Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra

14 Pages·0.21 MB·English

by  R. O. Hryniv

INVERSE SPECTRAL PROBLEMS FOR STURM-LIOUVILLE
OPERATORS WITH SINGULAR POTENTIALS, II.
RECONSTRUCTION BY TWO SPECTRA†
3
ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK
0
0
2
Abstract. We solve the inverse spectral problem of recovering the singular poten-
n tials q ∈W−1(0,1) of Sturm-Liouville operators by two spectra. The reconstruction
2
a
algorithmispresentedandnecessaryandsufficientconditionsontwosequencestobe
J
spectral data for Sturm-Liouville operators under consideration are given.
8
1

]
P
1. Introduction
S
h. Suppose that q is a real-valued distribution from the space W−1(0,1). A Sturm-
2
t
a Liouville operator T in a Hilbert space H := L2(0,1) corresponding formally to the
m differential expression
[
d2
(1.1) − +q
1
dx2
v
3 can be defined as follows [18]. We take a distributional primitive σ ∈ H of q, put
9
1 l (u) := −(u′ −σu)′ −σu′
σ
1
0 for absolutely continuous functions u whose quasi-derivative u[1] := u′ − σu is again
3
absolutely continuous, and then set
0
/
h Tu := lσ(u)
t
a for suitable u (see Section 2 for rigorous definitions). When considered on this natural
m
domain subject to the boundary conditions
:
v
(1.2) u[1](0)−Hu(0) = 0, u[1](1)+hu(1) = 0, H,h ∈ R∪{∞},
i
X
the operatorT = T(σ,H,h) becomes [17, 18] selfadjoint, bounded below, and possesses
r
a a simple discrete spectrum accumulating at +∞. (If H = ∞ or h = ∞, then the
corresponding boundary condition is understood as a Dirichlet one.)
Weobserve thatamongthesingular potentialsincluded inthisscheme thereare, e.g.,
the Dirac δ-potentials and Coulomb 1/x-like potentials that have been widely used in
quantum mechanics to model particle interactions of various types (see, e.g., [1, 2]). It
is also well known that singular potentials of this kind usually do not produce a single
Sturm-Liouvilleoperator. StillformanyreasonsT(σ,H,h)canberegardedasanatural
operator associated with differential expression (1.1) and boundary conditions (1.2).
For instance, l (u) = −u′′+qu in the sense of distributions and hence for regular (i.e.,
σ
locally integrable) potentials q the above definition of T(σ,H,h) coincides with the
classical one. Also T(σ,H,h) depends continuously in the uniform resolvent sense on
Date: February 8, 2008.
2000 Mathematics Subject Classification. Primary 34A55, Secondary 34B24,34L05, 34L20.
Key words and phrases. Inverse spectral problems, Sturm-Liouville operators,singular potentials.
†TheworkwaspartiallysupportedbyUkrainianFoundationforBasicResearchDFFDundergrant
No. 01.07/00172.
1
2 R. O. HRYNIV,YA.V. MYKYTYUK
σ ∈ H, which allows us to regard T(σ,H,h) for singular q = σ′ as a limit of regular
Sturm-Liouville operators.
Sturm-Liouville and Schr¨odinger operators with distributional potentials from the
space W−1(0,1) and W−1 (R) respectively have been shown to possess many prop-
2 2,unif
erties similar to those for operators with regular potentials, see, e. g., [9, 10, 17, 18].
In particular, just as in the regular case, the potential q = σ′ and boundary condi-
tions (1.2) of the Sturm-Liouville operator T(σ,H,h) can be recovered via the cor-
responding spectral data, the sequences of eigenvalues and so-called norming con-
stants [11].
The main aim of the present paper is to treat the following inverse spectral prob-
lem. Suppose that {λ2} is the spectrum of the operator T(σ,H,h ) and {µ2} is the
n 1 n
spectrum of the operator T(σ,H,h ) with h 6= h . Is it possible to recover σ, H,
2 2 1
h , and h from the two given spectra? More generally, the task is to find necessary
1 2
and sufficient conditions for two sequences (λ2) and (µ2) so that they could serve as
n n
spectra of Sturm-Liouville operators on (0,1) with some potential q = σ′ ∈ W−1(0,1)
2
and boundary conditions (1.2) for two different values of h and, then, to present an
algorithm recovering these two operators (i.e., an algorithm determining σ and the
boundary conditions).
