (−ζσ,13 + ζσ,23 + ζσ,33 − ζσ,43)(1, z) +O(z−σ−1) ) × ( 1 + (−ζσ,14 − iζσ,24 + iζσ,34 + ζσ,44)(1, z) +O(z−σ−1) ) = (1 + i)z5e−iασz−iβσz ( 1 + κσ,4(z) +O(z−σ−1) ) , (3.18) and ξ3(z) = −z3e−iασz−iβσz ( 1 + (−ζσ,13 + ζσ,23 + ζσ,33 − ζσ,43)(1, z) +O(z−σ−1) ) × ( 1 + (−ζσ,14 + iζσ,24 − iζσ,34 + ζσ,44)(1, z) +O(z−σ−1) ) + iz3e−iασz−iβσz ( 1 + (−iζσ,13 − ζσ,23 + ζσ,33 + iζσ,43)(1, z) +O(z−σ−1) ) × ( 1 + (ζσ,14 − ζσ,24 − ζσ,34 + ζσ,44)(1, z) +O(z−σ−1) ) = (−1 + i)z3e−iασz−iβσz ( 1 + κσ,5(z) +O(z−σ−1) ) , (3.19) where κσ,4 and κσ,5 satisfy (3.7). We substitute (3.17)–(3.19) into (3.16) and take out the factor z5e−iασz−iβσz. Then γ2(z) = z9e−iασz−iβσz ( c(1 + i) ( 1 + κσ,4(z) ) + c(−1 + i)p(1) z2 ( 1 + κσ,5(z) ) − 2a z3 ( 1 + κσ,3(z) ) + d(1 + i) z4 +O(z−σ−1) ) . The identity (2.34) implies that ζ1,lj(·, z) = O(z−1) and ζ4,lj(·, z) = O(z−2). This yields the formula (3.12) immediately. Arguments similar to this provide the asym- totics (3.13). EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 17 It remains only to prove formulas (3.14) and (3.15). If c = 0, then we rewrite (3.16) in the form γ2(z) = z4 ( aξ1(z) + b z4 ξ1(z) + d z4 ξ2(z) + dp(1) z4 ξ3(z) ) . Substituting (3.17)–(3.19) into this definition and using the relations ζ1,lj(·, z) = O(z−1) and ζ4,lj(·, z) = O(z−2), we obtain (3.14). Arguments similar to this pro- vide the equation (3.15). □ 3.2. Sharp eigenvalue asymptotics. Now we determine the eigenvalue asymp- totics for the case p ∈ W 1,1(0, 1). This corresponds to the case σ = 1. Recall that the eigenvalues λn of the operator H are zeros of the entire function D given by (1.3). The definition (2.6) and the asymptotics (2.36) show that the large eigenval- ues are zeros of the function detϕ. Therefore, using the formulas (2.10), we obtain the eigenvalue asymptotics of the operator H. Lemma 3.2. Let p ∈ W 1,1(0, 1). Then the eigenvalues λn satisfy the asymptotics (1.7) in the case c ̸= 0 and the asymptotics (1.8) in the case c = 0. Proof. Let c ̸= 0 and λ = z4 = λn, n → +∞. It follows from [14, Theorem 6.1] that z = −3π/2 + πn+ δn, δn = O(n−1). The relations (3.11)–(3.13) give γ1(z) = (1 + i)z ( 1 + κ1,1(z) +O(n−2) ) , γ3(z) = (−1 + i)z ( 1 + κ1,2(z) +O(n−2) ) , γ2(z) = c(1 + i)z9e−iα1z−iβ1z ( 1 + κ1,4(z) +O(n−2) ) , γ4(z) = c(1− i)z9eiα1z−iβ1z ( 1 + κ1,7(z) +O(n−2) ) , where κ1,j , j = 1, 2, 4, 7, have the form (3.3), (3.4), (3.7), and (3.8). Substituting these asymptotics into (2.10), we obtain detϕ(z) = 2icz10e−iβ1z ( e−iα1z ( 1 + κ1,1(z) + κ1,4(z) + κ1,1(z)κ1,4(z) ) + eiα1z ( 1 + κ1,2(z) + κ1,7(z) + κ1,2(z)κ1,7(z) ) +O(n−2) ) , (3.20) where α1 and β1 satisfy (3.2). The