Isospectral Dirac operators

Isospectral Dirac operators

Yuri Ashrafyan and Tigran Harutyunyan

Yerevan State University, Alex Manoogian 1, Yerevan, 0025, Armenia Received 30 December 2015, appeared 23 January 2017

Communicated by Miklós Horváth

Abstract. We give the description of self-adjoint regular Dirac operators, on[_0,π]_{, with} the same spectra.

Keywords: inverse spectral theory, Dirac operator, isospectral operators.

2010 Mathematics Subject Classification: 34A55, 34B30, 47E05.

1 Introduction and statement of result

Let pandqare real-valued, summable on [0,π]functions, i.e. p,q ∈ L¹_R[0,π]. By L(p,q,α) = L(_Ω,α)we denote the boundary-value problem for canonical Dirac system (see [5,6,9,13,14]):

`y≡

B d

dx+_Ω(x)

y=λy, x ∈(0,π), y= y₁

y₂

, λ∈_C, (1.1)

y₁(0)cosα+y₂(0)sinα=0, α∈−^π 2,π

2 i

, (1.2)

y₁(π) =0, (1.3)

where

B=

0 1

−1 0

, Ω(x) =

p(x) q(x) q(x) −p(x)

.

By the same L(p,q,α)we also denote a self-adjoint operator generated by differential expres- sion`in Hilbert space of two component vector-function L²([0,π];C²)on the domain

D=

y = y₁

y₂

;y_k ∈ AC[0,π],(`y)_k ∈ L²[0,π], k=1, 2;

y₁(0)cosα+y₂(0)sinα=0, y₁(π) =0

where AC[0,π]is the set of absolutely continuous functions on [0,π] (see, e.g. [13,16]). It is well known (see [1,5,9]) that under these conditions the spectra of the operator L(p,q,α) is purely discrete and consists of simple, real eigenvalues, which we denote byλ_n=λ_n(p,q,α) =

BCorresponding author. Email: hartigr@yahoo.co.uk

λ_n(_Ω,α), n ∈ Z, to emphasize the dependence ofλ_n on quantities p,q andα. It is also well known (see, e.g. [1,5,9]) that the eigenvalues form a sequence, unbounded below as well as above. So we will enumerate it asλ_k < λ_k+1,k ∈ _Z, λ_k > 0, whenk > 0 and λ_k < 0, when k<0, and the nearest to zero eigenvalue we will denote byλ₀. If there are two nearest to zero eigenvalue, then by λ₀ we will denote the negative one. With this enumeration it is proved (see [1,5,9]), that the eigenvalues have the asymptotics:

λn(_Ω,α) =n− ^α

π +rn, rn=o(1), n→ ±_∞. (1.4) In what follows, writingΩ ∈ Awill mean p,q ∈ A. IfΩ ∈ L²_R[_0,π], then we know, (see, e.g. [9]), that instead ofrn =o(1)we have:

∑

r²_n< _∞. (1.5)

Let ϕ(x,λ) = ϕ(x,λ,α,Ω)be the solution of the Cauchy problem

`ϕ=_λϕ, ϕ(_0,λ) =

sinα

−cosα

. (1.6)

Since the differential expression ` self-adjoint, then the components ϕ₁(x,λ) and ϕ₂(x,λ) of the vector-function ϕ(x,λ)we can choose real-valued for realλ. By an = an(_Ω,α)we denote the squares of theL²-norm of the eigenfunctions ϕ_n(x,Ω) = ϕ(x,λ_n(_Ω,α),α,Ω):

a_n=kϕ_nk² =

Z _π

0

|ϕ_n(x,Ω)|²dx, n∈_Z.

The numbers a_n are called norming constants. And by h_n(x,Ω)we will denote normalized eigenfunctions (i.e.khn(x)k=1):

hn(x,Ω) =hn(x) = ^ϕn(x,Ω)

pa_n(_Ω,α)^{. (1.7)}

It is known (see [5,9]) that in the case of Ω ∈ L²_R[_0,π] the norming constants have an asymptotic form:

an(_Ω) =π+cn,

∑

c²_n <_∞. (1.8)

Definition 1.1. Two Dirac operatorsL(_Ω,α)andL(_{Ω, ˜}^˜ α)are said to be isospectral, ifλn(_Ω,α)=

λ_n(_{Ω, ˜}^˜ α), for everyn∈_Z.