For the case of a regular (i.e., locally integrable) potential q the above problem was
treated by Levitan and Gasymov [14, 15] when h and h are finite and by Mar-
1 2
chenko [16, Ch. 3.4] when one of h ,h is infinite. Earlier, Borg [4] proved that two
1 2
such spectra determine the regular potential uniquely.
Observe that T(σ + h′,H − h′,h + h′) = T(σ,H,h) for any h′ ∈ C, so that it
is impossible to recover σ,H,h , and h uniquely; however, we shall show that the
1 2
potential q = σ′ and the very operators T(σ,H,h ) and T(σ,H,h ) are determined
1 2
uniquely by the two spectra. The reconstruction procedure consists in reduction of
the problem to recovering the potential q = σ′ and the boundary conditions based on
the spectrum and the sequence of so-called norming constants. The latter problem
has been completely solved in our paper [11], and this allows us to give a complete
description of the set of spectral data and to develop a reconstruction algorithm for
the inverse spectral problem under consideration.
The organization of the paper is as follows. In the next two sections we restrict
ourselves to the case where H = ∞ and h ,h ∈ R. In Section 2 refined eigenvalue
1 2
asymptotics is found and some other necessary conditions on spectral data are es-
tablished. These results are then used in Section 3 to determine the set of norming
constants and completely solve the inverse spectral problem. The case where one of h
1
and h is infinite is treated in Section 4, and in Section 5 we comment on the changes
2
to be made if H is finite. Finally, Appendix A contains some facts about Riesz bases
of sines and cosines in L (0,1) that are frequently used throughout the paper.
2
2. Spectral asymptotics
In this section (and until Section 4) we shall consider the case of the Dirichlet bound-
ary condition at the point x = 0 and the boundary conditions of the third type at the
point x = 1 (i. e., to the case where H = ∞ and h ,h ∈ R). We start with the precise
1 2
definition of the Sturm-Liouville operators under consideration.
Supposethatq isareal-valueddistributionfromtheclassW−1(0,1)andσ ∈ Hisany
2
of its real-valued distributional primitives. For h ∈ R we denote by T = T(σ,∞,h)
σ,h
RECONSTRUCTION BY TWO SPECTRA 3
the operator in H given by
T u = l (u) := −(u[1])′ −σu′
σ,h σ
and the boundary conditions
u(0) = 0, u[1](1)+hu(1) = 0, h ∈ R
(we recall that u[1] stands for the quasi-derivative u′ − σu of u). More precisely, the
domain of T equals
σ,h
D(T ) = {u ∈ W1[0,1] | u[1] ∈ W1[0,1], l (u) ∈ H, u(0) = 0, u[1](1)+hu(1) = 0}.
σ,h 1 1 σ
It follows from [18] that so defined T is a selfadjoint operator with simple discrete
σ,h
spectrum accumulating at +∞. The results of [19] also imply that the eigenvalues λ2
n
of T , when ordered increasingly, obey the asymptotic relation λ = π(n−1/2)+λ
σ,h n n
with an ℓ -sequence (λ ). However, we shall need a more precise form of λ , and shall
2 n n e
derive it next.
e e
To start with, we introduce an operator T defined by T u = l (u) on the domain
σ σ σ
D(T ) = {u ∈ W1[0,1] | u[1] ∈ W1[0,1], l (u) ∈ H, f(0) = 0}.
σ 1 1 σ
In other words, T is a one-dimensional extension of T discarding the boundary
σ σ,h
condition at the point x = 1. We invoke now the following results of [12] on the
operatorT . Just asintheclassical situationwithregular potentialq theoperatorsT
σ ±σ
turn out to be similar to the potential-free operator T ; the similarity is performed by
0
the transformation operators I + K±, where K± are integral Volterra operators of
σ σ
Hilbert-Schmidt class given by
x
K±u(x) = k±(x,t)u(t)dt.
