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NB Kerimov, ZS Aliev (2006)
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M Möller, B Zinsou (2011)
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The John Knopfmacher Centre for A regular fourth order diﬀerential equation with λ-dependent boundary conditions is Applicable Analysis and Number Theory, School of Mathematics, considered. For four distinct cases with exactly one λ-independent boundary University of the Witwatersrand, condition, the asymptotic eigenvalue distribution is presented. Johannesburg, South Africa MSC: 34L20; 34B07; 34B08; 34B09 Keywords: fourth order boundary value problems; self-adjoint; boundary conditions; eigenvalue distribution; pure imaginary eigenvalues; spectral asymptotics 1 Introduction Sturm-Liouville problems have attracted extensive attention due to their intrinsic mathe- matical challenges and their applications in physics and engineering. However, apart from classical Sturm-Liouville problems, also higher order ordinary linear diﬀerential equations occur in applications, with or without the eigenvalue parameter in the boundary condi- tions. Such problems are realized as operator polynomials, also called operator pencils. Some recent developments of higher order diﬀerential operators whose boundary condi- tions depend on the eigenvalue parameter, including spectral asymptotics and basis prop- erties, have been investigated in [–]. General characterizations of self-adjoint boundary conditions have been presented in [, ] for singular and (quasi-)regular problems. In all these cases, the minimal operator associated with an nth order diﬀerential equation must be symmetric, see [, ] for necessary and suﬃcient conditions. A more general discussion on the spectra of fourth order diﬀerential operators can be found in [, ]. The generalized Regge problem is realized by a second order diﬀerential operator which depends quadratically on the eigenvalue parameter and which has eigenvalue parameter dependent boundary conditions, see []. The particular feature of this problem is that the coeﬃcient operators of this pencil are self-adjoint, and it is shown in [] that this gives someapriori knowledge about the location of the spectrum. In [] this approach has been extended to a fourth order diﬀerential equation describing small transversal vibrations of a homogeneous beam compressed or stretched by a force g. Separation of variables leads to a fourth order boundary problem with eigenvalue parameter dependent boundary © 2012 Möller and Zinsou; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 2 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 conditions, where the diﬀerential equation () y – gy = λ y depends quadratically on the eigenvalue parameter. This problem is represented by a quadratic operator pencil, in a suitably chosen Hilbert space, whose coeﬃcient opera- tors are self-adjoint. In [] we have investigated a class of boundary conditions for which necessary and suﬃcient conditions were obtained such that the associated operator pencil consists of self-adjoint operators, while in []wehavecontinued theworkof[]inthe direction of [] to derive eigenvalue asymptotics associated with boundary conditions which lead to self-adjoint operator representations. We have considered the particular case of boundary conditions which do not depend on the eigenvalue parameter at the left endpoint and depend on the eigenvalue parameter at the right endpoint. In this paper, we extend the work of [] to a class of boundary conditions where exactly one of the left endpoint boundary conditions does not depend on the eigenvalue param- eter, while the remaining boundary conditions depend on the eigenvalue parameter. We deﬁne the operator pencil in Section and we discuss which boundary conditions are considered. In Section , the eigenvalue asymptotics for the case g = are derived. In Section , it is shown that the boundary value problems under consideration are Birkhoﬀ regular, which implies that the eigenvalues for general g are small perturbations of the eigenvalues for g = .Hence,inSection , the ﬁrst four terms of the eigenvalue asymptotics are found and are compared to those obtained in []. 