yerevan,september 7, 2019math.sci.am/sites/default/files/sammp19_harutyunyan.pdf · where [q] = 1...

About some inverse problems

Tigran Harutyunyan

Yerevan State University

Yerevan,September 7, 2019

Tigran Harutyunyan (YSU) About some inverse problems ... Yerevan,September 7, 2019 1 / 65

Let us denote by L(q, α, β) the Sturm-Liouville boundary-value problem

`y ≡ −y ′′ + q(x)y = µy , x ∈ (0, π), µ ∈ C, (1)

y(0) cosα + y ′(0) sinα = 0, α ∈ (0, π], (2)

y(π) cosβ + y ′(π) sinβ = 0, β ∈ [0, π), (3)

where q is a real-valued summable on [0, π] function (we writeq ∈ L1R[0, π]).

By L(q, α, β) we also denote the self-adjoint operator, generated byproblem (1)-(3) (see [Naimark:1969, Marchenko:1977,Levitan-Sargsyan:1988]). It is known, that under these conditions thespectra of the operator L(q, α, β) is discrete and consists of real, simpleeigenvalues [Naimark:1969, Marchenko:1977, Levitan-Sargsyan:1988,Harutyunyan:2008], which we denote by

µn = µn(q, α, β) = λ2n(q, α, β), n = 0, 1, 2, . . . ,

emphasizing the dependence of µn on q, α and β.


We assume that eigenvalues are enumerated in the increasing order, i.e.,

µ0(q, α, β) < µ1(q, α, β) < · · · < µn(q, α, β) < . . . .

The existence, countability and asymptotic formulae for the eigenvalues ofL(q, α, β), in the cases of smooth q, were investigated in XIX and inbegining of XX century (see, e.g., [Hobson:1908, Kneser:1903,Liouville:1836]).

The dependence of µn on q was investigated in [Isaacson-Trubowitz:1983,Isaacson-Mckean-Trubowitz:1984, Dahlberg-Trubowitz:1984,Poschel-Trubowitz:1987] for q ∈ L2R [0, π] and we will not concern to thisaspect.

The dependence of µn on α and β usually studied (see e.g.[Naimark:1969, Marchenko:1977, Levitan-Sargsyan:1988, Atkinson:1961,Zhikov:1967, Zettl:2005, Yurko:2007]) in the following sense: theboundary conditions are separated into four cases, and results, in particularthe asymptotics of the eigenvalues, are formulated separately for each case(more detailed list is in [Marchenko:1952], page 386), namely:


1) µn(q, α, β) = n2 +2

π(cotβ − cotα) + [q] + rn (q, α, β) ,

if α, β ∈ (0, π) , (4)

2) µn(q, π, β) =

(n +

1

2

)2

+2

πcotβ + [q] + rn (q, β) ,

if β ∈ (0, π) , (5)

3) µn(q, α, 0) =

(n +

1

2

)2

− 2

πcotα + [q] + rn (q, α) ,

if α ∈ (0, π) , (6)

4) µn (q, π, 0) = (n + 1)2 + [q] + rn (q) , (7)


where [q] =1

π

∫ π

0q (t) dt and rn = o(1) when n→∞, but this estimate

for rn is not uniform in α, β ∈ [0, π] and we cannot obtain (5), (6) and (7)from (4) by passing to the limit when α→ π or β → 0.

Our first aim to understand the nature of the dependence ofeigenvalues µn (q, α, β) on parameters α and β.

With this aim in the paper [Harutyunyan:2008] we have proved that thedependence of eigenvalues µn on α and β is smooth (analytic) and wehave derived one new formula (see below (8)), which takes into accountthis smooth dependence, which contains all formulae (4)–(7) as theparticular cases and in which the estimate of reminder is uniform withrespect to α, β and q.

More explicitly, in [Harutyunyan:2008] we have proved the followingassertion.