σ σ
Z
0
It follows that anyfunction ufromD(T ) hasthe formu = (I+K+)v forsome function
σ σ
v ∈ D(T ) (i.e., for some v ∈ W2[0,1] satisfying the boundary condition v(0) = 0) and
0 2
that T (I + K+)v = −(I + K+)v′′. Moreover, there exists a Hilbert-Schmidt kernel
σ σ σ
r (x,t) such that
σ
x
[(I +K+)v][1](x) = [(I +K−)v′](x)+ r (x,t)v(t)dt
σ σ σ
Z
0
for every v ∈ D(T ). Also, for every fixed x ∈ [0,1] the functions k±(x,·) and r (x,·)
0 σ σ
belong to H.
Due to the above similarity of T and T , for any nonzero λ ∈ C the function
σ 0
x
u(x,λ) := sinλx+ k+(x,t)sinλtdt
σ
Z
0
is an eigenfunction of the operator T corresponding to the eigenvalue λ2 ∈ C. If in
σ
addition u(x,λ) satisfies the boundary condition u[1](1)+hu(1) = 0, then u(x,λ) is also
an eigenfunction of the operator T corresponding to the eigenvalue λ2. Therefore the
σ,h
nonzero spectrum of T consists of the squared nonzero λ-solutions of the equation
σ,h
1 1 1
cosλ+ k−(1,t)cosλtdt+ r (1,t)sinλtdt
Z σ λ Z σ
(2.1) 0 0
h h 1
+ sinλ+ k+(1,t)sinλtdt = 0.
λ λ Z σ
0
4 R. O. HRYNIV,YA.V. MYKYTYUK
Recall that the operator T is bounded below and hence after addition to q a suitable
σ,h
constant T becomes positive. Therefore without loss of generality we may assume
σ,h
that all zeros of equation (2.1) are real. Observe also that due to the symmetry we
may consider only the positive zeros. The asymptotics of these zeros is given in the
following theorem.
Theorem 2.1. Suppose that σ ∈ H and h ∈ R are such that the operator T is
σ,h
positive and denote by λ2 < λ2 < ··· the eigenvalues of T . Then the numbers λ
1 2 σ,h n
satisfy the relation
h
λ = π(n−1/2)+ +δ (h)+γ ,
n n n
πn
in which the sequences δ (h) ∞ and (γ )∞ belong to ℓ and ℓ respectively with γ
n n=1 n n=1 1 2 n
independent of h. (cid:0) (cid:1)
Proof. Recall that λ → ∞ as n → ∞ and that the functions k±(1,t) and r (1,t)
n σ σ
belong to H. Therefore (2.1) and the Riemann lemma imply that
1
cosλ = − k−(1,t)cosλ tdt+O(1/λ ) = o(1)
n σ n n
Z
0
as n → ∞. By the standard Rouch´e-type arguments (see details, e.g., in [16, Ch. 1.3])
we conclude that λ = π(n − 1/2) + λ with λ → 0 as n → ∞. As a result (see
n n n
Appendix A), thesystem {sinλ x} turnsouttobeaRiesz basisofHandthesequences
n
e e
(c )∞ and (c′ )∞ with
n n=1 n n=1
1 1
c := k+(1,t)sinλ tdt, c′ := r (1,t)sinλ tdt
n σ n n σ n
Z Z
0 0
belong to ℓ . Relation (2.1) now implies that
2
1 cosλ hc +c′
sinλ = (−1)n+1 k−(1,t)cosλ tdt+h n +(−1)n+1 n n =
n Z σ n λ λ
0 ne n
e h
= γ + +δ′ +δ′′,
n πn n n
where
1
γ := (−1)n+1 k−(1,t)cosπ(n−1/2)tdt,
n σ
Z
0
1
(2.2) δ′ := (−1)n+1 k−(1,t)[cosλ t−cosπ(n−1/2)t]dt,
n σ n
Z
0
hcosλ h hc +c′
δ′′ := n − +(−1)n+1 n n.