2 The quadratic operator pencil L On the interval [, a], we consider the boundary value problem () y – gy = λ y,(.) B (λ)y =, j = ,,,, (.) where a >, g ∈ C [, a] is a real valued function and (.) are separated boundary condi- tions where the B (λ) are constant or depend on λ linearly. The boundary conditions (.) are taken at the endpoint for j = , and at the endpoint a for j = , . Further, we assume [p ] [q ] [p ] j j j for simplicity that either B (λ)y = y (a )+ iε αλy (a )or B (λ)y = y (a ), where a = j j j j j j j for j =,, a = a for j =,, α > and ε ∈{–, }. We recall that the quasi-derivatives j j associated with (.)are givenby [] [] [] [] () [] () y = y, y = y , y = y , y = y – gy , y = y – gy , see [,p.]. Deﬁne = s ∈{, , , } : B (λ)depends on λ , = {, , , }\ , s a = ∩{, }, = ∩{, }. Assumption . The numbers p , p , q for j ∈ aredistinctaswellasthe numbers p , j p , q for j ∈ . j Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 3 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 We denote by U the collection of the boundary conditions (.) and deﬁne the following operators related to U: [p ] [q ] j j U y = y (a ) and U y = ε y (a ) , y ∈ W (, a). (.) j j j j∈ j∈ We put k = | | and consider the linear operators A(U), K and M in the space L (, a) ⊕C with domains [p ] D A(U) = y = : y ∈ W (, a), y (a )= for j ∈ , j U y D(K)=D(M)= L (, a) ⊕ C , given by [] y I A(U) y = for y ∈D A(U) , K = and M = . U y I It is easy to check that K ≥ , M ≥ , M +K = I and M| > . We associate a quadratic D(A(U)) operator pencil L(λ, α)= λ M – iαλK – A(U), λ ∈ C (.) in the space L (, a) ⊕ C with the problem (.), (.). The conditions under which the diﬀerential operator A(U)is self-adjoint are given in [p] Theorem . ([], Theorem .) Denote by P the set of p in y () = for the λ- [p] independent boundary conditions and by P the corresponding set for y (a)=. Then the diﬀerential operator A(U) associated with this boundary value problem is self-adjoint if [p] [q] and only if p + q = for all boundary conditions of the form y (a )+ iαε λy (a )= and j j j ε = if q isevenincase a = or odd in case a = a, ε =– otherwise, {, }⊂ P , {, }⊂ P , j j j j {, }⊂ P and {, }⊂ P . a a Proposition . The operator pencil L(·, α) is a Fredholm valued operator function with index . The spectrum of the Fredholm operator L(·, α) consists of discrete eigenvalues of ﬁnite multiplicities, and all eigenvalues of L(·, α), α ≥ , lie in the closed upper half-plane and on the imaginary axis and are symmetric with respect to the imaginary axis. Proof As in [, Section ], we can argue that for all λ ∈ C, L(λ, α)is a relatively com- pact perturbation of L(, ), where L(, ) is well known to be a Fredholm operator. The statement on the location of the spectrum now follows as in [, Lemma .]. We now consider the particular cases that exactly one of the boundary conditions at depends on λ, whereas both boundary conditions at a depend on λ. Therefore, taking Assumption . and Theorem . into account, we have the four boundary conditions [p ] [p ] [q ] y () = , y () + iαε λy () = , Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 4 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 [p ] [q ] [p ] [q ] y (a)+ iαε λy (a)=, y (a)+ iαε λy (a)=, where ≤ p ≤ , ≤ p ≤ , ≤ q ≤ , p +q =,and p ∈{ / q , p },while {p , q } = {, } and {p , q } = {, }. Thus, we have and possible sets of boundary conditions at the end- point and a, respectively. Whence there are diﬀerent sets of boundary conditions. Re- call that the parameter λ emanates from derivatives with respect to the time variable in the original partial diﬀerential equation, and it is reasonable that the highest space derivative occurs in the term without time derivative. Thus, the most relevant boundary conditions would have q < p , q < p and q < p . This leaves us with four diﬀerent cases for the boundary conditions B (λ)y =. These four cases are uniquely determined by the value of p , so that we will consider Case : p =; Case : p =; Case : p =; Case : p =. The corresponding boundary operators are then [] B y = y () (Case ), B y = y() (Case ), (.) B y = y () (Case ), B y = y () (Case ), B (λ)y = y () – iαλy () (Cases and ), (.) [] B (λ)y = y () + iαλy() (Cases and ), B (λ)y = y (a)+ iαλy (a), (.) [] B (λ)y = y (a)– iαλy(a). (.) 