identity z = −3π/2+ πn+ δn and the formulas (3.2) give e±iα1z = e±iz∓ip0/(4z) = ±i(−1)ne±iδn∓ip0/(4z) = ±i(−1)n ( 1± iδn ∓ ip0 2π(2n− 3) +O(n−2) ) . Moreover, the identity (2.34) yields ζ1,kj(x, z) = O(n−1), ζ1,kj(x, z)ζ1,ls(y, z) = O(n−2), (3.21) for k, j, l, s = 1, 2, 3, 4, x, y ∈ [0, 1], and z ∈ Z+. Therefore, κ1,j(x, z)κ1,k(y, z) = O(n−2) (3.22) for k, j = 1, 2, 3, 4, 5, 6, 7, 8. Substituting these asymptotics into (3.20), we obtain detϕ(z) = 2(−1)n+1cz10e−3π/2+πn ( 2iδn − ip0 π(2n− 3) + κ1,2(z) + κ1,7(z)− κ1,1(z)− κ1,4(z) +O(n−2) ) . 18 D. M. POLYAKOV EJDE-2024/62 Then the equation detϕ(z) = 0 implies δn = p0 2π(2n− 3) − 1 2i ( κ1,2(z) + κ1,7(z)− κ1,1(z)− κ1,4(z) ) +O(n−2). Thus, the identity z = −3π/2 + πn+ δn yields z = −3π 2 + πn+ p0 2π(2n− 3) − 1 2i ( κ1,2(z) + κ1,7(z)− κ1,1(z) − κ1,4(z) ) +O(n−2). (3.23) Integrating by parts and using the identity z = −3π/2 + πn + δn, we obtain B1,21(0, z) = O(n−1). Similar arguments imply for B1,41(0, z) = O(n−1), B1,23(0, z) = O(n−1), B1,43(0, z) = O(n−1). Therefore, κ1,2(z) = O(n−2). Repeating this pro- cedure for κ1,1, κ1,4, and κ1,7, we obtain κ1,2(z) + κ1,7(z)− κ1,1(z)− κ1,4(z) = O(n−2). It remains to compute κ1,2(z) + κ1,7(z) − κ1,1(z) − κ1,4(z). The formulas (3.4), (3.8), the identity (2.34) with W = 0, asymptotics (3.21), and (2.31) give κ1,2(z) = 1 z ( (1− i)B1,21(0, z)− iB1,41(0, z) + iB1,23(0, z) + (1 + i)B1,43(0, z) ) +O(z−2). (3.24) The definitions (2.32) and (2.33) with (2.20) imply B1,23(0, z) = B1,32(1, z) = 0 and B1,21(0, z) = − ∫ 1 0 e(−1−i)zsF1,21(s) ds = −1 4 ∫ 1 0 e(−1−i)zsp(s) ds. It remains to substitute this expression into (3.23). Then z = −3π 2 + πn+ p0 2π(2n− 3) +O(n−2). This gives (1.7). Let c = 0 and λ = z4 = λn, n → +∞. It follows from [14, Theorem 6.1] that z = −3π/4 + πn+ δn, δn = O(n−1). The relations (3.11), (3.14), and (3.15) give γ1(z) = (1 + i)z ( 1 + κ1,1(z) +O(n−2) ) , γ2(z) = −2az6e−iαz−iβz ( 1 + κ1,3(z)− d(1 + i) 2az +O(n−2) ) , γ3(z) = (−1 + i)z ( 1 + κ1,2(z) +O(n−2) ) , γ4(z) = −2az6eiαz−iβz ( 1 + κ1,6(z)− d(1− i) 2az +O(n−2) ) , where κ1,j , j = 1, 2, 3, 6, have the form (3.3)–(3.6). Substituting these expressions into (2.10), we obtain detϕ(z) = −2az7e−iβ1z ( (1 + i)e−iα1z ( 1 + κ1,1(z) + κ1,3(z) + κ1,1(z)κ1,3(z)− d(1 + i) 2az ) + (−1 + i)eiα1z ( 1 + κ1,2(z) + κ1,6(z) + κ1,2(z)κ1,6(z)− d(1− i) 2az ) +O(n−2) ) , EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 19 where α1 and β1 satisfy (3.2). The identity z = −3π/4+ πn+ δn and the formulas (3.2) give e±iα1z = e±iz∓ip0/(4z) = (−1)n(−1∓ i) √ 2 2 e±iδn∓ip0/(4z) = (−1)n(−1∓ i) √ 2 2 ( 1± iδn ∓ ip0 π(4n− 3) +O(n−2) ) . Therefore, these