n λ πn λ
ne n
Next we observe that the systems {cosπ(n−1/2)t} and {cosλ t} form Riesz bases
n
of H, so that (γ ) and (δ′) belong to ℓ . It follows that (sinλ ) ∈ ℓ and therefore
n n 2 n 2
(λ ) ∈ ℓ . Lemma A.2 now implies that the numbers δ′ form an ℓ -sequence, and
n 2 n e 1
simple arguments justify the same statement about δ′′.
e n
Thus we have shown that
h
sinλ = γ + +δ′ +δ′′
n n πn n n
e
RECONSTRUCTION BY TWO SPECTRA 5
with the sequences (γ ) from ℓ and (δ′) and (δ′′) from ℓ . Applying arcsin to both
n 2 n n 1
parts of this equality, we arrive at
h
λ = γ + +δ
n n n
πn
with some ℓ -sequence (δ ), and tehe theorem is proved. (cid:3)
1 n
It was shown in [11] that any sequence (λ2) of pairwise distinct positive numbers
n
satisfying the relation λ = π(n−1/2)+λ with an ℓ -sequence (λ ) is the spectrum of
n n 2 n
some Sturm-Liouville operator T with potential q = σ′ from W−1(0,1), the Dirichlet
σ,h e e2
boundary condition at x = 0, and third type boundary condition (1.2) at x = 1 with
some h ∈ R. Suppose that (µ2) is the spectrum of a Sturm-Liouville operator with the
n
same potential q, the Dirichlet boundary condition at x = 0, and third type boundary
condition (1.2) at x = 1 with a different h ∈ R. We ask whether there exists any
relation between the two spectra.
The first restriction, just as in the regular case, is that the spectra should interlace.
Lemma 2.2. Suppose that σ ∈ H is real valued and that sequences {λ2} and {µ2} are
n n
the spectra of Sturm-Liouville operators T and T with distinct h ,h ∈ R. Then
σ,h1 σ,h2 1 2
these sequences interlace.
Proof. Denote by u = u(x,λ) a solution of equation l (u) = λu satisfying the boundary
σ
condition u(0) = 0. We recall that u being a solution of l (u) = λu means that
σ
d u[1] −σ −σ2 −λ u[1]
(2.3) =
dx(cid:18) u (cid:19) (cid:18) 1 σ (cid:19)(cid:18) u (cid:19)
and hence u enjoys the standard uniqueness properties of solutions to first order differ-
ential systems withregular (i.e. locallyintegrable) coefficients. Thenumbers λ andµ
n n
arethensolutionsoftheequationsu[1](1,λ)+h u(1,λ) = 0andu[1](1,λ)+h u(1,λ) = 0
1 2
respectively.
We introduce a continuous function θ(x,λ) through cotθ(x,λ) ≡ u[1]/u. After dif-
ferentiating both sides of this identity in x and using (2.3) we get
θ′ −σu′u−u[1]u′
− = −λ = −(cotθ+σ)2 −λ,
sin2θ u2
or
θ′ = λsin2θ+(cosθ +σsinθ)2.
It follows from [3, Ch. 8.4] or [7, proof of Theorem XI.3.1] that the function θ(1,λ)
is strictly increasing in λ; hence solutions of the equations cotθ(1,λ) = −h and
1
cotθ(1,λ) = −h interlace, and the proof is complete. (cid:3)
2
The next restriction concerns asymptotics of λ and µ .
n n
Lemma 2.3. Suppose that (λ2) and (µ2) are spectra of Sturm-Liouville operators T
n n σ,h1
and T with real-valued σ ∈ H and h ,h ∈ R. Then there exists an ℓ -sequence (ν )
σ,h2 1 2 2 n
such that
h −h ν
1 2 n
(2.4) λ −µ = + .
n n
πn n
Proof. Put λ := λ −π(n−1/2) and µ := µ −π(n−1/2) so that λ −µ = λ −µ .
