Theoretical and Mathematical Physics, 145(1): 1457–1461 (2005)

SPECTRAL ANALYSIS OF A CLASS OF NON-SELF-ADJOINT

DIFFERENTIAL OPERATOR PENCILS WITH A GENERALIZED

FUNCTION

R. F. Efendiev∗

We investigate the spectrum and solve the inverse problem for a pencil of non-self-adjoint second-order

differential operators with a generalized function in the space L2(−∞,+∞).

Keywords: Schrodinger equation, Dirac delta function, Jost solution, spectral singularities, inverse prob-lem

1. Introduction

Wave propagation in a one-dimensional nonconservative medium in a frequency domain is describedby the Schrodinger equation

y′′+(x, λ) + λ2y+(x, λ) =

[iλp(x) + q(x)

]y+(x, λ), x ∈ R,

where R is the real axis, λ is the wave number (known as the momentum), λ2 is the energy, p(x) describesthe joint effect of absorption and generation of energy, and q(x) describes the regeneration of the forcedensity. In the nonstationary case, this equation corresponds to the wave equation

∂2u

∂x2− ∂2u

∂t2− p(x)

∂u

∂t= q(x)u, t, x ∈ R,

where the wave velocity is unity. When p(x) ≤ 0, we have pure absorption, but we do not fix the sign ofp(x). Hence, the equation

y′′−(x, λ) + λ2y−(x, λ) =

[−iλp(x) + q(x)]y−(x, λ), x ∈ R,

with the opposite sign of p(x) is also important.We now begin studying the spectrum and solving the inverse problem for a pencil L of non-self-adjoint

differential operators generated by a formal differential expression

l

(d

dx, λ

)≡ − d2

dx2+ 2λp(x) + q(x) + βδ(x) − λ2 (1)

with a generalized function in the space L2(−∞,∞). Here, δ(x) is the Dirac delta function, β < 0 is real,λ is complex, and the coefficients p(x) and q(x) are

p(x) =∞∑

n=1

pneinx,∞∑

n=1

n|pn| < ∞,

q(x) =∞∑

n=1

qneinx,

∞∑

n=1

|qn| < ∞.

(2)

∗Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan, e-mail: rakibaz@yahoo.com.

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 102–107, October, 2005.

Original article submitted December 20, 2004; revised April 6, 2005.

0040-5779/05/1451-1457 c© 2005 Springer Science+Business Media, Inc. 1457

In relation to important applications in quantum mechanics, it is interesting to investigate the spectralcharacteristics of the operator pencil L, and if at least one of the colliding particles is a fermion, it is relevantto solve the inverse problem in the presence of central and spin–orbital potentials [1], [2]. As a rule, theproblem under consideration is related to discontinuities in the physical characteristics of a medium.

We note that at p(x) = 0 and under the condition that

∫ +∞

−∞(1 + x2)q(x) dx < ∞,

where q(x) is a scalar real-valued nonnegative function on (−∞, +∞), the spectral characteristics of thisproblem were studied in [3].

The inverse problem in the case p(x) = 0 and under the condition

∫ +∞

−∞

(1 + |x|)∣∣q(x)

∣∣ dx < ∞,

where q(x) is a real-valued function that vanishes for x < 0, was studied in [4].The case β = 0 was considered in [5].Assuming that potentials have form (2), we construct the corresponding Jost solutions, introduce

the notion of the generalized normalizing number, and investigate the spectrum of the operator pencil L

in Sec. 2. In Sec. 3, we solve the inverse problem of reconstructing potentials (2) from the normalizingnumbers.

2. Constructing the resolvent and studying the spectrum of theoperator pencil L

We call a solution of the system

− y′′(x) + 2λp(x)y(x) + q(x)y(x) = λ2y(x), (3)

y(+0) = y(−0) = y(0),

y′(+0) − y′(−0) = −βy(0)(4)

a solution of the equation L(y) = 0. For Eq. (3), we have the following theorem [5].