equalities and (3.22) imply detϕ(z) = 2 √ 2(−1)n+1az7e−3π/4+πn ( 2iδn − 2ip0a− 4id aπ(4n− 3) + κ1,2(z) + κ1,6(z) − κ1,1(z)− κ1,3(z) +O(n−2) ) . The equality detϕ(z) = 0 implies δn = p0a− 2d aπ(4n− 3) − 1 2i ( κ1,2(z) + κ1,6(z)− κ1,1(z)− κ1,3(z) ) +O(n−2). Thus, the identity z = −3π/4 + πn+ δn yields z = −3π 4 + πn+ p0 π(4n− 3) − 2d aπ(4n− 3) − 1 2i ( κ1,2(z) + κ1,6(z) − κ1,1(z)− κ1,3(z) ) +O(n−2). (3.25) Repeating the process as in the case c ̸= 0 and using the relation z = −3π/4 + πn+ δn, we obtain κ1,2(z) + κ1,6(z)− κ1,1(z)− κ1,3(z) = O(n−2), Substituting this expression into (3.25), we have z = −3π 4 + πn− 2d aπ(4n− 3) + p0 − ρ2,n + p̂sn(3/2) π(4n− 3) +O(n−2). This gives (1.8). □ 4. Eigenvalue asymptotics in the case p ∈W 3,1(0, 1) Now we determine sharp eigenvalue asymptotics of the operatorH in the case p ∈ W 3,1(0, 1). This corresponds to the case σ = 4. Again we obtain the asymptotics for the function detϕ of the form (2.10). Using this asymptotics, we get the eigenvalue asymptotics of the operator H. Lemma 4.1. Suppose that p ∈ W 3,1(0, 1) and p(0) = p(1). Then the eigenvalues λn satisfy (1.9) for c ̸= 0 and (1.11) for c = 0. Proof. Let c ̸= 0 and λ = z4 = λn, n→ +∞. It follows from Lemma 3.2 that z = −3π 2 + πn+ p0 2π(2n− 3) + δn, δn = O(n−2). (4.1) 20 D. M. POLYAKOV EJDE-2024/62 Substituting (3.11)–(3.13) with σ = 4 into (2.10), we obtain detϕ(z) = 2icz10e−iβ4z ( e−iα4z ( 1 + κ4,1(z) + κ4,4(z) + κ4,1(z)κ4,4(z) − 2a c(1 + i)z3 + d cz4 + ip(1) z2 ( 1 + κ4,1(z) + κ4,5(z) )) + eiα4z ( 1 + κ4,2(z) + κ4,7(z) + κ4,2(z)κ4,7(z) − ip(1) z2 ( 1 + κ4,2(z) + κ4,8(z) ) − 2a c(1− i)z3 + d cz4 ) +O(z−5) ) , (4.2) where α4, β4 satisfy (3.2) and κ4,j , j = 1, 2, 4, 5, 7, 8, have the form (3.3)–(3.8). Now we compute κ4,1(z) + κ4,4(z) + κ4,1(z)κ4,4(z). Formulas (3.3), (3.7), iden- tity (2.34) with σ = 4, and (2.31) give κ4,1(z) = 1 z2 ( W22 − iW32 + (1− i)W42 +W11 + (1 + i)W31 + iW41 ) (0, z) + 1− i 2 ( M1 + M̃1 ) (0, z) + 1 z4 ( − iB4,32(0, z) + (1− i)B4,42(0, z) + (1 + i)B4,31(0, z) + iB4,41(0, z) ) +O(z−5), (4.3) and κ4,4(z) = 1 z2 ( iW14 + (−1− i)W24 +W44 + (−1 + i)W13 − iW23 +W33 ) (1, z) + 1− i 2 ( M3 + M̃3 ) (1, z) + 1 z4 ( iB4,14(1, z) + (−1− i)B4,24(1, z) + (−1 + i)B4,13(1, z)− iB4,23(1, z) ) +O(z−5), (4.4) where W, Mj , M̃j , j = 1, 3, have the form (2.22), (3.9), (3.10), respectively. Direct calculations imply 1 z2 ( W22 − iW32 + (1− i)W42 +W11 + (1 + i)W31 + iW41 ) (0, z) = p(1) z2 Υ(W1) + p′(1) z3 Υ(W2) + p′′(1) 32z4 Υ(Q1)− p2(1) 64z4 Υ(Q2), where Υ(A) = A1,22 − iA1,32 + (1− i)A1,42 +A1,11 + (1 + i)A1,31 + iA1,41 and W1, W2, Q1, Q2 have the form (2.23) and (2.24). Therefore, these formulas yield 1 z2 ( W22 − iW32 + (1− i)W42 +W11 + (1 + i)W31 + iW41 ) (0, z) = − ip(1) 4z2 + (3− 3i)p′(1) 16z3 + p′′(1) 4z4 − (1 + i)p2(1) 32z4 . (4.5) Similar arguments give 1 z2 ( iW14 + (−1− i)W24 +W44 + (−1 + i)W13 − iW23 +W33 ) (1, z) = 3ip(1) 4z2 + (5− 5i)p′(1) 16z3 − p′′(1) 4z4 + (1 + i)p2(1) 32z4 . EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 21 Moreover, using the identities (3.9), (3.10), and (2.23), we obtain 1− i 2 ( M1 + M̃1 ) (0, z) = (1− i)p2(1) 128z4 (( −W1,12 + iW1,22 − iW1,32 +W1,42 ) × ( W1,11 +W1,21 +W1,31 +W1,41 ) + ( W1,12 +W1,22 +W1,32 +W1,42 )( W1,11 − iW1,21 + iW1,31 −W1,41 )) = p2(1) 64z4 (4.6) and 1− i 2 ( M3 + M̃3 ) (1, z) = (1− i)p2(1) 128z4 (( −W1,13 − iW1,23 + iW1,33 +W1,43 ) × ( W1,14 −W1,24 −W1,34 +W1,44 ) + ( −W1,13 +W1,23 +W1,33 −W1,43 )( −W1,14 − iW1,24 + iW1,34 +W1,44 )) = 9p2(1) 64z4 . The definitions (2.32) and (2.33) with (2.27) imply B1,23(0, z) = B1,32(1, z) = 0 and B4,32(0, z) = − ∫ 1 0 ei2zsF4,32(s) ds = 1 32 ∫ 1 0 ei2zsp′′′(s) ds+O(z−1). (4.7) Similar arguments yield B4,42(0, z) = 1 + i 16 ∫ 1 0 e(−1+i)zsp′′′(s) ds+O(z−1), (4.8) B4,31(0, z) = 1− i 16 ∫ 1 0 e(−1+i)zsp′′′(s) ds+O(z−1), (4.9) B4,41(0, z) = 1 32 ∫ 1 0 e−2zsp′′′(s) ds+O(z−1), (4.10) B4,14(1, z) = − 1 32 ∫ 1 0 e−2z(1−s)p′′′(s) ds+O(z−1), (4.11) B4,24(1, z) = −1 + i 16 ∫ 1 0 e(−1+i)z(1−s)p′′′(s) ds+O(z−1), (4.12) B4,23(1, z) = − 1 32 ∫ 1 0 ei2z(1−s)p′′′(s) ds+O(z−1), (4.13) B4,13(1, z) = −1− i 16 ∫ 1 0 e(−1+i)z(1−s)p′′′(s) ds+O(z−1). (4.14) Substituting these expressions into (4.3) and (4.4), we obtain κ4,1(z) + κ4,4(z) + κ4,1(z)κ4,4(z) = ip(1) 2z2 + (1− i)p′(1) 2z3 + 11p2(1) 32z4 + η1(z) z4 +O(z−5), (4.15) 22 D. M. POLYAKOV EJDE-2024/62 where η1(z) = i 32 ∫ 1 0 e−2zs ( p′′′(s)− p′′′(1− s) ) ds+ 1 4 ∫ 1 0 e(−1+i)zs ( p′′′(s) + p′′′(1− s) ) ds + i 32 ∫ 1 0 p′′′(s) ( ei2z(1−s) − ei2zs ) ds. (4.16) Arguments similar to this provide κ4,1(z)+κ4,5(z) = − ip(1) 2z2 +O(z−3), κ4,2(z)+κ4,8(z) = ip(1) 2z2 +O(z−3), (4.17) and κ4,2(z) + κ4,7(z) + κ4,2(z)κ4,7(z) = − ip(1) 2z2 + (1 + i)p′(1) 2z3 + 11p2(1) 32z4 + η2(z) z4 +O(z−5), (4.18) where η2(z) = i 32 ∫ 1 0 e−2zs ( p′′′(1− s)− p′′′(s) ) ds + 1 4 ∫ 1 0 e(−1−i)zs ( p′′′(s) + p′′′(1− s) ) ds. (4.19) Now we substitute (4.15), (4.17), and (4.18) into (4.2). Then detϕ(z) = 2icz10e−iβ4z ( e−iα4z ( 1 + 3ip(1) 2z2 + (1− i)p′(1) 2z3 + 27p2(1) 32z4 + η1(z) z4 − 2a c(1 + i)z3 + d cz4 ) + eiα4z ( 1− 3ip(1) 2z2 + (1 + i)p′(1) 2z3 + 27p2(1) 32z4 + η2(z) z4 − 2a c(1− i)z3 + d cz4 ) +O(z−5) ) , (4.20) where α4 and β4 have the form (3.2). Now we compute the asymptotic behavior of the eigenvalues λn, as n → +∞. Recall that