n n n n n n n n
Recall that
e e
e e
h −h
sinλ −sinµ = 1 2 +δ′(h )−δ′(h )+δ′′(h )−δ′′(h )
n n πn n 1 n 2 n 1 n 2
e
e
6 R. O. HRYNIV,YA.V. MYKYTYUK
with the quantities δ′(h),δ′′(h) of (2.2). Relation
n n
1
δ′(h )−δ′(h ) = (−1)n+1 k−(1,t)[cosλ t−cosµ t]dt
n 1 n 2 σ n n
Z
0
and Lemma A.2 show that δ′(h ) − δ′(h ) = (λ −µ )ν′ for some ℓ -sequence (ν′).
n 1 n 2 n n n 2 n
Also there exists an ℓ -sequence (ν′′) such that δ′′(h )−δ′′(h ) = ν′′/n. Finally, using
2 n en 1 n 2 n
the relations e
sinλ −sinµ = (λ −µ )cosν ,
n n n n n
where ν are points betweeneλ and µ (soethat (1 − cosν ) is an ℓ -sequence), and
n n en e e n 2
combining the above relations we arrive at
e
e e e
h −h ν′′
(λ −µ )(cosν −ν′) = 1 2 + n,
n n n n πn n
which implies (2.4). e e e (cid:3)
Corollary 2.4. Under the above assumptions, lim (λ2 −µ2) = 2(h −h ).
n→∞ n n 1 2
3. Reduction to the inverse spectral problem by one spectrum and
norming constants
Suppose that σ ∈ H is real-valued and that {λ2} and {µ2} are eigenvalues of the
n n
operators T and T respectively introduced in the previous section. We shall show
σ,h1 σ,h2
how the problem of recovering σ, h , and h based on these spectra can be reduced
1 2
to the problem of recovering σ and h based on the spectrum {λ2} of T and the
1 n σ,h1
so-called norming constants {α } defined below. This latter problem for the class of
n
Sturm-Liouville operators with singular potentials from W−1(0,1) is completely solved
2
in [11]; see also [13, 16] for the regular case of integrable potentials.
Denote by u (x,λ) and u (x,λ) solutions of the equation l (u) = λ2u satisfying the
1 2 σ
[1]
initial conditions u (1,λ) = u (1,λ) = 1 and u (1,λ) = −h , j = 1,2. Then u (x,λ )
1 2 j j 1 n
are the eigenfunctions of the operator T corresponding to the eigenvalues λ2, and
σ,h1 n
we put
1
α := 2 |u (x,λ )|2dx.
n 1 n
Z
0
Our next aim is to show that α can be expressed in terms of the spectra {λ2,µ2} only.
n k k
Set Φ (λ) := u (0,λ), j = 1,2; then zeros of Φ and Φ are precisely the numbers
j j 1 2
±λ and ±µ respectively. We show that Φ and Φ uniquely determine α and are
n n 1 2 n
uniquely determined by their zeros.
Lemma 3.1. The norming constants α satisfy the following equality:
n
h −h Φ′(λ )
(3.1) α = 1 2 1 n .
n
λ Φ (λ )
n 2 n
Proof. The proof is rather standard and the details can be found, e.g., in [13], so we
shall only sketch its main points here.
Put m(λ) := −Φ (λ)/Φ (λ); then the function f(x,λ) := u (x,λ) + m(λ)u (x,λ)
2 1 2 1
for all λ ∈ C not in the spectrum of T satisfies the equation l (f) = λ2f and the
σ,h1 σ
boundary condition f(0,λ) = 0. In other words, f(x,λ) is an eigenfunction of the
RECONSTRUCTION BY TWO SPECTRA 7
operator T corresponding to the eigenvalue λ2. Since u (x,λ ) is an eigenfunction of
σ 1 n
the operator T corresponding to the eigenvalue λ2, we have
σ n
1
(λ2 −λ2) f(x,λ)u (x,λ )dx = (T f,u )−(f,T u )
n 1 n σ 1 σ 1
Z
0
= −f[1](1,λ)u (1,λ)+f(1,λ)u[1](1,λ ) = h −h .