Theorem 1 (Harutyunyan:2008)

The lowest eigenvalue µ0(q, α, β) has the property:limα→0

µ0(q, α, β) = −∞, limβ→π

µ0(q, α, β) = −∞. For eigenvalues

µn(q, α, β), n = 2, 3, . . . , hold the formula

µn(q, α, β) = [n + δn(α, β)]2 + [q] + rn(q, α, β), (8)

where δn (·, ·) is the solution of the following equation

δn(α, β) =1

πarccos

cosα√[n + δn(α, β)]2 sin2 α + cos2 α

−

1

πarccos

cosβ√[n + δn(α, β)]2 sin2 β + cos2 β

, (9)

and rn = rn(q, α, β) = o(1), when n→∞, uniformly by α, β ∈ [0, π] andq from the bounded subsets of L1R [0, π] (we will write q ∈ BL1R [0, π]).


In the next picture we give the graph of µ0 (α, β) on (0, π)× (0, π) .

Picture 1


Let us note that during the proof of (8) in [Harutyunyan:2008] was provedformula µn (0, α, β) = [n + δn (α, β)]2 , n ≥ 2, i.e. for the first time wasobtained the explicit formula for zero potential.

In many respect these advantages of formula (8) stipulated by introducing(in consideration) the sequence of functions {δn (α, β)}∞n=2 , which wedefine for n ≥ 2 as

δn (α, β) :=√µn (0, α, β)− n = λn (0, α, β)− λn

(0,π

2,π

2

),

and after that we prove that δn (α, β) is the solution of (9).

Also we must note that the presence of δn (α, β) in principal term ofasymptotics of µn (q, α, β) doing the estimate of reminder term uniformwith respect to q, α, β.

Although (9) is not a representation of δn (α, β) , but only an(transcendental) equation, it is sufficiently convenient for investigation. Inparticular, using the program MATHEMATICA, we construct the graphs ofδn (α, β) for different n. Below we show the graphs of δ10 and δ100.


Picture 2


Picture 3


Picture 4


In what follows let ϕ(x , µ, γ) and ψ(x , µ, δ) denote the solutions of (1),satisfying the initial conditions

ϕ(0, µ, γ) = sin γ, ϕ′(0, µ, γ) = − cos γ, γ ∈ C, (10)

ψ(π, µ, δ) = sin δ, ψ′(π, µ, δ) = − cos δ, δ ∈ C, (11)

correspondingly.

The eigenvalues µn = µn(q, α, β), n = 0, 1, 2, . . . , of L(q, α, β) are thesolutions of the equation

Φ(µ) = Φ(µ, α, β)def= ϕ(π, µ, α) cosβ + ϕ′(π, µ, α) sinβ = 0, (12)

or the equation

Ψ(µ) = Ψ(µ, α, β)def= ψ(0, µ, β) cosα + ψ′(0, µ, β) sinα = 0. (13)


According to the well-known Liouville formula, the WronskianW (x) = W (x , ϕ, ψ) = ϕ · ψ′ − ϕ′ψ of the solutions ϕ and ψ is constant.It follows that W (0) = W (π) and

Ψ(µ, α, β) = −Φ(µ, α, β).

The functions ϕn(x) = ϕ(x , µn, α) and ψn(x) = ψ(x , µn, β),n = 0, 1, 2, . . . , are the eigenfunctions, corresponding to the eigenvalue µn.

The squares of the L2-norms of these eigenfunctions:

an = an(q, α, β) =

π∫0

|ϕn(x)|2dx , bn = bn(q, α, β) =

π∫0

|ψn(x)|2dx , (14)

are called the norming constants.


In [Harutyunyan:2008] we have proved that

∂µn(q, α, β)

∂α=

1

an(q, α, β)(15)

and∂µn(q, α, β)

∂β= − 1

bn(q, α, β)(16)

To prove Theorem 1 and in order to investigate the dependence of thespectral data on parameters α and β in papers[Harutyunyan-Navasardyan:2000, Harutyunyan:2008, Harutyunyan:2010]we introduce the conception of ”the eigenvalues function (EVF)”.

In order to give the definition, we note that arbitrary positive number γ wecan represent in the form γ = α + πn, where α ∈ (0, π] andn = 0, 1, 2, . . . ; and arbitrary δ ∈ (−∞, π) we can represent asδ = β − πm, where β ∈ [0, π) and m = 0, 1, 2, . . .