3 Asymptotics of eigenvalues for g =0 In this section, we consider the boundary value problem (.), (.)with g =. We count all eigenvalues with their proper multiplicities and develop a formula for the asymptotic distribution of the eigenvalues, which is used to obtain the corresponding formula for [j] general g.Observe that for g = , the quasi-derivatives y coincide with the standard (j) derivatives y .Wetakethe canonicalfundamental system y (·, λ), j = ,...,, of (.)with (m) y () = δ if j ≥ for m = ,...,. It is well known that the functions y (·, λ) are analytic j,m+ j on C with respect to λ. Putting M(λ)= B (λ)y (·, λ) , i j i,j= the eigenvalues of the boundary value problem (.), (.) are the eigenvalues of the ana- lytic matrix function M, where the corresponding geometric and algebraic multiplicities coincide, see [, Theorem ..]. Setting λ = μ and y(x, μ)= sinh(μx)– sin(μx) μ μ it is easy to see that (–j) y (x, λ)= y (x, μ), j = ,...,. The second row of M(λ) has exactly two non-zero entries (for λ = ), and these non-zero entries are: Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 5 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 In Cases and , B (λ)y (·, λ)=–iαμ and B (λ)y (·, λ)=; In Cases and , B (λ)y (·, λ)= iαμ and B (λ)y (·, λ)=. Since the ﬁrst row of M(λ) has exactly one entry and all other entries zero, an expansion of M(λ) with respect to the second row shows that det M(λ)= ±φ(μ), where B (μ )y (·, μ) B (μ )y (·, μ) σ () σ () φ(μ)= iαμ det B (μ )y (·, μ) B (μ )y (·, μ) σ () σ () B (μ )y (·, μ) B (μ )y (·, μ) σ () σ () + det , B (μ )y (·, μ) B (μ )y (·, μ) σ () σ () with (,,,) in Case , (,,,) in Case , σ (), σ (), σ (), σ () = (,,,) in Case , (,,,) in Case . In view of (.), (.)thisgives () φ(μ)= iαμ y (a, μ)+ iαμ y (a, μ) y (a, μ)– iαμ y (a, μ) σ () σ () σ () σ () () – y (a, μ)+ iαμ y (a, μ) y (a, μ)– iαμ y (a, μ) σ () σ () σ () σ () () + y (a, μ)+ iαμ y (a, μ) y (a, μ)– iαμ y (a, μ) σ () σ () σ () σ () () – y (a, μ)+ iαμ y (a, μ) y (a, μ)– iαμ y (a, μ) σ () σ () σ () σ () () () = iαμ iαμ y (a, μ)y (a, μ)– y (a, μ)y (a, μ) σ () σ () σ () σ () + y (a, μ)y (a, μ)– y (a, μ)y (a, μ) σ () σ () σ () σ () + α μ y (a, μ)y (a, μ)– y (a, μ)y (a, μ) σ () σ () σ () σ () () () + y (a, μ)y (a, μ)– y (a, μ)y (a, μ) σ () σ () σ () σ () () () + iαμ y (a, μ)y (a, μ)– y (a, μ)y (a, μ) σ () σ () σ () σ () + y (a, μ)y (a, μ)– y (a, μ)y (a, μ) σ () σ () σ () σ () + α μ y (a, μ)y (a, μ)– y (a, μ)y (a, μ) σ () σ () σ () σ () () () + y (a, μ)y (a, μ)– y (a, μ)y (a, μ). σ () σ () σ () σ () Each of the summands in φ is aproduct of apower in μ and a product of two sums of a trigonometric and a hyperbolic functions. The terms with the highest μ-powers in φ(μ) are non-zero constant multiples of () () ⎨ μ (y (a, μ)y (a, μ)– y (a, μ)y (a, μ)) in Cases , , σ () σ () σ () σ () φ (μ)= () () ⎩ μ (y (a, μ)y (a, μ)– y (a, μ)y (a, μ)) in Cases , . σ () σ () σ () σ () Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 6 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 For the above four cases, we obtain: Case : φ (μ)= μ sinh(μa)– sin(μa) – sinh(μa)+ sin(μa) =–μ sin(μa) sinh(μa). Case : φ (μ)= μ sinh(μa)+ sin(μa) cosh(μa)+ cos(μa) – sinh(μa)– sin(μa) cosh(μa)– cos(μa) = μ sinh(μa) cos(μa)+ cosh(μa) sin(μa) . Case : φ (μ)= μ sinh(μa)– sin(μa) – sinh(μa)+ sin(μa) =–μ sin(μa) sinh(μa). Case : φ (μ)= μ sinh(μa)– sin(μa) cosh(μa)– cos(μa) – sinh(μa)+ sin(μa) cosh(μa)+ cos(μa) =–μ sinh(μa) cos(μa)+ cosh(μa) sin(μa) . We next give the asymptotic distributions of the zeros of φ (μ) with proper counting. Lemma . Case : φ has a zero of multiplicity at , simple zeros at μ ˜ =(k –) , k = ,,..., simple zeros at –μ ˜ , μ ˜ = iμ ˜ and –iμ ˜ for k = ,,..., and no other zeros. k –k k k Case : φ has a zero of multiplicity at , exactly one simple zero μ ˜ in each interval k π π ((k – ) ,(k + ) ) for positive integers k with asymptotics a a μ ˜ =(k –) + o(), k = ,,..., a simple zeros at –μ ˜ , μ ˜ = iμ ˜ and –iμ ˜ for k = ,,... , and no other zeros. k –k k k Case : φ has a zero of multiplicity at , simple zeros at μ ˜ =(k –) , k = ,,..., simple zeros at –μ ˜ , μ ˜ = iμ ˜ and –iμ ˜ for k = ,,..., and no other zeros. k –k k k Case : φ has a zero of multiplicity at , exactly one simple zero μ ˜ in each interval k π π ((k – ) ,(k + ) ) for positive integers k with asymptotics a a μ ˜ =(k –) + o(), k = ,,..., a simple zeros at –μ ˜ , μ ˜ = iμ ˜ and –iμ ˜ for k = ,,..., and no other zeros. k –k k k Proof The result is obvious in Cases and . Cases and only diﬀer in the factor with the power of μ, and the multiplicity of the corresponding zero of φ at is easy to verify. Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 7 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 The choice of the indexing for the non-zero zeros of φ in each case will become apparent later. It, therefore, remains to describe the behavior of the non-zero zeros of φ in Case . First, we are going to ﬁnd the zeros of φ on the positive real axis. One can observe that for μ =, φ (μ)= implies cosh(μa) = and cos(μa) = , whence the positive zeros of φ are those μ > for which tan(μa)+tanh(μa)=.Since tan (μa) ≥ and tanh (μa) > for all x ∈ R where the functions are deﬁned, the function μ → tan(μa)+ tanh(μa)is increasing π π with a positive derivative on each interval ((k – ) ,(k + ) ), k ∈ Z.Oneachofthese a a intervals, the function moves from –∞ to ∞, thus we have exactly one simple zero μ ˜ of π π tan(μa)+ tanh(μa)in each interval((k – ) ,(k + ) ), where k is a positive integer, and a a no zero in (, ). Since tanh(μa) → as μ →∞,wehave a μ ˜ =(k –) + o(), k = ,,.... a The location of the zeros on the other three half-axes follows by repeated application of φ (iμ)=–φ (μ). The proof will be complete if we show that all zeros of φ lie on the real or the imaginary axis. To this end, we observe that the product-to-sum formula for trigonometric functions gives φ (μ)= μ cosh(μa) sin(μa)+ sinh(μa) cos(μa) = μ sin ( + i)μa + sin ( – i)μa – i sin ( + i)μa + i sin ( – i)μa = μ ( – i) sin ( + i)μa +(+ i) sin ( – i)μa.(.) Putting ( + i)μa = x + iy, x, y ∈ R, it follows for μ = that φ (μ)= ⇒ sin ( + i)μa = sin ( – i)μa ⇔ sin(x + iy) = sin(y – ix) ⇔ cosh y – cos x = cosh x – cos y ⇔ cosh |y| + cos |y| = cosh |x| + cos |x|.(.) Since cosh x + cos x = cosh(x)+ cos(x) + has a positive derivative on (, ∞), this function is strictly increasing, and φ (μ) = therefore, implies by (.)that |y| = |x| and thus y = ±x.Then x + iy ± i x μ = = ( + i)a + i a is either real or pure imaginary. Proposition . For g =, there exists a positive integer k such that the eigenvalues λ , k counted with multiplicity, of the problem (.), (.)-(.), where k ∈ Z\{} in Cases and and k ∈ Z in Cases and , can be enumerated in such a way that the eigenvalues λ are k Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 8 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 ˆ ˆ ˆ pure imaginary for |k| < k , and λ =–λ for k ≥ k . For k >, we can write λ = μ ˆ , where –k k k the μ ˆ have the following asymptotic representation as k →∞: Case : μ ˆ =(k –) + o(). Case : μ ˆ =(k –) + o(). a Case : μ ˆ =(k –) + o(). Case : μ ˆ =(k –) + o(). a In particular, the number of pure imaginary eigenvalues is even in Cases and and odd in Cases and . Proof Case : A straightforward calculation gives φ(μ)=– i α + α μ cosh(μa) sin(μa)– sinh(μa) cos(μa) – iαμ sinh(μa) cos(μa)+ cosh(μa) sin(μa) – α μ cosh(μa) cos(μa)+ – μ cosh(μa) cos(μa)– – α μ sin(μa) sinh(μa). (.) Up to the constant factor iα,the second term equals φ (μ). It follows that for μ outside the zeros of φ , cos(·a)and cosh(·a), we have φ(μ)– iαφ (μ) α – φ (μ)= = iαφ (μ) iαμ cosh(μa) cos(μa) tan(μa)+ tanh(μa) α + α tan(μa) tanh(μa) + + iαμ tan(μa)+ tanh(μa) iμ tan(μa)+ tanh(μa) ( + α ) tanh(μa) + – .