Lemma 1.2. LetΩ, ˜Ω∈ L¹_R[0,π]and the operators L(_Ω,α)and L(_{Ω, ˜}^˜ α)are isospectral. Thenα˜ =α.

Proof. The proof follows from the asymptotics (1.4):

α π

= _lim

n→_∞(n−λ_n(_Ω,α)) = _lim

n→_∞(n−λ_n(_{Ω, ˜}^˜ α)) = ^α˜ π.

So, instead of isospectral operators L(_Ω,α) and L(_{Ω, ˜}^˜ α), we can talk about “isospectral potentials”Ωand ˜Ω.

Theorem 1.3(Uniqueness theorem). The map (_Ω,α)∈ L²_R[0,π]×−^π

2,π 2

i←→ {λn(_Ω,α), an(_Ω,α); n∈_Z} is one-to-one.

Remark 1.4. It is natural to call this a Marchenko theorem, since it is an analogue of the famous theorem of V. A. Marchenko [15], in the case for Sturm–Liouville problem. The proof of this theorem for the case p,q ∈ AC[0,π]there is in the paper [18]. The detailed proof for the case p,q∈ L²_R[0,π]there is in [7] (see also [4–6,8,10,19]).

Let us fix someΩ∈ L²_R[0,π]and consider the set of all canonical potentials ˜Ω= ^p_q^˜_˜−^q˜_p˜, with the same spectra as Ω:

M²(_Ω) ={_Ω^˜ ∈ L²_R[0,π]:λ_n(_{Ω, ˜}^˜ α) =λ_n(_Ω,α), n∈_Z}.

Our main goal is to give the description of the set M²(_Ω)as explicit as it possible.

From the uniqueness theorem the next corollary easily follows.

Corollary 1.5. The map

Ω˜ ∈ M²(_Ω)↔ {a_n(_Ω^˜), n∈_Z} is one-to-one.

Since ˜Ω ∈ M²(_Ω)_{, then} a_n(_Ω^˜) have similar to (1.8) asymptotics. Sincea_n(_Ω)_and a_n(_Ω^˜) are positive numbers, there exist real numbers t_n = t_n(_Ω^˜), such that ^an(Ω)

an(_Ω^˜) = e^tn. From the latter equality and from (1.8) follows that

e^tn =₁+d_n,

∑

d²_n<_{∞. (1.9)}

It is easy to see, that the sequence {t_n; n ∈ _Z} is also from l², i.e. ∑^∞n=−_∞t²_n < _∞. Since all an(_Ω) are fixed, then from the corollary1.5 and the equalityan(_Ω^˜) = an(_Ω)e^−tn we will get the following corollary.

Corollary 1.6. The map

Ω˜ ∈ M²(_Ω)↔ {tn(_Ω^˜), n∈_Z} ∈l² is one-to-one.

Thus, each isospectral potential is uniquely determined by a sequence{tn;n ∈_Z}. Note, that the problem of description of isospectral Sturm–Liouville operators was solved in [3,11, 12,17].

For Dirac operators the description ofM²(_Ω)is given in [8]. This description has a “recur- rent” form, i.e. at the first in [8] is given the description of a family of isospectral potentials Ω(x,t), t∈ R, for which only one norming constanta_m(_Ω(·,t))different froma_m(_Ω)(namely, a_m(_Ω(·_,t)) =a_m(_Ω)e^−t), while the others are equal, i.e. a_m(_Ω(·_,t)) =a_m(_Ω)_{, when}n6=m.

Theorem 1.7([8]). Let t∈ _R,α∈ −^π₂,^π₂ and

Ω(x,t) =_Ω(x) + ^e

t−1

θ_m(x,t,Ω){Bhm(x,Ω)h^∗_m(x,Ω)−hm(x,Ω)h^∗_m(x,Ω)B}, where θ_n(x,t,Ω) = ₁+ (e^t−1)Rx

0 |h_n(s,Ω)|²ds, and ∗ is a sign of transponation, e.g. h^∗_m = _h

m1

hm2

∗

= (h_m1,h_m2). Then, for arbitrary t ∈_R,λ_n(_Ω,t) =λ_n(_Ω)for all n∈_Z, a_n(_Ω,t) =a_n(_Ω) for all n ∈ _Z\{m} and a_m(_Ω,t) = a_m(_Ω)e^−t. The normalized eigenfunctions of the problem L(_Ω(·,t)_,α)are given by the formulae:

h_n(x,Ω(·,t)) =











e^−t/2

θm(x,t,Ω)^hm(x,Ω), if n=m, hn(x,Ω)−(e^t−1)Rx

0 h^∗_m(s,Ω)hn(s,Ω)ds

θm(x,t,Ω) ^hm(x,Ω), if n6= m.