1 n 2 1
On the other hand,
1 1
(λ2 −λ2) f(x,λ)u (x,λ )dx = (λ2 −λ2) u (x,λ)u (x,λ )dx
n 1 n n 2 1 n
Z Z
0 0
Φ (λ) 1
−(λ2 −λ2) 2 u (x,λ)u (x,λ )dx,
n Φ (λ) Z 1 1 n
1 0
and after combining the two relations and letting λ → λ we arrive at
n
h −h Φ′(λ )
α = 1 2 1 n .
n
λ Φ (λ )
n 2 n
(cid:3)
The proof is complete.
The following statement has appeared in many variants in numerous sources, but we
include its proof here for the sake of completeness.
Lemma 3.2. In order that a function f(λ) admit the representation
1
(3.2) f(λ) = cosλ+ g(t)cosλtdt
Z
0
with an H-function g, it is necessary and sufficient that
∞ (f2 −λ2)
(3.3) f(λ) = k ,
π2(k −1/2)2
kY=1
where f := π(k −1/2)+f and (f ) is an ℓ -sequence.
k k k 2
Proof. Necessity. The funcetion f oef (3.2) is an even entire function of order 1, and the
standard Rouch´e-type arguments (see the analysis of Section 2) show that zeros ±f
k
of f have the asymptotics f = π(k−1/2)+f with (f ) ∈ ℓ . On the other hand, up
k k k 2
to a scalar factor C, the function f is recovered from its zeros as
e e
∞ λ2 ∞ (f2 −λ2)
(3.4) f(λ) = C 1− = C k ,
f2 1 π2(k −1/2)2
kY=1(cid:16) k(cid:17) Yk=1
where
∞ π2(k −1/2)2
C = C .
1 f2
kY=1 k
To determine C , we observe that, in view of (3.2), lim f(iν)/cosiν = 1 and that
1 ν→∞
∞ (π2(k −1/2)2 −λ2)
(3.5) cosλ = ;
π2(k −1/2)2
Yk=1
comparing now (3.4) and (3.5), we conclude that C = 1 and necessity is justified.
1
Sufficiency. Suppose now that f has a representation of the form (3.3), in which
f = π(k − 1/2) + f with (f ) ∈ ℓ . We assume first that the zeros f are pairwise
k k k 2 k
distinct. Then duetotheasymptotics off thesystem (cosf x)∞ isaRiesz basisofH
e e k k k=1
8 R. O. HRYNIV,YA.V. MYKYTYUK
(see Lemma A.1) and the numbers c := −cosf = (−1)ksinf define an ℓ -sequence.
k k k 2
Therefore there exists a unique function g ∈ H with Fourier coefficients c in the Riesz
k
e
basis (cosf x)∞ . Now the function
k k=1
1
(3.6) f(λ) := cosλ+ g(x)cosλxdx
Z
0
e
is an even entire function of order 1 and has zeros at the points ±f . It follows that f
k
and f differ by a scalar factor, which as before is shown to equal 1.
e
Due to the asymptotics of the zeros f only finitely many of them can repeat. If,
k
e.g., f = f = ··· = f is a zero of f of multiplicity p = p(n), then we include
n n+1 n+p−1
to the above system the functions cosf x,xsinf x,··· ,xp−1sin(πp/2 − f x). After
n n n
this modification has been done for every multiple root, we again get a Riesz basis
of H and can find a function g ∈ H with Fourier coefficients c := −cosf ,c =
n n n+1
−sinf ,··· ,c := −sin(πp(n)/2−f x). Then ±f becomes a zero of the function
n n+p(n) n n
f in (3.6) of multiplicity p(n), and the proof is completed as above. (cid:3)
Leemma 3.3. The following equalities hold:
Φ (λ) = π−2(k −1/2)−2(λ2 −λ2),
1 k
(3.7) Y
Φ (λ) = π−2(k −1/2)−2(µ2 −λ2).
2 k
Y
Proof. We recall [12] that the function u has a representation via a transformation
1
operator connected with the point x = 1 of the form
1
u (x,λ) = cosλ(1−x)+ k(x,t)cosλ(1−t)dt,
1
Z
x
which implies that
1
Φ (λ) = cosλ+ k (t)cosλtdt,
1 1
Z
0
where k (t) := k(0,1−t) is a function from H. The claim now follows from Lemma 3.2.