(.) μ tan(μa)+ tanh(μa) Fix ε ∈ (, )and for k = ,,... let R be the squares determined by the vertices (k – k,ε a π π ) ± ε ± iε, k ∈ Z. These squares do not intersect due to ε < .Since tan z =– if and a a only if z = jπ – and j ∈ Z, it follows from the periodicity of tan that the number C (ε)= min tan(μa)+ : μ ∈ R k,ε is positive and independent of ε.Since tanh(μa) → uniformly in the strip {μ ∈ C : Re μ ≥ , | Im μ|≤ } as |μ|→∞, there is an integer k (ε)such that a tan(μa)+ tanh(μa) ≥ C (ε) for all μ ∈ R with k > k (ε). k,ε By periodicity, there are numbers C (ε)> and C (ε)> such that | tan(μa)| < C (ε)and | cos(μa)| > C (ε) for all μ ∈ R and all k.Observing | cosh(μa)|≥| sinh((Re μ)a)|,itfol- k,ε lows that there is k (ε) ≥ k (ε) such that for all μ on the squares R with k > k (ε)the k,ε Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 9 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 estimate |φ (μ)| < holds. Further, we may assume by Lemma . that μ ˜ is inside R k k,ε for k > k (ε) and that no other zero of φ has this property. Hence, it follows by Rouché’s theorem that there is exactly one (simple) zero μ ˆ of φ in each R for k ≥ k (ε). Replacing k k μ with iμ only changes the sign of the second term in (.) and thus the sign of φ .Hence, the same estimates apply to corresponding squares along the other three half-axes, and we therefore have that φ has zeros ±ˆ μ , ±ˆ μ for k > k (ε) with the same asymptotic behavior k –k as the zeros ±˜ μ , ±iμ ˜ of φ as discussed in Lemma .. k k π π Next, we are going to estimate φ on the squares S , k ∈ N,whose vertices are ±k ±ik . k a a For k ∈ Z and γ ∈ R, kπ tan + iγ a = tan(iγ a)= i tanh(γ a) ∈ iR.(.) kπ Therefore, we have for μ = + iγ,where k ∈ Z and γ ∈ R,that tan(μa) < and tan(μa) ± ≥ . (.) For μ = x + iy, x, y ∈ R and x =, we have (ax+iay) –(ax+iay) e – e tanh(μa)= →± (ax+iay) –(ax+iay) e + e ˆ ˆ uniformly in y as x →±∞.Hence,there exists k ∈ N such that for all k ∈ Z, |k|≥ k and γ ∈ R, kπ tanh + iγ a – sgn(k) <.(.) a kπ It follows from (.)and (.)for μ = + iγ , k ∈ Z, |k|≥ k and γ ∈ R that tan(μa)+ tanh(μa) ≥.(.) Furthermore, we are going to make use of the estimates kπ cosh + iγ a ≥ sinh(kπ),(.) kπ cos + iγ a = cosh(γ a) ≥ , (.) which hold for all k ∈ Z with |k|≥ k and all γ ∈ R. Therefore, it follows from (.), (.)- (.) and the corresponding estimates with μ replaced by iμ that there is k such that |φ (μ)| < for all μ ∈ S with k > k . Again from the deﬁnition of φ in (.)and Rouché’s k theorem, we conclude that the functions φ and φ have the same number of zeros in the square S ,for k ∈ N with k ≥ k . k Since φ has k + zeros inside S and thus k ++ zeros inside S , it follows that φ k k+ has no large zeros other than the zeros ±ˆ μ found above for |k| suﬃciently large, and that k Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 10 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 λ = μ ˆ account for all eigenvalues of the problem (.)-(.) since each of these eigenval- ues gives rise to two zeros of φ, counted with multiplicity. By Proposition ., all eigenval- ˆ ˆ ues with non-zero real part occur in pairs λ ,–λ , which shows that we can index all such k k ˆ ˆ eigenvalues as λ =–λ . Since there is an odd number of remaining indices, the number –k k of pure imaginary eigenvalues must be odd. Case : The function φ in this case is φ(μ)=– α + μ cosh(μa) sin(μa)– sinh(μa) cos(μa) – α μ sinh(μa) cos(μa)+ cosh(μa) sin(μa) + iαμ cosh(μa) cos(μa)+ + ia μ cosh(μa) cos(μa)– + iαμ sin(μa) sinh(μa). Then all the estimates are as in Case , and the result in Case immediately follows from that in Case ifweobserve that each S for k large enough contains two fewer zeros of φ than in Case . Case : A straightforward calculation gives φ(μ)= α μ sin(μa) sinh(μa)– +α μ cos(μa) cosh(μa) – i α + α μ sin(μa) cosh(μa)+ cos(μa) sinh(μa) – iαμ sin(μa) cosh(μa)– cos(μa) sinh(μa) + – α μ . Then φ(μ)+ α φ (μ) φ (μ)= φ (μ) +α (α + α )i = cot(μa) coth(μa)+ coth(μa)+ cot(μa) μ μ iα – α + coth(μa)– cot(μa) – . μ μ sin(μa) sinh(μa) The result follows with reasonings and estimates as in the proof of Case , replacing μ by π π μ ± and μ ± i ,respectively. Case : The function φ in this case is φ(μ)=–iαμ sin(μa) sinh(μa)+ i α + α μ cos(μa) cosh(μa) – α + μ sin(μa) cosh(μa)+ cos(μa) sinh(μa) – α μ sin(μa) cosh(μa)– cos(μa) sinh(μa) + i α – α μ , and a reasoning as in Case completes the proof. Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 11 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 4 Birkhoff regularity We referto[, Deﬁnition ..] for the deﬁnition of the Birkhoﬀ regularity. Proposition . The boundary value problem (.), (.)