Definition 1

The function µ(γ, δ) of two arguments defined on (0,∞)× (−∞, π) byformula

µ(γ, δ) = µ(α + πn, β − πm)def= µn+m(q, α, β), (17)

where µk(q, α, β), k = 0, 1, 2, . . . , are the eigenvalues of L(q, α, β),enumerated in the increasing order, we shall call the eigenvalues’ function(EVF) of the family of the problems {L(q, α, β), α ∈ (0, π], β ∈ [0, π)}.

Remark 2

To emphasize that the EVF is generated by potential q, sometimes we willuse the notations µq (γ, δ) = µ (q, γ, δ) = µ (γ, δ) .


The meaning of this definition easy to see in following picture

Picture 5


At the next pictures we show the graph of EVF, which contain the partwith µ0

Picture 6


Next picture shows the part of the graph of function λ (γ, δ) =√µ (γ, δ)

which far from µ0

Picture 8


Of course, these graphs constructed for the case q (x) ≡ 0, but when weadd q (x) 6≡ 0 the change of graphs are not principal. Thus, we understandthe nature of dependence of the eigenvalues from parameters α and β inmeans of pictures of graphs. More deep and strong understanding of thisdependence we give in the following theorem.

Theorem 2

The EVF function µ(γ, δ) has the properties:a) for arbitrary fixed β ∈ [0, π) the function µ+(γ) = µ(γ, β) is stronglyincreasing (on γ ∈ (0,∞)) and its range of values is whole real axis(−∞,∞); for arbitrary fixed α ∈ (0, π] the function µ−(δ) = µ(α, δ) isstrongly decreasing (on δ ∈ (−∞, π)) and its range of values is (−∞,∞);b) for arbitrary (γ, δ) ∈ (0,∞)× (−∞, π) exists a neighborhoodUγ,δ ⊂ C2, on which defined one-valued analytic function µ(γ, δ) (of twocomplex variables γ and δ), which coincide with µ(γ, δ) for real argumentsγ and δ (from Uγ,δ). In other words, µ(γ, δ) is a real analytic function on(0,∞)× (−∞, π).c) the square root of µn (for µn > 0) has the form (k + m = n)


Theorem 2 (cont.)

λn (α, β) =√µ(α + πk, β − πm) =

=√µn(q, α, β) = n + δn(α, β) +

c0

2[n + δn(α, β)]+ ln

where c0 = [q], and the reminders ln = ln(q, α, β) are such that

ln = o

(1

n

)and the function

l(x)def=

∞∑n=1

ln sin[n + δn(α, β)] x (18)

is absolutely continuous on arbitrary [a, b] ⊂ (0, 2π) , i.e. l ∈ AC (0, 2π)for each (α, β) ∈ (0, π]× [0, π) and q ∈ L1R[0, π].d+) For arbitrary ε ∈ (0, π), ε 6= α


Theorem 2 (cont.)

∂µ(γ, β)

∂γ

∣∣∣∣γ=α+πn

=sin ε

sinα· µn(α, β)− µn(ε, β)

sin(α− ε)·

·∞∏k=0k 6=n

µk(ε, β)− µn(α, β)

µk(α, β)− µn(α, β), if α, β ∈ (0, π), (19)

∂µ(γ, β)

∂γ


=π(n + 1

2

)24n2

·

· [µ0(ε, β)− µn(π, β)] · [µn(π, β)− µn(ε, β)]

µ0(π, β)− µn(π, β)·

·∞∏k=1k 6=n

(k + 1

2

)2k2

· µk(ε, β)− µn(π, β)

µk(π, β)− µn(π, β),

if α = π, β ∈ (0, π) and n 6= 0, (20)Tigran Harutyunyan (YSU) About some inverse problems ... Yerevan,September 7, 2019 27 / 65

Theorem 2 (cont.)