1
(cid:3)
The representation for Φ is derived analogously, and the lemma is proved.
2
Now, giventwospectra(λ2)and(µ2)oftwoSturm-LiouvilleoperatorsT andT
n n σ,h1 σ,h2
respectively, with unknown σ ∈ H and h ,h ∈ R, we proceed as follows. First, we
1 2
identify h −h as lim (λ2 −µ2)/2 (see Corollary 2.4), then construct the functions
1 2 n→∞ n n
Φ and Φ of (3.7) and determine the norming coefficients (α ) via (3.1). The spectral
1 2 n
data {(λ2),(α )} determine σ and h up to an additive constant by means of the
n n 1
algorithm of [11]. This gives the required function σ and two numbers h and h up to
1 2
an additive constant, and the reconstruction is complete.
The second part of the inverse spectral problem is to identify those pairs ofsequences
(λ2) and (µ2) that are spectra of Sturm-Liouville operators T and T for some
n n σ,h1 σ,h2
real-valued σ ∈ H and some real h ,h , h 6= h . We established in Section 2 the
1 2 1 2
necessary conditions on the two spectra given by Theorem 2.1 and Lemmata 2.2 and
2.3; it turns out that these conditions are sufficient as well. Namely, the following
statement holds true.
Theorem 3.4. Suppose that sequences (λ2) and (µ2) of positive pairwise distinct num-
n n
bers satisfy the following assumptions:
(1) the sequences (λ2) and (µ2) interlace;
n n
RECONSTRUCTION BY TWO SPECTRA 9
(2) λ = π(n−1/2)+λ and µ = π(n−1/2)+µ with some ℓ -sequences (λ )
n n n n 2 n
and (µ );
n
e e
(3) there exist a real number h and an ℓ -sequence (νe ) such that λ −µ = h +νn.
2 n n n πn n
e
Then there exist a function σ ∈ H and two real numbers h and h such that (λ2)
1 2 n
and (µ2) are the spectra of the Sturm-Liouville operators T and T respectively.
n σ,h1 σ,h2
Moreover, the operators T and T are recovered uniquely.
σ,h1 σ,h2
In order to prove the theorem we have to show first that the numbers α constructed
n
for the two sequences through formula (3.1), with the functions Φ ,Φ of (3.7) and
1 2
h − h := h with h of item (3), are positive and obey the asymptotics α = 1 +α
1 2 n n
for some ℓ -sequence (α ). Then the algorithm of [11] uses (λ2) and (α ) to determine
2 n n n
(unique up to an additive constant) function σ ∈ H and h ∈ R such that {λ2} is tehe
1 n
spectrum of the Sturme-Liouville operator T and α are the corresponding norming
σ,h1 n
constants. The second and final step is to verify that {µ2} is the spectrum of the
n
Sturm-Liouville operator T with h := h −h, where h is the number of item (3).
σ,h2 2 1
These two steps are performed in the following two lemmata.
Lemma 3.5. Assume (1)–(3) of Theorem 3.4. Then the numbers α constructed
n
through relation (3.1) with Φ ,Φ of (3.7) and h − h := h with h of item (3) are
1 2 1 2
all positive and obey the asymptotics α = 1+α , where (α ) is an ℓ -sequence.
n n n 2
Proof. ByLemma3.2thereexistH-functionsf1e,f2 suchthaetthefunctionsΦj,j = 1,2,
of (3.7) admit the representation
1
Φ (λ) = cosλ+ f (t)cosλtdt.
j j
Z
0
It follows that
1
Φ′(λ ) = −sinλ − tf (t)sinλ tdt
1 n n 1 n
Z
0
1
= (−1)ncosλ − tf (t)sinλ tdt = (−1)n +λˆ ,
n 1 n n
Z
0
e
where (λˆ )∞ ∈ ℓ , and by similar arguments
n n=1 2
Φ (λ ) = (λ −µ )Φ′(µ )+O(|λ −µ |2)
2 n n n 2 n n n
= (λ −µ )[(−1)n +µˆ ]
n n n
for some ℓ -sequence (µˆ ). Next, assumptions (2) and (3) easily imply the relation
2 n
h
= 1+νˆ ,
n
λ (λ −µ )
n n n
for some ℓ -sequence (νˆ ). Therefore the numbers
2 n
h Φ′(λ )
α := 1 n
n
λ Φ (λ )
n 2 n
obey the required asymptotics.