-(.) is Birkhoﬀ regular for α > with respect to the eigenvalue parameter μ given by λ = μ . Proof The characteristic function of (.)asdeﬁnedin[, (..)] is π(ρ)= ρ –, and its k– zeros are i , k = ,...,. We can choose (k–)(j–) C(x, μ)= diag , μ, μ , μ i k,j= according to [, Theorem ...A]. The boundary condition (.)-(.)can be writtenin the form () B (λ)y = B (μ) y(a ), y (a ), y (a ), y (a ) , j = ,,,, j j j j j j where (, –g(), , ) in Case , B (μ)= ⎩ T ε in Cases r = ,,, r– ⎨ (, –iαμ ,,) in Cases and , B (μ)= ⎩ (–iαμ ,–g(), , ) in Cases and , B (μ)= , iαμ ,, , B (μ)= –iαμ ,–g(a), , , and where ε denotes the νth unit vector in C . Thus the boundary matrices deﬁned in [, (..)] are given by ⎛ ⎞ ⎛ ⎞ B ⎜ ⎟ ⎜ ⎟ B (μ) ⎜ ⎟ ⎜ ⎟ () () W (μ)= ⎜ ⎟ C(, μ), W (μ)= ⎜ ⎟ C(a, μ). ⎝ ⎠ ⎝ B (μ)⎠ B (μ) (j) p – (j) – Choosing C (μ)= diag(μ , μ , μ , μ ), it follows that C (μ) W (μ)= W + O(μ ), where ⎛ ⎞ ⎛ ⎞ r– (r–) (r–) i i i ⎜ ⎟ ⎜ ⎟ θ θ θ θ () ⎜ ⎟ () ⎜ ⎟ W = ⎜ ⎟ , W = ⎜ ⎟ , ⎝ ⎠ ⎝ iα –α –iαα⎠ –i – i j j– for Case r and θ =–i α for Cases and , while θ =(–i) for Cases and . The Birkhoﬀ j j matrices are () () W + W (I – ), (.) j j Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 12 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 where , j = ,,, are the × diagonal matrices with consecutive ones and con- secutive zeros in the diagonal in a cyclic arrangement, see [, Deﬁnition .. and Propo- sition ..]. It is easy to see that after a permutation of columns, the matrices (.)are block diagonal matrices consisting of × blocks taken from two consecutive columns () (inthe senseofcyclicarrangement)ofthe ﬁrst tworowsof W and the last two rows of () W , respectively. Hence the determinants of the Birkhoﬀ matrices (.)are (j–)(r–) j(r–) j+ j+ i i i α i α j(r–) –r ± = ±i + i α = j j+ j+ j+ –i α –i α (–i) (–i) in Cases and , i.e., r ∈{, },whereas (j–)(r–) j(r–) j+ j+ i i i α i α j(r–) –r ± = ±i i – i α =, (j–) j j+ j+ (–i) (–i) (–i) (–i) in Cases and . Thus, the problem (.), (.)-(.)isBirkhoﬀ regular. 5 Asymptotic expansions of eigenvalues Let D, as a function of μ with λ = μ , be the characteristic function of the problem (.), [m] (.)-(.) with respect to the fundamental system y , j = ,,,, with y () = δ for j j,m+ m = ,,,, where δ is the Kronecker delta. Denote by D the corresponding characteris- tic function for g = . Note that the characteristic functions D and φ considered in Sec- tion have the same zeros counted with multiplicity. Due to the Birkhoﬀ regularity, g only inﬂuences lower order terms in D. Therefore, it can be inferred that outside the interior of the small squares R ,–R , iR ,–iR around the zeros of D , |D(μ)– D (μ)| < |D (μ)| if k k k –k |μ| is suﬃciently large. Since the fundamental system y , j = ,,,, depends analytically on μ,also D and D are analytic functions. Hence, applying Rouché’s theorem both to the large squares S and to the small squares which are suﬃciently far away from the origin, it follows that the eigenvalues of the boundary value problem for general g have the same asymptotic distribution as for g = . Whence Proposition . leads to Proposition . For g ∈ C [, a], there exists a positive integer k such that the eigenval- ues λ , counted with multiplicity, of the problem (.), (.)-(.), where k ∈ Z \{} in Cases and and k ∈ Z in Cases and , canbeenumeratedinsuchawaythatthe eigenvalues λ arepureimaginary for |k| < k , and λ =–λ for k ≥ k . For k >, we can k –k k write λ = μ , where the μ have the following asymptotic representation as k →∞: k k Case : μ =(k –) + o(). Case : μ =(k –) + o(). a Case : μ =(k –) + o(). Case : μ =(k –) + o(). a In particular, the number of pure imaginary eigenvalues is even in Cases and and odd in Cases and . Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 13 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 In the remainder of the section, we are going to establish more precise asymptotic ex- pansions of the eigenvalues. According to [,Theorem ..],(.) has an asymptotic fundamental system {η , η , η , η } of the form ν– (j) i μx η (x, μ)= δ (x, μ)e ; ν = ,...,; j = ,...,; (.) ν,j where d ν– ν– –r ν– i μx –i μx –+j δ (x, μ)= μi ϕ (x)e e + o μ,(.) ν,j r dx r= and [ ] means that we omit those terms of the Leibniz expansion which contain a func- dx (k) () tion ϕ with k > – r. Since the coeﬃcient of y in (.)iszero, we have ϕ (x)=, see r [, (..)]