EIGENFUNCTION EXPANSIONS OF A QUADRATIC PENCIL OF DIFFERENTIAL OPERATORS WITH PERIODIC GENERALIZED POTENTIAL

In this article we obtain the eigenfunction expansions of a quadratic pencil of Sturm Liouville operators with periodic coe cients. The important point to note here is the given potential is a rst order generalized function.


introduction
The idea of expanding an arbitrary function in terms of the solutions of a secondorder dierential equation goes back to the time of Sturm and Liouville, more than a hundred years ago.The rst satisfactory proofs were constructed by various authors early in the twentieth century.The second-order linear dierential equation with real periodic coecients, commonly known as Hill's equation, has been investigated by many mathematicians.An account of much of this theory is given in [9].Further results relating to spectral theory are given in [4].A characterization of the spectrum of Hill's operator is studied in [11].The spectral problems of the quadratic pencil of dierential operators with periodic potential are investigated in [7].In the same place one can nd wide bibliography.Dierential operators with periodic generalized potential are widely used in applications to quantum and atomic physics to produce exact solvable models of complicated physical phenomena in [13,5,8,12,15].
In this paper, we study the eigenfunction problems of the following quadratic pencil of dierential equation with generalized potential (1.1) α [y] := −y + 2αλ where q(x) is a 1-periodic, real, non-negative and piecewise continuous function; δ(x) is the Dirac's delta function; α = 0 is a real number and λ is a spectral parameter.
In order to obtain an eigenfunction expansion of (1.1) we have to know the structure of spectrum and we will expose this in Sections 24.

Hill's discriminant and Floquet theory
The Hill discriminant is at the heart of the spectral theory of the periodic Sturm Liouville operator.If y(x) is a solution of (1.1), then so is also y(x + 1).But generally, y(x + 1) is not the same as y(x) and, indeed, (1.1) does not need to have a non-trivial solution with period 1, (see [4]).From (1.3), we give below that (1.1) has the property that there is a non-zero constant ρ and a non-trivial solution y(x) such that from the properties of the delta function Let θ(x, λ) and ϕ(x, λ) be linearly independent solutions of (1.2) which satisfy the initial conditions Since θ(x + 1, λ) and ϕ(x + 1, λ) are also linearly independent solutions of (1.2), they can be written as a linear combination of θ(x, λ) and ϕ(x, λ).Furthermore, every solution of (1.2) has the form where c 1 and c 2 are constants.To obtain a non-trivial solution of the system with respect to c 1 and c 2 by using the facts above and condition (2.1), the following equality must be satised This is a quadratic equation for ρ and it is satised by at least one non-zero value of ρ.Suppose that this equation has distinct solutions ρ 1 and ρ 2 .Since ρ 1 and ρ 2 are non-zero, we can dene µ 1 and µ 2 such that e µ k = ρ k , (k = 1, 2).Now dene Y k (x, λ) = e −µ k x y k (x, λ).Thus the general solution of (1.1) has the Floquet form The function F (λ) dened by is called a discriminant of (1.1) and we consider ve cases as follows (see [4, pp.67]): (1) There is a real number µ = 0 such that ρ = e µ , ρ = e −µ .Thus (2) F (λ) < −2.The situation here is the same as in (1).Here µ must now be replaced by µ + iπ in (2.4). ( There is a real number ν with 0 < ν < π (or EJQTDE, 2013 No. 76, p. 2 (4) F (λ) = ±2.Then there is only one non-trivial solution y 1 (x, λ).Let us denote the other solution by y 2 (x, λ), dened as below, such that y 1 (x, λ) and y 2 (x, λ) are linearly independent.
(5) If λ is not a real number, then the possible alternatives are: If F (λ) is real, then one of the above cases is valid.If F (λ) is not real, then there is a complex number µ such that ρ = e µ , ρ = e −µ and (1.1) has two linearly independent solutions From this denition and ve cases above, we obtain the following theorem.

Stability and instability intervals
For 0 ≤ x ≤ 1 the equation and the boundary conditions are called a t-quasi-periodic boundary problem, where t ∈ [0, 2π).
First we dene the linear operator in the Hilbert space H = L 2 (0, 1) by L t as follows: L t y := −y + q(x)y with domain Theorem 3.1.The eigenvalues of the operator L t are real and the eigenvalues λ n (t) are the values of λ which satisfy the equation F (λ) = 2 cos t.Proof.Suppose that λ is an eigenvalue of the operator L t and that y(x) is a corresponding eigenfunction such that (y, y) = 1.Taking the inner product of both sides of (3.1) with y(x) and using (3.2) we get Since α is a real number and q(x) ≥ 0, the roots of this equation are real numbers.Substituting ρ = exp(it) into (2.3),we obtain F (λ) = 2 cos t.
Proof.We suppose that t = mπ (m = 0, ±1, ±2, . ..) and y 1 (x) and y 2 (x) are linearly independent eigenfunctions corresponding to the eigenvalue λ of the operator L t .Thus, for all λ, the solutions of the equation (3.1) especially θ(x, λ) and ϕ(x, λ) can be written as a linear combination of the functions y 1 (x) and y 2 (x) and these solutions satisfy the boundary conditions (3.2).It follows that From Theorem 3.1, we arrive at cos t = e it .But this equality holds only for t = mπ (m = 0, ±1, ±2, . ..) which contradicts the assumption.
The periodic and quasi-periodic problems associated with (3.1) and (3.2) correspond to the cases when m is an even (resp.odd) number and their eigenvalues are the zeros of Theorem 3.3.The eigenvalues of the operator L t are of the second order if and only if Proof.The proof of the theorem is immediately obtained from the fact that ϕ(x, λ) and θ(x, λ) satisfy the conditions (3.2).
In addition a) Let EJQTDE, 2013 No. 76, p. 4 The proofs of these lemmas are seen by using the method in [16, p. 290].Considering all lemmas above, we can derive the following results: Corollary 3.7.The functions F (λ) ∓ 2 do not have a zero of order higher than second.
Corollary 3.8.The zeros of the function F (λ) − 2 are of the second order if and only if F (λ) has a maximum value at these zeros.The zeros of the function F (λ)+2 are of the second order if and only if F (λ) has a minimum value at these zeros.Theorem 3.9.1) Let α ± 2k , α ± 2k+1 (k = 0, ±1, ±2, . ..) be eigenvalues of the periodic and quasiperiodic boundary problem respectively.Then the numbers α ± 2k and α ± 2k+1 occur in the order In the intervals Thus the stability intervals of (1.1) are (α + k−1 , α − k ) and that the conditional stability intervals are the closures of these intervals.The instability intervals of (1.1) are (α − k , α + k ).