∂µ(γ, β)

∂γ


=π

4[µ0(π, β)− µ0(ε, β)] ·

·∞∏k=1

(k + 1

2

)2k2

· µk(ε, β)− µ0(π, β)

µk(π, β)− µ0(π, β),

if α = π, β ∈ (0, π) and n = 0, (21)

∂µ(γ, β)

∂γ


=(n + 1)2(n + 1

2

)2 · µn(π, 0)− µn(ε, 0)

π·

·∞∏k=0k 6=n

(k + 1)2(k + 1

2

)2 · µk(ε, 0)− µn(π, 0)

µk(π, 0)− µn(π, 0),

if α = π, β = 0. (22)


Theorem 2 (cont.)

e+)an (q, α, β) =

[∂µ(γ, β)

∂γ


]−1=π

2[1 + sn1 (q, α, β)] sin2 α+

+π

2 [n + δn(α, β)]2[1 + sn2 (q, α, β)] cos2 α, (23)

where sni = o

(1

n

)and the functions (i = 1, 2)

Si (x)def=

∞∑n=2

sni cos[n + δn(α, β)] x (24)

are absolutely continuous functions on arbitrary segment [a, b] ⊂ (0, 2π) ,i.e. Si ∈ AC (0, 2π) .


Theorem 2 (cont.)

f +) 1

a0sin2 α− 1

π+∞∑n=1

(1

ansin2 α− 2

π

)= cotα. (25)

d−) For arbitrary η ∈ (0, π), η 6= β

∂µ(α, δ)

∂δ

∣∣∣∣δ=β−πn

=sin η

sinβ· µn(α, β)− µn(α, η)

sin(β − η)·

·∞∏k=0k 6=n

µk(α, η)− µn(α, β)

µk(α, β)− µn(α, β), if α, β ∈ (0, π), (26)


Theorem 2 (cont.)

∂µ(α, δ)

∂δ


=π(n + 1

2

)24n2

·

· [µn(α, 0)− µn(α, η)] · [µ0(α, η)− µn(α, 0)]

µ0(α, 0)− µn(α, 0)·

·∞∏k=1k 6=n

(k + 1

2

)2k2

· µk(α, η)− µn(α, 0)

µk(α, 0)− µn(α, 0),

if α ∈ (0, π) , β = 0, n 6= 0, (27)


Theorem 2 (cont.)

∂µ(α, δ)

∂δ


=π

4[µ0(α, 0)− µ0(α, η)] ·

·∞∏k=1

(k + 1

2

)2k2

· µk(α, η)− µn(α, 0)

µk(α, 0)− µn(α, 0),

if α ∈ (0, π) , β = 0, n = 0, (28)

∂µ(α, δ)

∂δ


=(n + 1)2(n + 1

2

)2 · µn(π, 0)− µn(π, η)

π·

·∞∏k=0k 6=n

(k + 1)2(k + 1

2

)2 · µk(π, η)− µn(π, 0)

µk(π, 0)− µn(π, 0),

if α = π, β = 0. (29)


Theorem 2 (cont.)

e−)bn (q, α, β) =

[∂µ(α, δ)

∂δ


]−1=π

2[1 + pn1 (q, α, β)] sin2 β+

+π

2 [n + δn(α, β)]2[1 + pn2 (q, α, β)] cos2 β, (30)

where pni = o

(1

n


Pi (x)def=

∞∑n=2

pni cos[n + δn(α, β)] x (31)

are absolutely continuous functions on arbitrary segment [a, b] ⊂ (0, 2π) ,i.e. Pi ∈ AC (0, 2π) .


Theorem 2 (cont.)

f −) 1

b0sin2 β − 1

π+∞∑n=1

(1

bnsin2 β − 2

π

)= − cotβ. (32)

g)

(∂µn(q, α, β)

∂α

∂µn(q, α, β)

∂β

)−1= −π2 sin2 α sin2 β·

·

( ∞∏k=1

µk(α, β)− µ0(α, β)

k2

)2

, if α, β ∈ (0, π) , n = 0, (33)

(∂µn(q, α, β)

∂α

∂µn(q, α, β)

∂β

)−1= −π

2

n4[µ0(α, β)− µn(α, β)]2 ·

· sin2 α sin2 β

∞∏k=1k 6=n

µk(α, β)− µn(α, β)

k2

2

, if α, β ∈ (0, π) , n 6= 0. (34)


Theorem 2 corresponds to the direct Sturm-Liouville problem. Therepresentations (19)–(22) and (26)–(29) of the derivatives of the EVF are,at the same time, according to (15) and (16), the representations of thenorming constants an and bn by two spectra. These representations wasinvestigated in detail in papers [Levitan:1964, Gasymov-Levitan:1964], butonly for the case sinα 6= 0, sinβ 6= 0, i.e. they had the formula (19) witha small difference: instead of our an (q, α, β) they usually consider

an =an (q, α, β)

sin2 α. The cases (20)–(22) and (26)–(29) have not been

considered.