. We will now determine the functions ϕ and ϕ . In this regard, observe that n = and l = in the notation of [, (..) and (..)], see [,Theorem ..].From[, (..)], we know that T [r] ϕ = ϕ = ε VQ ε,(.) r ,r (j–)(k–) [r] where ε is the νth unit vector in C , V =(i ) ,and Q are × matrices given j,k= [] by [, (..), (..) and (..)], that is, Q = I , [] [] [] Q – Q = Q =, (.) [] [] [] T – [] Q – Q = Q – g εε Q , (.) ––j T [] T [–j] = ε Q + k εε Q ε (ν = ,,,), (.) –j ν ν j= where k =–g, k =–g , = diag(, i, –, –i)and ε = (, , , ). Let G(x)= g(t) dt. A lengthy but straightforward calculation gives ϕ = G, ϕ = G – g,(.) and thus ν– –ν+ – ν– – i μx η (x, μ)= + i G(x)μ +(–) G(x) – g(x) μ e ν– – i μx + o μ e (.) for ν = ,,,, where {o(·)} means that the estimate is uniform in x. The characteristic function of (.), (.)-(.)is D(μ)= det γ exp(ε ) , jk jk j,k= Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 14 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 where k– ε = ε =, ε = ε = i μa, k k k k δ (, μ)– g()δ (, μ)inCase, k, k, γ = k δ (, μ)inCases r with r = ,,, k,r– ⎨ δ (, μ)– iαμ δ (, μ)inCasesand, k, k, γ = k ⎩ δ (, μ)– g()δ (, μ)+ iαμ δ (, μ) inCases and , k, k, k, γ = δ (a, μ)+ iαμ δ (a, μ), k k, k, γ = δ (a, μ)– g(a)δ (a, μ)– iαμ δ (a, μ). k k, k, k, Note that ω μa D(μ)= ψ (μ)e,(.) m= where ω = + i, ω =– + i, ω =– – i, ω = – i, ω = , and each of the functions k k– k k ψ ,..., ψ has asymptotic representations of the form c μ +c μ +···+c μ +o(μ ). k k– k It follows from (.)that –ω μa (ω –ω )μa D (μ):= D(μ)e = ψ (μ)+ ψ (μ)e ,(.) m m= π π where ω – ω =–, ω – ω =––i, ω – ω =–i, ω – ω =–– i.If arg μ ∈ [– , ], we (ω –ω )μa – sin |μ|a (ω –ω )μa m m have |e |≤ e for m = ,, and the terms ψ (μ)e for m = ,, –s can be absorbed by ψ (μ) astheyare of theform o(μ ) for any integer s.Hence,for arg μ ∈ π π [– , ], (ω –ω )μa –iμa D (μ)= ψ (μ)+ ψ (μ)e = ψ (μ)+ ψ (μ)e,(.) where ψ (μ)=[γ γ – γ γ ][γ γ – γ γ ], (.) ψ (μ)=[γ γ – γ γ ][γ γ – γ γ ]. (.) A straightforward calculation gives γ γ – γ γ =αμ +(– i)μ + α +αϕ (a) –iμ α + ϕ (a) + + α ϕ (a) + o μ,(.) γ γ – γ γ =–αμ +(+ i)μ + α –αϕ (a) –iμ α + ϕ (a) – + α ϕ (a) + o μ . (.) Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 15 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 For the other two factors in (.)and (.), we have to consider the four diﬀerent cases. Case : γ γ – γ γ =αμ +(– i)μ + o μ , γ γ – γ γ =αμ –(+ i)μ + o μ . Therefore, ψ (μ)=α μ +(– i)α α + αG(a)+ μ – i α G (a)+ α α + G(a)+ +α μ + o μ,(.) ψ (μ)=–α μ +(+ i)α α – αG(a)+ μ – i α G (a)– α α + G(a)+ +α μ + o μ.(.) Case : γ γ – γ γ =–( + i)αμ –μ + ( – i)αg()μ + o(μ), γ γ – γ γ =–( – i)αμ +μ + ( + i)αg()μ + o(μ). Thus, we have ψ (μ)=–( + i)α μ – α G(a)+ α + α μ – ( – i)μ α G (a)+ α +α G(a) – α g() + α + + o μ,(.) ψ (μ)=( – i)α μ + α G(a)– α + α μ + ( + i)μ α G (a)– α +α G(a) – α g() + α + + o μ.(.) Case : γ γ – γ γ =–iμ –(+ i)αμ + o μ , γ γ – γ γ =–iμ –(– i)αμ + o μ . Hence, we get ψ (μ)=–iαμ –(+ i) α + αG(a)+ μ – αG (a)+ α + G(a)+α +α μ + o μ,(.) ψ (μ)=iαμ +(– i) α – αG(a)+ μ – αG (a)– α + G(a)+α +α μ + o μ.(.) Case : γ γ – γ γ =–( – i)μ +iαμ + ( + i)g()μ + o μ , Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 16 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 γ γ – γ γ =( + i)μ –iαμ – ( – i)g()μ + o μ . Thus, we have ψ (μ)=–( – i)αμ + i αG(a)+α + μ +(+ i) αG (a)+ α + G(a) + αg() + α +α μ + o μ,(.) ψ (μ)=–( + i)αμ – i αG(a)–α – μ +(– i) αG (a)– α + G(a) + αg() + α +α μ + o μ.(.) We already know by Proposition . that the zeros μ of D satisfy the asymptotic repre- sentations μ = k + τ + o() as k →∞. In order to improve on these asymptotic repre- k sentations, write –m –n μ = k + τ(k), τ(k)= τ k + o k , k = ,,.... (.) k m m= Because of the symmetry of the eigenvalues, we will only need to ﬁnd the asymptotic ex- pansions as k →∞.Weknow τ from Proposition ., and it is our aim to ﬁnd τ and τ .To this end, we will substitute (.)into D (μ ) = and we will then compare the coeﬃcients k – – of k , k and k . Observe that τ τ –iμ a –iτ(k)a –iτ a – k e = e = e exp –ia + + o k k k –iτ a – = e –iaτ – a τ +iaτ + o k,(.) k k while – a aτ(k) a a τ – = + = – + o k.