the zeros of the function (4.20) are the eigenvalues λn. Then (4.1) and (3.2) imply e±iα4z = e±iz±i∥p∥2/(32z3)∓ip0/(4z) = ±i(−1)ne±iδn+O(z−3) = ±i(−1)n ( 1± iδn +O(n−3) ) . Substituting these into (4.20), we obtain detϕ(z) = 2icz10e−3π/2+πn ( e−iα4z ( 1 + 3ip(1) 2z2 ) + eiα4z ( 1− 3ip(1) 2z2 ) +O(n−3) ) = 4icz10(−1)n+1e−3π/2+πn ( δn − 6p(1) π2(2n− 3)2 +O(n−3) ) . The equation detϕ(z) = 0 yields δn = 6p(1) π2(2n− 3)2 +O(n−3). EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 23 Then (4.1) yields z = −3π 2 + πn+ p0 2π(2n− 3) + 6p(1) π2(2n− 3)2 + δn, δn = O(n−3). (4.21) Now we improve the asymptotics (4.21). The formulas (4.21) and (3.2) give α4z = z − p0 4z + ∥p∥2 32z3 = −3π 2 + πn+ δn + p0 2π(2n− 3) + 6p(1) π2(2n− 3)2 + ∥p∥2 32z3 − p0 4z +O(n−4) = −3π 2 + πn+ δn + 6p(1) π2(2n− 3)2 + 2p20 + ∥p∥2 4π3(2n− 3)3 +O(n−4). Then e±iα4z = ±i(−1)n ( 1± iδn ± 6ip(1) π2(2n− 3)2 ± i ( ∥p∥2 + 2p20 ) π3(2n− 3)3 +O(n−4) ) . Substituting these into (4.20), we obtain detϕ(z) = 2cz10(−1)n+1e−3π/2+πn ( 2iδn + 8ip′(1) π3(2n− 3)3 + i(∥p∥2 + 2p20) 2π3(2n− 3)3 − 16ia cπ3(2n− 3)3 +O(n−4) ) . The equation detϕ(z) = 0 implies δn = − 4p′(1) π3(2n− 3)3 − ∥p∥2 + 2p20 4π3(2n− 3)3 + 8a cπ3(2n− 3)3 +O(n−4). Then identity (4.21) yields z = −3π 2 + πn+ p0 2π(2n− 3) + 6p(1) π2(2n− 3)2 − 4p′(1) π3(2n− 3)3 − ∥p∥2 + 2p20 4π3(2n− 3)3 + 8a cπ3(2n− 3)3 + δn, δn = O(n−4). (4.22) Now we improve this asymptotic behavior. Relations (4.22) and (3.2) give α4z = z − p0 4z + ∥p∥2 32z3 = −3π 2 + πn+ δn + p0 2π(2n− 3) + 6p(1) π2(2n− 3)2 − 4p′(1) π3(2n− 3)3 − ∥p∥2 + 2p20 4π3(2n− 3)3 + 8a cπ3(2n− 3)3 + ∥p∥2 32z3 − p0 4z = −3π 2 + πn+ δn + 6p(1) π2(2n− 3)2 − 4p′(1) π3(2n− 3)3 + 8a cπ3(2n− 3)3 + 6p0p(1) π4(2n− 3)4 +O(n−5). Then e±iα4z = ±i(−1)n ( 1± iδn ± 6ip(1) π2(2n− 3)2 ∓ 4ip′(1) π3(2n− 3)3 ± 8ia cπ3(2n− 3)3 24 D. M. POLYAKOV EJDE-2024/62 ± 6ip0p(1) π4(2n− 3)4 − 18p2(1) π4(2n− 3)4 +O(n−5) ) . Substituting these expression into (4.20), we obtain detϕ(z) = 2cz10(−1)n+1e−3π/2+πn ( 2iδn + 36ip0p(1) π4(2n− 3)4 + 16 ( η2(z)− η1(z) ) π4(2n− 3)4 +O(n−5) ) . The relations (4.16), (4.19), and (4.22) imply η2(z)− η1(z) = i 16 ∫ 1 0 e−2zs ( p′′′(1− s)− p′′′(s) ) ds + 1 4 ∫ 1 0 ( p′′′(s) + p′′′(1− s) )( e(−1−i)zs − e(−1+i)zs ) ds − i 32 ∫ 1 0 p′′′(s) ( ei2z(1−s) − ei2zs ) ds = iρ1,n 16 + i 16 ∫ 1 0 p′′′(s) cos 2zs ds+O(n−1) = i ( ρ1,n + p̂′′′cn(3) ) 16 +O(n−1), where ρ1,n has the form (1.10). Then detϕ(z) = 2cz10(−1)n+1e−3π/2+πn ( 2iδn+ 36ip0p(1) π4(2n− 3)4 + i ( ρ1,n + p̂′′′cn(3) ) π4(2n− 3)4 +O(n−5) ) . The equation detϕ(z) = 0 implies δn = − 18p0p(1) π4(2n− 3)4 − ρ1,n + p̂′′′cn(3) 2π4(2n− 3)4 +O(n−5). Then identity (4.22) yields z = −3π 2 + πn+ p0 2π(2n− 3) + 6p(1) π2(2n− 3)2 − 4p′(1) π3(2n− 3)3 − ∥p∥2 + 2p20 4π3(2n− 3)3 + 8a cπ3(2n− 3)3 − 18p0p(1) π4(2n− 3)4 − ρ1,n + p̂′′′cn(3) 2π4(2n− 3)4 +O(n−5). This gives (1.9). Let c = 0 and λ = z4 = λn, n→ +∞. It follows from Lemma 3.2 that z = −3π 4 + πn+ p0 π(4n− 3) − 2d aπ(4n− 3) + δn, δn = O(n−2). (4.23) EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 25 Substituting (3.11), (3.14), (3.15) with σ = 4 into (2.10), we obtain detϕ(z) = −2az7e−iβ4z ( (1 + i)e−iα4z ( 1 + κ4,1(z) + κ4,3(z) + κ4,1(z)κ4,3(z) − d(1 + i) 2az ( 1 + κ4,1(z) + κ4,4(z) ) − d(−1 + i)p(1) 2az3 + b az4 ) + (−1 + i)eiα4z ( 1 + κ4,2(z) + κ4,6(z) + κ4,2(z)κ4,6(z) − d(1− i) 2az ( 1 + κ4,2(z) + κ4,7(z) ) − d(−1− i)p(1) 2az3 + b az4 ) +O(z−5) ) , (4.24) where α4, β4 satisfy (3.2) and κ4,j , j = 1, 2, 3, 4, 6, 7, have the form (3.3)–(3.8). Now we compute κ4,1(z) + κ4,3(z) + κ4,1(z)κ4,3(z). The formula (3.5), iden- tity (2.34) with σ = 4, and (2.31) give κ4,3(z) = 1 z2 ( W14 +W44 +W23 +W33 ) (1, z) + 1 z4 ( B4,14(1, z) + B4,23(1, z) ) + p2(1) 64z4 ( (W1,23 +W1,33)(W1,14 +W1,44) − (W1,13 +W1,43)(W1,24 +W1,34) ) +O(z−5), (4.25) where W and W1 have the form (2.22), (2.23), respectively. Direct calculations imply 1 z2 ( W14 +W44 +W23 +W33 ) (1, z) + p2(1) 64z4 ( (W1,23 +W1,33)(W1,14 +W1,44)− (W1,13 +W1,43)(W1,24 +W1,34) ) = (−1 + i)p′(1) 16z3 + p′′(1) 16z4 + (1 + 2i)p2(1) 64z4 . This statement, the identities (4.8), (4.14), and (4.25) imply κ4,3(z) = (−1 + i)p′(1) 16z3 + p′′(1) 16z4 + (1 + 2i)p2(1) 64z4 − 1 32z4 ∫ 1 0 e−2z(1−s)p′′′(s) ds− 1 32z4 ∫ 1 0 ei2z(1−s)p′′′(s) ds+O(z−5). Therefore, using this asymptotics and the formulas (4.5)–(4.14), we obtain κ4,1(z) + κ4,3(z) + κ4,1(z)κ4,3(z) = − ip(1) 4z2 + (1− i)p′(1) 8z3 + 5p′′(1) 16z4 + η3(z) z4 +O(z−5), (4.26) where η3(z) = 1 4 ∫ 1 0 e(−1+i)zsp′′′(s) ds− i 32 ∫ 1 0 ei2zsp′′′(s) ds− 1 32 ∫ 1 0 ei2z(1−s)p′′′(s) ds + i 32 ∫ 1 0 e−2zsp′′′(s) ds− 1 32 ∫ 1 0 e−2z(1−s)p′′′(s) ds. (4.27) 26 D. M. POLYAKOV EJDE-2024/62 Arguments similar to this provide the asymtotics κ4,1(z) + κ4,4(z) = ip(1) 2z2 + (1− i)p′(1) 2z3 +O(z−4), κ4,2(z) + κ4,7(z) = − ip(1) 2z2 + (1 + i)p′(1) 2z3 +O(z−4), (4.28) and κ4,2(z) + κ4,6(z) + κ4,2(z)κ4,6(z) = ip(1) 4z2 + (1 + i)p′(1) 8z3 + 5p′′(1) 16z4 + η4(z) z4 +O(z−5), (4.29) where η4(z) = − i 32 ∫ 1 0 e−2zsp′′′(s) ds− 1 32 ∫ 1 0 e−2z(1−s)p′′′(s) ds + 1 4 ∫ 1 0 e(−1−i)zsp′′′(s) ds. (4.30) Now we substitute (4.26), (4.28), and (4.29) into (4.24). Then detϕ(z) = −2az7e−iβ4z ( (1 + i)e−iα4z ( 1− d(1 + i) 2az − ip(1) 4z2 + (1− i)p′(1) 8z3 + 3dp(1)(1− i) 4az3 + 5p′′(1) 16z4 − dp′(1) 2az4 + η3(z) z4 + b az4 ) + (−1 + i)eiα4z ( 1− d(1− i) 2az + ip(1) 4z2 + (1 + i)p′(1) 8z3 + 3dp(1)(1 + i) 4az3 + 5p′′(1) 16z4 − dp′(1) 2az4 + η4(z) z4 + b az4 ) +O(z−5) ) , (4.31) where α4 and β4 have the form (3.2). Now we compute the asymptotic behavior of the eigenvalues λn, as n → +∞. Recall that the zeros of the function (4.31) are the eigenvalues λn. Then formulas (4.23) and (3.2) imply e±iα4z = e±iz±i∥p∥2/(32z3)∓ip0/(4z) = (−1)n √ 2(−1∓ i) 2 ( 1± iδn ∓ 2d aπ(4n− 3) +O(n−3) ) . Substituting these into (4.31), we obtain detϕ(z) = −2az7e−3π/4+πn ( (1 + i)e−iα4z ( 1− d(1 + i) 2az − ip(1) 4z2 ) + (−1 + i)eiα4z ( 1− d(1− i) 2az + ip(1) 4z2 ) +O(z−3) ) = 2 √ 2(−1)n+1az7e−3π/4+πn ( 2iδn + 8ip(1) π2(4n− 3)2 + 8id2 a2π2(4n− 3)2 +O(n−3) ) . The equation detϕ(z) = 0 yields δn = − 4p(1) π2(4n− 3)2 − 4d2 a2π2(4n− 3)2 +O(n−3). EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 27 Then identity (4.23) yields z = −3π 4 + πn+ p0 π(4n− 3) − 2d aπ(4n− 3) − 4p(1) π2(4n− 3)2 − 4d2 a2π2(4n− 3)2 + δn, δn = O(n−3). (4.32) Now we improve the asymptotic (4.32). The formulas (4.32) and (3.2) give α4z = z − p0 4z + ∥p∥2 32z3 = −3π 4 + πn+ δn + p0 + A π(4n− 3) + B π2(4n− 3)2 + ∥p∥2 32z3 − p0 4z +O(n−4) = −3π 4 + πn+ δn + A π(4n− 3) + B π2(4n− 3)2 + C π3(4n− 3)3 +O(n−4), where A := −2d a , B := −4p(1)− 4d2 a2 , C := 2∥p∥2 + 4p20 − 8p0d a . (4.33) Then e±iα4z = (−1)n √ 2(−1∓ i) 2 ( 1± iδn ± iA π(4n− 3) ± iB π2(4n− 3)2 ± iC π3(4n− 3)3 + 1 2 ( iA π(4n− 3) + iB π2(4n− 3)2 )2 ∓ iA3 6π3(4n− 3)3 +O(n−4) ) . Substituting these expressions into (4.31), we obtain detϕ(z) = 2 √ 2(−1)n+1az7e−3π/4+πn ( 2iδn − 2iD π3(4n− 3)3 +O(n−4) ) , where D := −2∥p∥2 − 4p20 − 8p′(1)− 56p(1)d a + 16p0d a − 16d2 a2 − 16d3 3a3 . (4.34) The equation detϕ(z) = 0 implies δn = D π3(4n− 3)3 +O(n−4). Then identity (4.32) yields z = −3π 4 + πn+ p0 + A π(4n− 3) + B π2(4n− 3)2 + D π3(4n− 3)3 + δn, δn = O(n−4). (4.35) Now we improve this asymptotics. The relations (4.35) and (3.2) give α4z = z − p0 4z + ∥p∥2 32z3 = −3π 4 + πn+ δn + p0 π(4n− 3) + A π(4n− 3) + B π2(4n− 3)2 28 D. M. POLYAKOV EJDE-2024/62 + D π3(4n− 3)3 + ∥p∥2 32z3 − p0 4z = −3π 4 + πn+ δn + A π(4n− 3) + B π2(4n− 3)2 + F π3(4n− 3)3 + 4Bp0 π4(4n− 3)4 +O(n−5), where F := −56p(1)d a + 8p0d a − 16d2 a2 − 16d3 3a3 − 8p′(1). Then e±iα4z = (−1)n √ 2(−1∓ i) 2 ( 1± iδn ± iA π(4n− 3) ± iB π2(4n− 3)2 ± iF π3(4n− 3)3 ± 4Bp0 π4(4n− 3)4 + 1 2 ( iA π(4n− 3) + iB π2(4n− 3)2 + iF π3(4n− 3)3 )2 ± 1 6 ( iA π(4n− 3) + iB π2(4n− 3)2 )3 + A4 24π4(4n− 3)4 +O(n−5) ) . Substituting these expressions into (4.31), we obtain detϕ(z) = 2 √ 2(−1)n+1az7e−3π/4+πn ( 2iδn − 96id2p0 a2π4(4n− 3)4 + 192id3 a3π4(4n− 3)4 − 96ip0p(1) π4(4n− 3)4 + 192idp(1) aπ4(4n− 