Nature of the spectrum of the operator L(λ)
We denote the pencil operator in L 2 (R) of the dierential expression by L(λ) and D is a maximal domain such that We note that the functions y and 2αλ (see [10]).
Let us denote the set consisting of the conditional stability intervals of (1.1) by S. We prove rst in this section that the spectrum of L(λ) denoted by σ is continuous, that is, L(λ) has no eigenvalues, and then that σ coincides with S. Proof.If L(λ) had an eigenvalue λ 0 with corresponding eigenfunction ψ(x), we would have L(λ)ψ(x) = 0. Then ψ(x) would be a non-trivial solution of (1.1).But from cases 15 of §2, (1.1) has no such non-trivial solution ψ(x) in L 2 (−∞, ∞) for any complex number λ and this nishes the proof.Theorem 4.2.The sets σ and S are identical.
Proof.We show rst that S ⊂ σ.We suppose then that if λ 0 is any point in S then λ 0 is also in σ.Referring to cases 15 of §2, there is, for λ 0 in S, at least one non-trivial solution ψ(x) of (1.1), with λ = λ 0 , such that |ρ| = 1.
On the other hand, let g(x) be any function with a continuous second derivative in [0, 1] such that Now dene a sequence {f n (x)} as follows except for any interval of length 1 at each end, we have as n → ∞.In particular, where K is a nonnegative number and does not depend on n.From (4.1), we get that L(λ 0 )f n (x) → 0 as n → ∞.It follows from Theorem 5.2.2 in [4, p. 81] that λ 0 is in σ, and therefore S ⊂ σ.
To prove the reverse inclusion σ ⊂ S, we suppose now that λ 0 is not in S and prove that λ 0 is not in σ.For λ 0 is not in S, the following three possibilities can occur: i) The analysis for F (λ 0 ) > 2 and F (λ 0 ) < −2 is virtually the same and so we write it out only for F (λ 0 ) > 2. Then there are solutions ψ 1 (x) and ψ 2 (x) of (1.1), with λ = λ 0 .Thus, we dene the Green's function G(x, ξ; λ 0 ) for the equation L(λ 0 ) = f (x) from using the method in [13] and then we dene the linear operator R by It can be seen that R is a bounded operator.This means that λ 0 is in the resolvent set of L(λ 0 ) and so λ 0 is not in σ. ii) It is enough to show that |F (λ 0 )| > 2, then the same proof works for this case.
Indeed, if |F (λ 0 )| ≤ 2, then there's at least one t 0 ∈ (−∞, ∞) which satises the equality F (λ 0 ) = 2 cos t 0 .This means that λ 0 is the eigenvalue and from Theorem 3.1 this eigenvalue is a real number.This contradicts our assumption Im λ 0 = 0. iii) There are two linearly independent solutions of (1.1) for λ = λ 0 .Hence the same proof as in i) works for this case, too.
Theorem 4.5.The number of gaps in the spectrum of L(λ) is innite and the lengths of gaps tend to innity as n → ∞, (q(x) = 0).
Proof.If we apply arguments stated in [16, p. 296] for the function
Consequently, these functions are the solutions of the t-quasi periodic boundary problem (3.1), (3.2).
Theorem 5.2.We suppose that the function f (x) is twice (continuously) dierentiable and supp f (x) ⊂ (0, 1).Then as |λ| → ∞ (5.9) So by using contour integration method, Parseval's equality, (5.5) and (5.9) we arrive at These equalities can also be derived for f (x) ∈ L 2 [0, 1].Now we will obtain the eigenfunction expansion on the real axis.Let f (x) be a continuous function and vanish except on a nite interval.Let us consider the following function (see [6]) Thus from (5.11) and (5.12), we get (5.13) Replacing f t (x) by f (x) in equalities (5.10) we obtain Furthermore, from (5.7) and (5.13) we have the following equality where λ k = λ k (t) and (5.15) Without loss of generality we can assume that f (x) is a real function.After some operations we have We integrate both sides of the equations (5.18) with respect to t over [0, π] and take into account the conditional stability set of the equation (1.1).Then substitute λ for λ k (t) in all integrals we arrive at the following expansion formulas where the functions F 1 (λ) and F 2 (λ) are obtained from formulas (5.15) and υ (λ) is given in (5.19).We note that these results have been given for dierential operators but not for the quadratic pencil with regular potential in [16] and for higher order self-adjoint dierential operators in [17].