(.) μ πk kπ kπ k π Using (.), D (μ )= can be written as k –γ –γ –iτ a μ ψ (μ )+ μ ψ (μ )e =, (.) k k k k where γ is the highest μ-power in ψ (μ)and ψ (μ). Substituting (.)and (.)into – – (.) and comparing the coeﬃcients of k , k and k ,weget Theorem . For g ∈ C [, a], there exists a positive integer k such that the eigenvalues λ , k ∈ Z, counted with multiplicity, of the problem (.), (.)-(.), where k ∈ Z \{} in Cases and and k ∈ Z in Cases and , canbeenumeratedinsuchawaythatthe Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 17 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 eigenvalues λ arepureimaginary for |k| < k , and λ =–λ for k ≥ k , where λ = μ and k –k k k the μ have the asymptotic representations π τ τ – μ = k + τ + + + o k k a k k and the numbers τ , τ , τ are as follows: π i + α G(a) Case : τ =– , τ = + , a πα π ( + α )i a( – α + α ) G(a) τ = – + . πα π α π π i + α G(a) Case : τ =– , τ = + , a πα π i + α a –α + α a g() G(a) τ = – – + . πα α π π π π i +α G(a) Case : τ =– , τ = + , a πα π i( + α ) a( – α +α ) G(a) τ = – + . πα π α π π i α + G(a) Case : τ =– , τ = + , a πα π G(a) ag() i α + a α –α + τ = + + – . π π απ α π In particular, the number of pure imaginary eigenvalues is even in Cases and and odd in Cases and . Remark . In [] we have considered the diﬀerential equation (.)withthe same boundary conditions B , B at a as in this paper but with λ-independent boundary con- ditions at , that is, the boundary conditions B also occur in []. Whereas in []the number of pure imaginary eigenvalues is odd in each case, this number is even in Cases and of this paper. We observe that in Cases and , the λ-dependent part is the ‘dominat- ing’ part of the boundary condition B , in the sense that it has the highest μ-power arising j+k j [k] as μ from λ y ,whereas in Casesandthe λ-independent part is dominating. It may be interesting to investigate if, in general, the parity of the number of pure imaginary eigenvalues can be determined by the number of dominating λ-dependent parts in the boundary conditions. We can observe that the functions φ in theCases and are respectively thesameas in [] since the corresponding dominating terms in the boundary conditions coincide. However, the numbers τ and τ diﬀer from those of []ineachcase, whichisdue to the λ-term in the boundary condition B . Competing interests The authors do not have any competing interests. Authors’ contributions The subject of this paper is part of the PhD thesis of BZ. The subject has been suggested and supervised by MM, and the initial version of the paper has been written by BZ. The submitted version has been veriﬁed and discussed by MM and BZ. Möller and Zinsou Boundary Value Problems 2012, 2012:106 Page 18 of 18 http://www.boundaryvalueproblems.com/content/2012/1/106 Acknowledgements This research was partially supported by a grant from the NRF of South Africa, Grant number 69659. Various of the above calculations have been veriﬁed with Sage. Received: 8 May 2012 Accepted: 17 September 2012 Published: 4 October 2012 References 1. Kerimov, NB, Aliev, ZS: Basis properties of a spectral problem with a spectral parameter in the boundary condition (Russian). Mat. 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Möller, M, Zinsou, B: Spectral asymptotics of self-adjoint fourth order diﬀerential operators with eigenvalue parameter dependent boundary conditions. Complex Anal. Oper. Theory 6, 799-818 (2012). doi:10.1007/s11785-011-0162-1 15. Mennicken, R, Möller, M: Non-self Adjoint Boundary Eigenvalue Problems. North-Holland Mathematics Studies, vol. 192. Elsevier, Amsterdam (2003) doi:10.1186/1687-2770-2012-106 Cite this article as: Möller and Zinsou: Spectral asymptotics of self-adjoint fourth order boundary value problems with eigenvalue parameter dependent boundary conditions. Boundary Value Problems 2012 2012:106.
Boundary Value Problems – Springer Journals
Published: Oct 4, 2012
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