Theorem1.7shows that it is possible to change exactly one norming constant, keeping the others. As examples of isospectral potentialsΩand ˜Ωwe can presentΩ(x)≡0= ^{0 0}_{0 0} and

Ω˜(x) =_Ωm,t(x) = ^π(e^t−1) π+ (e^t−1)x

−sin 2mx cos 2mx cos 2mx sin 2mx

, wheret∈_Ris an arbitrary real number andm∈ _Zis an arbitrary integer.

Changing successively eacha_m(_Ω)bya_m(_Ω)e^−tm, we can obtain any isospectral potential, corresponding to the sequence{tm;m∈_Z} ∈ l². It follows from the uniqueness Theorem1.3 that the sequence, in which we change the norming constants, is not important.

In [8] were used the following designations:

T−1 ={. . . , 0, . . .},

T₀ ={. . . , 0, . . . , 0,t₀, 0, . . . , 0, . . .}, T₁ ={. . . , 0, . . . , 0, 0,t0,t₁, 0, . . . , 0, . . .}, T₂ ={. . . , 0, . . . , 0,t₋₁,t₀,t₁, 0, . . . , 0, . . .},

...

T2n ={. . . , 0, 0,t−n, . . . ,t−1,t0,t₁, . . . ,tn−1,tn, 0, . . .}, T_2n+1 ={. . . , 0,t−n,t_−n+1, . . . ,t₋₁,t₀,t₁, . . . ,t_n,t_n+1, 0, . . .},

... LetΩ(x,T−1)≡_Ω(x)and

Ω(x,Tm) =_Ω(x,Tm−1) +4_Ω(x,Tm), m=0, 1, 2, . . . , where

4_Ω(x,Tm) = ^e

tm˜ −1

θ_m(x,t_m˜,Ω(·,T_m−1))[Bhm˜(x,Ω(·,T_m−1))h^∗_m˜(·)−hm˜(·)h^∗_m˜(·)B],

where ˜m = ^m+₂¹, if m is odd and ˜m = −^m₂, if mis even. The arguments in others h_m˜(·)and h^∗_m˜(·)are the same as in the first. And after that in [8] the following theorem was proved.

Theorem 1.8([8]). Let T= {t_n,n∈_Z} ∈l²andΩ∈ L²_R[0,π]. Then Ω(x,T)≡_Ω(x) +

∑

4_Ω(x,Tm)∈ M²(_Ω). (1.10) We see, that each potential matrix 4_Ω(x,Tm) defined by normalized eigenfunctions h_m˜(x,Ω(x,T_m−1)) of the previous operator L(_Ω(·,T_m−1),α). This approach we call “recur- rent” description.

In this paper, we want to give a description of the set M²(_Ω)only in terms of eigenfunc- tions hn(x,Ω)of the initial operatorL(_Ω,α)and sequenceT∈l². With this aim, let us denote byN(T_m)the set of the positions of the numbers in T_m, which are not necessary zero, i.e.

N(T₀) ={0}, N(T₁) ={0, 1}, N(_T2) ={−1, 0, 1},

...

N(T2n) ={−n,−(n−1), . . . , 0, . . . ,n−1,n}, N(T_2n+1) ={−n,−(n−1), . . . , 0, . . . ,n,n+1},

...

in particular N(T)≡_{Z. By}S(x,Tm)we denote the(m+1)×(m+1)square matrix S(x,T_m) =

δ_ij+ (e^tj−1)

Z _x

0 h^∗_i(s)h_j(s)ds

i,j∈N(Tm)

(1.11)

whereδ_ij is a Kronecker symbol. ByS⁽_p^k)(x,Tm)we denote a matrix which is obtained from the matrixS(x,T_m)by replacing thekth column ofS(x,T_m)byH_p(x,T_m) ={−(e^tk−1)h_kp(x)}_k∈N(Tm) column, p=1, 2, Now we can formulate our result as follows.