What is new in the properties of EVF?a) 1) From this property follows that µ0 (q, α, β)→ −∞ when α→ 0 andµ0 (q, α, β)→ −∞ when β → π and this, in particular, implynonsemiboundness of the family of operators{L(q, α, β), α ∈ (0, π], β ∈ [0, π)}, i.e. although each operator L(q, α, β)semibounded from below for each fixed α ∈ (0, π] and β ∈ [0, π), but thefamily - not.


2) From this property follows also the alternation of eigenvalues{µn(q, α, β)}∞n=0 and {µn(q, α1, β)}∞n=0 (i.e. if 0 < α < α1 ≤ π, then· · · < µn(q, α, β) < µn(q, α1, β) < µn+1(q, α, β) < µn+1(q, α1, β) < . . . )

and also the alternation of eigenvalues {µn(q, α, β)}∞n=0 and{µn(q, α, β1)}∞n=0 (i.e. if 0 ≤ β < β1 < π, then· · · < µn(q, α, β1) < µn(q, α, β) < µn+1(q, α, β1) < µn+1(q, α, β) < . . . )

It is well-known that alternation of the two spectra corresponding to thepoint (α, β) and (α1, β) [or (α, β) and (α, β1)] it is one of necessary andsufficient conditions for solvability the inverse problem by two spectra (see[Borg:1946, Chudov:1949, Gasymov-Levitan:1964]). So, today, instead ofalternation, we can speak on increasing or decreasing.


b) It is a new property, it has not been investigated before. But thisproperty allow us to prove new uniqueness theorem for EVF:

Ifµ(q, γk , β) = µ(q, γk , β)

for all k = 1, 2, . . . , where 0 < γk → γ0 when k →∞ and γk 6= γm, ifk 6= m, then

q(x) = q(x) a.e. on [0, π] and β = β.

Id est µ(q, γ, δ) ≡ µ(q, γ, δ).

c) New asymptotics for eigenvalues which absorbed all the known results,added uniformity of estimates of the reminders, combined four formulaeinto one and allow to pass to the limit for all α, β ∈ [0, π]. Concerning lnwe have the expression


ln(q, α, β) =1

[n + δn(α, β)]

∫ π

0q(t) cos[n + δn(α, β)]tdt + O

(1

n2

)which shows that {ln} ∈ l2 for q ∈ L2[0, π]. But for q ∈ L1R[0, π], sincethere are no general estimates for the rate of decrease of Fouriercoefficients for L1[0, π] function, we get (18) as a characterization (insome sense) of the rate of decrease of ln.

d) First of all we obtain that

∂µn(q, α, β)

∂α=

1

an(q, α, β),∂µn(q, α, β)

∂β= − 1

bn(q, α, β),

that is why the formulae (19)–(22) and (26)–(29) are the representationsof the norming constants by two spectra.


Such representation was received by Gasymov and Levitan forα, β ∈ (0, π), i.e. for interior points.

We added the cases α = π, β ∈ (0, π) (α ∈ (0, π), β = 0) and α = π,β = 0.

These formulas play an important role in the constructive solution ofinverse problem.

e) We derive new asymptotic formulae for the norming constants ofSturm-Liouville problem with summable potentials, which generalize andmake more precise previously known formulae.

Moreover, our formulae take into account the smooth dependence ofnorming constants on boundary conditions.

We also find some new properties of the remainder terms of asymptotics.


By formula (17) we construct the map

L1R[0, π] 3 q → µ (·, ·) ∈ M (35)

where we state the correspondence between L1R[0, π] and surfaces µ (·, ·)defined on (0,∞)× (−∞, π) , which have the properties (17) and a), b),c), d±), e±), f ±), g) of Theorem 2.

The principal result of our work is that the map (35) is one to one.

Our second aim is to solve inverse problem using EVF.