Theorem 1. Let p(x) and q(x) have form (2). Then Eq. (3) has special solutions of the form

f±(x, λ) = e±iλx

(1 +

∞∑

n=1

ν±n einx +

∞∑

n=1

∞∑

α=n

ν±nα

n ± 2λeiαx

), (5)

where the numbers ν±n and ν±

nα are determined from the recursive relations

α2ν±α + α

α∑

n=1

ν±nα +

α−1∑

s=1

(qα−sν

±s − pα−s

s∑

n=1

ν±ns

)+ qα = 0, (6a)

α(α − n)ν±nα +

α−1∑

s=n

(qα−s ∓ npα−s)ν±ns = 0, (6b)

αν±α ±

α−1∑

s=1

ν±s pα−s ± pα = 0, (6c)

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and series (5) admits double termwise differentiation.

The functions f+(x, λ) and f−(x, λ) are linearly independent; their Wronskian W[f+(x, λ), f−(x, λ)

]=

2iλ. This follows because the Wronskian W is independent of x, the functions f±(x, λ) admit continuationsholomorphic in x to the respective lower and upper half-planes, and

limIm x→∞

f(ν)± (x, λ)e∓iλx = (±iλ)ν , ν = 0, 1.

Setting

f±n (x) = lim

λ→∓n/2(n ± 2λ)f±(x, λ) =

∞∑

α=n

V ±nαeiαxe−i(n/2)x,

we find from relation (6b) that if V +nn = 0, then V +

nα = 0 for all α > n and hence f+n (x) ≡ 0 (an analogous

relation holds for f−n (x)). Then the points ±n/2 are not singular points for the respective functions f±(x, λ).

Then

W

[f±

n (x), f∓

(x,∓n

2

)]= 0,

and the functions f∓(x,∓n/2) and f±n (x), which are solutions of the equation L(y) = 0 for λ = ∓n/2, are

hence linearly dependent. Therefore,

f±n (x) = S±

n f∓

(x,∓n

2

). (7)

Comparing the analytic expressions for these functions, we observe that

S±n = V ±

nn.

We divide the λ plane into the sectors Sν ={νπ < argλ < (ν + 1)π

}, ν = 0, 1, and redesignate f±(x, λ)

such that the conditions f+(x, λ) ∈ L2(0, +∞) and f−(x, λ) ∈ L2(−∞, 0) are satisfied. We note that everysolution y(x, λ) of Eq. (3) is a linear combination of the functions f±(x, λ) and can be written in the form

y(x, λ) =

{c0(x)f+(x, λ) + c1(x)f−(x, λ), x ∈ (−∞, 0),

c2(x)f+(x, λ) + c3(x)f−(x, λ), x ∈ (0, +∞),

where cj(x), j = 0, 3, are such that conditions (4) hold for y(x, λ).We construct the resolvent of the operator pencil L for λ ∈ Sν . For this, we solve the problem

− y′′(x) + 2λp(x)y(x) + q(x)y(x) = λ2y(x) + f(x), x �= 0, (8)

y(+0) = y(−0) = y(0),

y′(+0) − y′(−0) = −βy(0)(9)

in the space L2(−∞, +∞). Here, f(x) is an arbitrary function belonging to the space L2(−∞, +∞). Tofind the functions cj(x), j = 0, 1, we write the system of equations

c′0(x)f+(x, λ) + c′1(x)f−(x, λ) = 0,

c′0(x)f ′+(x, λ) + c′1(x)f ′

−(x, λ) = f(x),

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whence we have

c0(x) = − 12iλ

∫ x

−∞f−(t, λ)f(t) dt + c0,

c1(x) = − 12iλ

∫ 0

x

f+(t, λ)f(t) dt + c1,

where x ∈ (−∞, 0) and cj , j = 0, 1, are arbitrary numbers. Calculating analogously for the functions cj(x),j = 2, 3, we obtain the solution of Eq. (8) for λ ∈ Sν :

y(x, λ) = − 12iλ

− ∫ x

−∞ f+(x, λ)f−(t, λ)f(t) dt − ∫ 0

x f−(x, λ)f+(t, λ)f(t) dt +

+ c0f+(x, λ) + c1f−(x, λ), x ∈ (−∞, 0),

− ∫ x

0 f+(x, λ)f−(t, λ)f(t) dt − ∫ ∞x f−(x, λ)f+(t, λ)f(t) dt +

+ c2f+(x, λ) + c3f−(x, λ), x ∈ (0, +∞).By virtue of the condition y( · , λ) ∈ L2(−∞, +∞), f+(x, λ) ∈ L2(0,∞), and f−(x, λ) ∈ L2(−∞, 0), we findthat c0 = c3 = 0. We then have the formula for the solution of Eq. (8):

y(x, λ) = − 12iλ

∫ +∞

−∞G(x, t, λ)f(t) dt +

{c1f−(x, λ), x ∈ (−∞, 0),

c2f+(x, λ), x ∈ (0, +∞),λ ∈ Sν ,

with

G(x, t, λ) =

{f+(x, λ)f−(t, λ), t ≤ x,

f−(x, λ)f+(t, λ), t ≥ x.