Theorem 1.9. Let T= {t_k}_k∈Z∈ l² andΩ∈ L²_R[0,π]. Then the isospectral potential from M²(_Ω), corresponding to T, is given by the formula

Ω(x,T) =_Ω(x) +G(x,x,T)B−BG(x,x,T) =

p(x,T) q(x,T) q(x,T) −p(x,T)

, (1.12)

where

G(x,x,T) = ¹

detS(x,T)

∑

k∈_Z

detS⁽₁^k)(x,T) detS⁽₂^k)(x,T)

! h^∗_k(x), anddetS(x,T) =limm→_∞detS(x,Tm)(the same fordetS^k_p(x,T), p=1, 2).

In addition, for p(x,T)and q(x,T)we get explicit representations:

p(x,T) = p(x)− ¹

detS(x,T)

∑

k∈_Z

∑

detS⁽_p^k)(x,T)h_k(3−p)(x),

q(x,T) =q(x) + ¹

detS(x,T)

∑

k∈_Z

∑

(−1)^p−1S⁽_p^k)(x,T)h_kp(x).

2 Proof of Theorem 1.9

The spectral function of an operatorL(_Ω,α)_{defined as}

ρ(λ) =















0<

∑

1

a_n(_Ω)^{, λ}>0,

−

∑

λ<λn≤0

1

a_n(_Ω)^{, λ}<0,

i.e.ρ(λ)is left-continuous, step function with jumps in pointsλ= λ_nequals _a¹

n andρ(0) =0.

LetΩ, ˜Ω∈ L²_R[0,π]and they are isospectral. It is known (see [1,2,6,13]), that there exists a functionG(x,y)such that:

ϕ(x,λ,α, ˜Ω) = ϕ(x,λ,α,Ω) +

Z _x

0 G(x,s)ϕ(s,λ,α,Ω)dt. (2.1) It is also known (see, e.g. [1,6,13]), that the function G(x,y) satisfies to the Gelfand–Levitan integral equation:

G(x,y) +F(x,y) +

Z _x

0

G(x,s)F(s,y)ds=_{0, 0}≤y≤ x, (2.2) where

F(x,y) =

Z _∞

−_∞ϕ(x,λ,α,Ω)ϕ^∗(y,λ,α,Ω)d[ρ˜(λ)−ρ(λ)]. (2.3) If the potential ˜Ω from M²(_Ω) is such that only finite norming constants of the opera- tor L(_Ω,^˜ α) are different from the norming constants of the operator L(_Ω,α)_{, i.e.} a_n(_Ω^˜) = an(_Ω)e^−tn, n∈ N(Tm)and the others are equal, then it means, that

dρ˜(λ)−dρ(λ) =

∑

k∈N(Tm)

1 a˜_k − ¹

a_k

δ(λ−λ_k)dλ=

∑

k∈N(Tm)

e^tk−1 a_k

δ(λ−λ_k)dλ, (2.4) whereδ is Diracδ-function. In this case the kernelF(x,y)can be written in a form of a finite sum (using notation (1.7)):

F(x,y) =F(x,y,Tm) =

∑

k∈N(Tm)

(e^tk−1)h_k(x,Ω)h^∗_k(y,Ω), (2.5) and consequently, the integral equation (2.2) becomes to an integral equation with degener- ated kernel, i.e. it becomes to a system of linear equations and we will look for the solution in the following form:

G(x,y,T_m) =

∑

k∈N(Tm)

g_k(x)h^∗_k(y), (2.6)

whereg_k(x) =^g_g^k1(x)

k2(x)

is an unknown vector-function. Putting the expressions (2.5) and (2.6) into the integral equation (2.2) we will obtain a system of algebraic equations for determining the functionsg_k(x)_:

g_k(x) +

∑

i∈N(Tm)

s_ik(x)g_i(x) =−(e^tk−1)h_k(x), k ∈N(Tm), (2.7) where

s_ik(x) = (e^tk−1)

Z _x

0 h^∗_i(s)h_k(s)ds.