Theorem 3

Let some function ν (·, ·) of two variables γ and δ, where γ ∈ (0,∞) andδ ∈ (−∞, π) (let us note that the arbitrary γ from (0,∞) can berepresented in the form γ = α + πk , where α ∈ (0, π] and k = 0, 1, 2, . . . ,and arbitrary δ from (−∞, π) can be represented in the form δ = β − πm,where β ∈ [0, π) and m = 0, 1, 2, . . . ,) has the properties 1)–8):

1) ν (α + πk , β − πm) = ν (α + πn, β) = ν (α, β − πn) := νn (α, β)

for arbitrary k,m = 0, 1, 2, . . . , such that k + m = n and for arbitraryα ∈ (0, π] and β ∈ [0, π) . That means ν(·, ·) corresponds to the picture 4(where instead of µn state νn);2) for each fixed δ ∈ (−∞, π) this function ν (·, δ) is strongly increasing byγ on (0,∞) and for arbitrary β ∈ [0, π) the range of ν (·, β) is the wholereal axis (−∞,∞) ;3) for each fixed γ ∈ (0,∞) this function ν (γ, ·) is strongly decreasing byδ on (−∞, π) and for arbitrary α ∈ (0, π] the range of ν (α, ·) is the wholereal axis (−∞,∞) ;


Theorem 3 (cont.)

4) ν (·, ·) is a real analytic function on (0,∞)× (−∞, π) ;5) there exist a constant c and a sequence {ln}∞n=2 such that theasymptotics (n→∞)√

νn(α, β) = n + δn(α, β) +c

2[n + δn(α, β)]+ ln (36)

hold, where ln = o

(1

n

)and the function

l(x)def=

∞∑n=1

ln sin[n + δn(α, β)] x (37)

is absolutely continuous on arbitrary [a, b] ⊂ (0, 2π) , i.e.

l ∈ AC (0, 2π) . (38)


Theorem 3 (cont.)

61) for arbitrary ε ∈ (0, π), ε 6= α

∂ν(γ, β)

∂γ


=sin ε

sinα· νn(α, β)− νn(ε, β)

sin(α− ε)·

·∞∏k=0k 6=n

νk(ε, β)− νn(α, β)

νk(α, β)− νn(α, β), if α, β ∈ (0, π), (39)

∂ν(γ, β)

∂γ


=π(n + 1

2

)24n2

·

· [ν0(ε, β)− νn(π, β)] · [νn(π, β)− νn(ε, β)]

ν0(π, β)− νn(π, β)·

·∞∏k=1k 6=n

(k + 1

2

)2k2

· νk(ε, β)− νn(π, β)

νk(π, β)− νn(π, β), if α = π, β ∈ (0, π) and n 6= 0,

(40)Tigran Harutyunyan (YSU) About some inverse problems ... Yerevan,September 7, 2019 43 / 65

Theorem 3 (cont.)

∂ν(γ, β)

∂γ


=π

4[ν0(π, β)− ν0(ε, β)] ·

·∞∏k=1

(k + 1

2

)2k2

· νk(ε, β)− ν0(π, β)

νk(π, β)− ν0(π, β),

if α = π, β ∈ (0, π) and n = 0, (41)

∂ν(γ, β)

∂γ


=(n + 1)2(n + 1

2

)2 · νn(π, 0)− νn(ε, 0)

π·

·∞∏k=0k 6=n

(k + 1)2(k + 1

2

)2 · νk(ε, 0)− νn(π, 0)

νk(π, 0)− νn(π, 0),

if α = π, β = 0. (42)


Theorem 3 (cont.)

62) for arbitrary η ∈ (0, π), η 6= β

∂ν(α, δ)

∂δ


=sin η

sinβ· νn(α, β)− νn(α, η)

sin(β − η)·

·∞∏k=0k 6=n

νk(α, η)− νn(α, β)

νk(α, β)− νn(α, β), if α, β ∈ (0, π), (43)

∂ν(α, δ)

∂δ


=π(n + 1

2

)24n2

·

· [νn(α, 0)− νn(α, η)] · [ν0(α, η)− νn(α, 0)]

ν0(α, 0)− νn(α, 0)·

·∞∏k=1k 6=n

(k + 1

2

)2k2

· νk(α, η)− νn(α, 0)

νk(α, 0)− νn(α, 0), if α ∈ (0, π) , β = 0, n 6= 0,

(44)Tigran Harutyunyan (YSU) About some inverse problems ... Yerevan,September 7, 2019 45 / 65

Theorem 3 (cont.)