3)4 + 256 ( η4(z)− η3(z) ) π4(4n− 3)4 +O(n−5) ) . Now we obtain η4(z)− η3(z). Then equations (4.27), (4.30), and (4.35) give η4(z)− η3(z) = − i 16 ∫ 1 0 e−2zsp′′′(s) ds+ 1 4 ∫ 1 0 e−zsp′′′(s) ( e−izs − eizs ) ds + 1 32 ∫ 1 0 p′′′(s) ( iei2zs + ei2z(1−s) ) ds = − iρ2,n 16 + i 16 ∫ 1 0 p′′′(s) cosπ ( 2n− 3 2 ) s ds+O(n−1) = − i(ρ2,n − p̂′′′cn(3/2)) 16 +O(n−1), where ρ2,n has the form (1.12). The equation detϕ(z) = 0 implies δn = 48d2p0 a2π4(4n− 3)4 − 96d3 a3π4(4n− 3)4 + 48p0p(1) π4(4n− 3)4 − 96dp(1) aπ4(4n− 3)4 + 8 ( ρ2,n − p̂′′′cn(3/2) ) π4(4n− 3)4 +O(n−5). Then identity (4.35) yields z = −3π 4 + πn+ p0 π(4n− 3) + A π(4n− 3) + B π2(4n− 3)2 + D π3(4n− 3)3 + 48d2p0 a2π4(4n− 3)4 − 96d3 a3π4(4n− 3)4 + 48p0p(1) π4(4n− 3)4 − 96dp(1) aπ4(4n− 3)4 + 8 ( ρ2,n − p̂′′′cn(3/2) ) π4(4n− 3)4 +O(n−5), EJDE-2024/62 ASYMPTOTIC BEHAVIOR OF EIGENVALUES 29 where A, B, and D have the form (4.33) and (4.34). This implies (1.11). □ Proof of Theorem 1.1. It follows from [14, Lemmas 4.1, 4.2] that the eigenvalues λn are real and simple. The asymptotics (1.7), (1.8), (1.9), and (1.11) are proved in Lemmas 3.2 and 4.1. The proof is complete. □ References [1] Aliyev, Z. S.; Basis properties of a fourth order differential operator with spectral parameter in the boundary condition. Cent. Eur. J. Math. 2010. V. 8(2). P. 378–388. [2] Aliev, Z. S.; Basis properties in Lp of systems of root functions of a spectral problem with spectral parameter in a boundary condition. Differ. Equat. 2011. V. 47(6). P. 766–777. [3] Aliyev, Z. S., Gulieva, S. 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[24] Roseau, M.; Vibrations in Mechanical Systems. Analytical Methods and Applications. Springer, Berlin, 1987. [25] Shkalikov, A. A.; Boundary value problems for ordinary differential equations with a param- eter in the boundary conditions. J. Sov. Math. 1986. V. 33. P. 1311–1342. [26] Tretter, C.; Boundary eigenvalue problems for differential equations Nη = λPη with λ- polynomial boundary conditions. J. Differ. Equ. 2001. V. 170. P. 408–471. [27] Zinsou, B.; Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problems. Arab. J. Math. 2023. V. 12. 711–732. Dmitry M. Polyakov Southern Mathematical Institute, Vladikavkaz Scientific Center of RAS, 362025, 53 Vatutin str., Vladikavkaz, Russia Email address: DmitryPolyakow@mail.ru, ORCID: 0000-0003-0263-056X 1. Introduction and main results 2. Properties of the fundamental solutions 3. Eigenvalue asymptotics in the case pW1, 1(0, 1) 3.1. Asymptotics of the functions j, j=1, 2, 3, 4 3.2. Sharp eigenvalue asymptotics 4. Eigenvalue asymptotics in the case pW3, 1(0, 1) References