The numbers cj , j = 1, 2, are here determined from conditions (9), namely,

c1f−(0, λ) − c2f+(0, λ) = 0,

c1

[f ′−(0, λ) − βf−(0, λ)

] − c2f′+(0, λ) = − β

2iλ

∫ +∞

−∞G(0, t, λ)f(t) dt.

Hence,

c1,2 = −βf±(0, λ)2iλD(λ)

∫ +∞

−∞G(0, t, λ)f(t) dt,

D(λ) = −∣∣∣∣∣

f−(0, λ) f+(0, λ)

f ′−(0, λ) − βf−(0, λ) f ′

+(0, λ)

∣∣∣∣∣.

As the result, the solution of problem (8), (9) becomes

y(x, λ) = − 12iλ

∫ +∞

−∞G(x, t, λ)f(t) dt +

+β

2iλD(λ)

{f+(0, λ)f−(x, λ)

∫ +∞−∞ G(0, t, λ)f(t) dt, x ∈ (−∞, 0),

f−(0, λ)f+(x, λ)∫ +∞−∞ G(0, t, λ)f(t) dt, x ∈ (0,∞).

Theorem 2. The spectrum of the operator pencil L consists of the continuum spectrum filling the

axis {−∞ < λ < +∞} on which there may exist spectral singularities coinciding with the numbers ±n/2,

n ∈ N , and no more than one eigennumber λ1 defined as a root of the equation D(λ) = 0.

Proof. It is known [6] that for β = 0, the spectrum of the operator pencil L (denoted here by L0) ispurely continuous and fills the axis {−∞ < λ < +∞}, on which spectral singularities coinciding with thenumbers ±n/2, n ∈ N , may occur. Following [7], the spectrum of the operator L may differ from that ofthe operator pencil L0 at most by a finite number of eigenvalues. Moreover, the number of these eigenvaluesdoes not exceed the operator rank Rλ − Rλ(β=0), i.e., unity. The theorem is thus proved.

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3. Inverse problem

It follows from relation (7) that the numbers S±n play the role of “normalizing” numbers for the

functions corresponding to the spectral singularities ±n/2, n = 1, 2, . . . .We pose the problem of finding the functions q(x), p(x), and β from the numbers {S±

n }. We first findthe explicit relations between the sequences {S±

n } and {V ±nα}, {V ±

n }. For this, we use identities (7) in theopen form:

∞∑

α=m

V ±nαeiαxe−i(n/2)x = S±

n ei(n/2)x

(1 +

∞∑

n=1

V ∓n einx +

∞∑

n=1

∞∑

α=n

V ∓nα

n + neiαx

).

We then find that

V ±mm = S±

m, V ±m,α+m = S±

m

(V ∓

α +α∑

n=1

V ∓nα

n + m

),

and β can be determined from the equality

β = −y′(+0) − y′(−0)y(0)

.

These relations are basic equations for determining {qα}, {pα}, and β from the numbers {S±n }.

Theorem 3. For the numbers {S±n } to be “normalizing” numbers of the operator pencil of type L

with potentials of form (2), it suffices that the conditions

∞∑

m=1

m|S∗m| = δ < ∞,

∞∑

m=1

|S∗m|

1 + m= p < 1,

where |S∗m| = max

{|S+m|, |S−

m|}, be satisfied.

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1. S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer,

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2. P. C. Sabatier, J. Math. Phys., 9, 1241–1258 (1968).

3. R. I. Kadiev Jr., Izv. Vyssh. Uchebn. Zaved. Ser. Mat., No. 7, 26–31 (1998).

4. T. Aktosun, Inverse Problems, 20, 859–876 (2004).

5. R. F. Efendiev, Dokl. Akad. Nauk Azerb., No. 4–6, 15–20 (2001).

6. R. F. Efendiev, Mat. Fiz. Anal. Geom., 11, No. 1, 114–121 (2004).

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