∂ν(α, δ)

∂δ


=π

4[ν0(α, 0)− ν0(α, η)] ·

·∞∏k=1

(k + 1

2

)2k2

· νk(α, η)− νn(α, 0)

νk(α, 0)− νn(α, 0),

if α ∈ (0, π) , β = 0, n = 0, (45)

∂ν(α, δ)

∂δ


=(n + 1)2(n + 1

2

)2 · νn(π, 0)− νn(π, η)

π·

·∞∏k=0k 6=n

(k + 1)2(k + 1

2

)2 · νk(π, η)− νn(π, 0)

νk(π, 0)− νn(π, 0),

if α = π, β = 0. (46)


Theorem 3 (cont.)

71)

[∂ν(γ, β)

∂γ


]−1=π

2[1 + sn1] sin2 α+

+π

2 [n + δn(α, β)]2[1 + sn2] cos2 α, (47)

where sni = o

(1

n


Si (x)def=

∞∑n=2

sni cos[n + δn(α, β)] x (48)

are absolutely continuous functions on arbitrary [a, b] ⊂ (0, 2π) , i.e.

Si ∈ AC (0, 2π) . (49)


Theorem 3 (cont.)

72)

[∂ν(α, δ)

∂δ


]−1=π

2[1 + pn1] sin2 β+

+π

2 [n + δn(α, β)]2[1 + pn2] cos2 β, (50)

where pni = o

(1

n


Pi (x)def=

∞∑n=2

pni cos[n + δn(α, β)] x (51)

are absolutely continuous functions on arbitrary [a, b] ⊂ (0, 2π) , i.e.

Pi ∈ AC (0, 2π) . (52)


Theorem 3 (cont.)

8) if α ∈ (0, π) , then

∂ν0 (α, β)

∂αsin2 α− 1

π+∞∑n=1

(∂νn (α, β)

∂αsin2 α− 2

π

)= cotα. (53)

and if β ∈ (0, π) , then

∂ν0 (α, β)

∂βsin2 β − 1

π+∞∑n=1

(∂νn (α, β)

∂βsin2 β − 2

π

)= − cotβ. (54)

9)

(∂νn(α, β)

∂α

∂νn(α, β)

∂β

)−1= −π2 sin2 α sin2 β·

·

( ∞∏k=1

νk(α, β)− ν0(α, β)

k2

)2

, if α, β ∈ (0, π) , n = 0, (55)


Theorem 3 (cont.)

(∂νn(α, β)

∂α

∂νn(α, β)

∂β

)−1= −π

2

n4[ν0(α, β)− νn(α, β)]2 ·

· sin2 α sin2 β

∞∏k=1k 6=n

νk(α, β)− νn(α, β)

k2

2

, if α, β ∈ (0, π) , n 6= 0. (56)

Then there exists unique q ∈ L1R [0, π] (unique in L1 sense), such that

ν (γ, δ) = µq (γ, δ) (57)

for arbitrary (γ, δ) ∈ (0,∞)× (−∞, π) , i.e. function ν (·, ·) be the EVF ofthe family {L(q, α, β);α ∈ (0, π] , β ∈ [0, π)} .


In particular we have the algorithm of construction of the potential q(·) bya function ν which has the properties 1)–9) of the Theorem 3.

Gelfand and Levitan [Gelfand-Levitan:1951, Yurko:2007] state the inverseproblem (for regular Sturm-Liouville problem) in a such way:What kind must be the sequences {µn}∞n=0 and {an}∞n=0 to be the spectraldata of a Sturm-Liouville problem L(q, h,H)?

And the answer was the following: If holds the asymptotics

√µn = n +

c

n+

æn

n, an =

π

2+

æn1

n,

where c = const, {æn}∞n=0, {æn1}∞n=0 ∈ l2, then there exist q, h,H suchthat µn = µn(q, h,H), an = an(q, h,H), n = 0, 1, . . . .