Finally, the interlacing property of the two sequences implies that all α are of the
n
same sign, and thus are all positive in view of the asymptotics established. The proof
(cid:3)
is complete.
10 R. O. HRYNIV,YA.V. MYKYTYUK
With the above-stated properties of the sequences (λ2) and (α ) we can employ the
n n
result of [11] that guarantees existence of a unique Sturm-Liouville operator T with
σ,h1
a real-valued function σ ∈ H and a real number h such that {λ2} coincides with
1 n
the spectrum of T and α are the corresponding norming constants. Now we put
σ,h1 n
h := h −h with h of assumption (3) and expect {µ2} to be the spectrum of T .
2 1 n σ,h2
Lemma 3.6. The spectrum of the Sturm-Liouville operator T with the above σ ∈ H
σ,h2
and h ∈ R coincides with (µ2).
2 n
Proof. Denotebyv (x,λ)asolutionoftheequationl (v) = λ2v satisfyingtheboundary
1 σ
conditions v(1) = 1,v[1](1) = −h . Then v (x,λ ) is an eigenfunction of the operator
1 1 n
T corresponding to the eigenvalue λ2, and by construction
σ,h1 n
1 h Φ′(λ )
2 |v (x,λ )|2dx = α := 1 n
1 n n
Z λ Φ (λ )
0 n 2 n
with the functions Φ ,Φ of (3.7).
1 2
Denote by (ν2) the spectrum of T and put
n σ,h2
∞
Φ (λ) := π−2(k −1/2)−2(ν2 −λ2).
2 n
kY=1
e
By Lemma 3.1 we have
h Φ′(λ )
α = 1 n ,
n
λnΦ2(λn)
whence Φ2(λn) = Φ2(λn) for all n ∈ N. Lemema 3.2 implies that
1
e
Φ (λ)−Φ (λ) = f(t)cosλtdt
2 2
Z
0
e
for some f ∈ H. Equality Φ (λ ) = Φ (λ ) means that the function f is orthogonal to
2 n 2 n
cosλ t, n ∈ N. Since the system {cosλ t} forms a Riesz basis of H, we get f ≡ 0 and
n n
e
Ψ ≡ Ψ . Thus ν = µ , and the lemma is proved. (cid:3)
2 2 n n
e
4. Reconstruction by Dirichlet and Dirichlet-Neumann spectra
The analysis of the previous two sections does not cover the case where one of h ,h
1 2
is infinite. In this case the other number may be taken 0 without loss of generality
(recall that T = T ), i.e., the boundary conditions under considerations become
σ,h σ+h,0
Dirichlet and Dirichlet-Neumann ones.
Suppose therefore that σ ∈ H is real valued and that (λ2) and (µ2) are spectra of
n n
the operators T and T respectively; without loss of generality we assume that λ
σ,0 σ,∞ n
and µ are positive and strictly increase with n. The reconstruction procedure remains
n
the same as before; namely, we use the two spectra to determine a sequence of norming
constants (α ) and then recover σ by the spectral data {(λ2),(α )}.
n n n
Denote by u (·,λ) and u (·,λ) solutions of the equation l u = λu satisfying the
− + σ
[1] [1]
initial conditions u (0,λ) = 0,u (0,λ) = λ and u (1,λ) = 1,u (1,λ) = 0 respec-
− − + +
tively. Then u (·,λ ) is an eigenfunction of the operator T corresponding to the
+ n σ,0
eigenvalue λ2 and
n
1
α := 2 |u (x,λ )|2dx
n + n
Z
0