Our statement is the following:What kind must be the surface ν(·, ·) (defined on (0,∞)× (−∞, π)) to bethe EVF of a family of Sturm-Liouville problems?

Change of α and β changes the sequences {µn}∞n=0 and {an}∞n=0, andmaybe they have the similar asymptotics, but each pair of sequences{µn}∞n=0 and {an}∞n=0 generate one triple (q, h,H).

Is the function q is the same in all cases?The properties of EVF allow us to prove that the q is the same.


One of the remarkable theorems in inverse problems (and historically thefirst) is the theorem of Ambarzumian. It’s known that the eigenvaluesµn(0, π/2, π/2) of problem L(0, π/2, π/2) are n2, n ≥ 0. The classicalAmbarzumian’s theorem states:

Theorem 4 (Ambarzumian:1929)

If µn(q, π/2, π/2) = µn(0, π/2, π/2) = n2, for all n ≥ 0, then q(x) ≡ 0.

The last generalization of Ambarzumian theorem is:

Theorem 5 (Isaacson-McKean-Trubowitz:1984)

Let q ∈ L2R(0, π).If µn(q, α, π − α) = µn(0, α, π − α), for all n ≥ 0, then q(x) ≡ 0.


Let L = L(q, α, β) and L0 = L(q0, α0, β0) be two operators. The followingassertion is usually called uniqueness theorem of Marchenko:

Theorem 6 (Marchenko:1950)

Let q ∈ L1R(0, π). If

µn(q, α, β) = µn(q0, α0, β0), (58)

an(q, α, β) = an(q0, α0, β0), (59)

for all n ≥ 0, then α = α0, β = β0 and q(x) = q0(x) almost everywhere.

Recently [Ashrafyan:2017] was proved the result, which, in some sense, isa generalization of Marchenko uniqueness theorem:


Theorem 7 (Ashrafyan:2017)

Let q′ ∈ L2R(0, π). If

µn(q, α0, β) = µn(q0, α0, β0), (60)

an(q, α0, β) ≥ an(q0, α0, β0), (61)

for all n ≥ 0, then β = β0 and q(x) ≡ q0(x).

This kind of uniqueness theorem has not been considered before. Themain difference between Theorems 6 and 7 is that the equality in (59) isreplaced with inequality in (61). Note, it is assumed q′ ∈ L2R(0, π) insteadof general q ∈ L1R(0, π), since the proof is based on the results of[Jodeit-Levitan:1997]. And the parameter α of boundary condition is inadvance fixed α = α0.


Thanks for your attention!


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Marchenko, V.A. The Sturm-Liouville Operators and theirApplications, Naukova Dumka, Kiev, (in Russian), 1977.

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Marchenko, V.A. “Some questions of the theory of one-dimensionallinear differential operators of the second order.” Trudy Moskov. Mat.Obsh., 1, (in Russian), (1952): 327–420.

Harutyunyan, T.N. Eigenvalue Functions of Family of Sturm-Liouvilleand Dirac Operators, Doctoral Thesis, Yerevan, (in Russian), 2010.

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Jodeit, Max, Jr. and Levitan B.M. “The isospectrality problem for theclassical Sturm-Liouville equation.” Adv. Differential Equations, 2, no.2, (1997): 297–318.

Harutyunyan, T.N. “Isospectral Dirac operators.” Izv. Nats. Akad.Nauk Armenii, Mathematika, 29, no. 2, (in Russian), (1994): 3–14.


References VII

Harutyunyan, T.N. “On an inverse problem for the canonical Diracsystem.” Izv. Nats. Akad. Nauk Armenii, Mathematika, 41, no. 1, (inRussian), (2006): 5–14.

Yang, Chuan-Fu and Huang, Zhen-You “Inverse spectral problems for2m-dimensional canonical Dirac operators.” Inverse Problems, 23, no.6, (2007): 2565–2574.

Horvath, M. “On a theorem of Ambarzumian.” Proceedings of theRoyal Society of Edinburgh A, 131, no. 4, (2001): 809–907.

Ashrafyan, Yu.A. and Harutyunyan, T.N. “Isospectral Diracoperators.” Electronic Journal of Qualitative Theory of DifferentialEquations, 2017, no. 4, (2017): 1–9.


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