It would be better if we consider the equations (2.7) for the vectors g_k =^g_g^k_k¹

2

by coordinates g_k1 andg_k2 to be a system of scalar linear equations:

g_kp(x) +

∑

i∈N(Tm)

s_ik(x)g_ip(x) =−(e^tk−1)h_kp(x), k∈ N(Tm), p=1, 2. (2.8) The systems (2.8) might be written in matrix form

S(x,Tm)gp(x,Tm) =Hp(x,Tm), p=1, 2, (2.9) where the column vectors g_p(x,T_m) = {g_kp(x,T_m)}_k∈N(Tm), p = 1, 2, and the solution can be found in the form (Cramer’s rule):

g_kp(x,Tm) = ^detS

(k) p (x,T_m)

detS(x,T_m) ^{, k}∈ N(Tm), p =1, 2.

Thus we have obtained forg_k(x)the following representation:

g_k(x,T_m) = ¹ detS(x,Tm)

detS₁^(k)(x,T_m) detS₂^(k)(x,T_m)

!

(2.10) and then by putting (2.10) into (2.6) we find theG(x,y,Tm)function. If the potentialΩis from L¹_R, then such is also the kernel G(x,x,T_m) (see [8]), and the relation between them gives as follows:

Ω(x,T_m) =_Ω(x) +G(x,x,T_m)B−BG(x,x,T_m). (2.11) On the other hand we have

Ω(x,T_m) =_Ω(x) +

∑

4_Ω(x,T_k)_{. (2.12)}

So, using the Theorem 1.8 and the equality (2.12) we can pass to the limit in (2.11), when m→_∞:

Ω(x,T) =_Ω(x) +G(x,x,T)B−BG(x,x,T). (2.13) The potentialsΩ(x,T)in (1.10) and (2.13) have the same spectral data{λ_n(T),a_n(T)}_n∈Z, and therefore they are the same andΩ(·_,T)defined by (2.13) is also from M²(_Ω)_.

Using (2.6) and (2.10) we calculate the expression G(x,x,T_m)B−BG(x,x,T_m)and pass to the limit, obtaining for the p(x,T)_andq(x,T)the representations:

p(x,T) = p(x)− ¹

detS(x,T)

∑

k∈N(T)

∑

detS⁽_p^k)(x,T)h_k(3−p)(x),

q(x,T) =q(x) + ¹

detS(x,T)

∑

k∈N(T)

∑

(−1)^p−1S⁽_p^k)(x,T)h_kp(x). Theorem1.9 is proved.

For example, when we change just one norming constant (e.g. forT₀) we get two indepen- dent linear equations:

(1+s₀₀(x))g₀₁(x) =−(e^t0−1)h₀₁(x), (1+s00(x))g0₂(x) =−(e^t0−1)h0₂(x). For the solutions we get:

g₀₁(x) =−(e^t0 −1)h0₁(x) 1+s₀₀(x) ^, g₀₂(x) =−(e^t0 −1)h₀₂(x)

1+s00(x) ^, and for the potentialsp(x,T0)andq(x,T0):

p(x,T₀) = p(x) + ^e

t0−1

1+s₀₀(x)(2h₀₁(x)h₀₂(x)), q(x,T0) =q(x) + ^e

t0−1

1+s₀₀(x)(h²₀₂(x)−h²₀₁(x)).

Acknowledgements

This work was supported by the RA MES State Committee of Science, in the frames of the research project No.15T-1A392.

References

[1] S. Albeverio, R. Hryniv, Ya. Mykytyuk, Inverse spectral problems for Dirac op- erators with summable potential, Russ. J. Math. Phys. 12(2005), No. 5, 406–423.

arXiv:math/0701158;MR2201307

[2] T. N. Arutyunyan (T. N. Harutyunyan), Transformation operators for the canonical Dirac system (in Russian),Differ. Uravn.44(2008), No. 8, 1011–1021.MR2479963;url [3] B. E. Dahlberg, E. Trubowitz, The inverse Sturm–Liouville problem. III, Comm. Pure

Appl. Math.37(1983), No. 2, 255–168.MR733718;url

[4] G. Freiling, V. A. Yurko, Inverse Sturm–Liouville problems and their applications, Nova Science Publishers, Inc., New York, 2001.MR2094651

[5] M. G. Gasymov, T. T. Dzhabiev, Determination of the system of Dirac differential equa- tions from two spectra (in Russian), in: Proceedings of the Summer School in the Spectral Theory of Operators and the Theory of Group Representation (Baku, 1968), Izdat. “Èlm”, Baku, 1975, pp. 46–71.MR0402164

[6] M. G. Gasymov, B. M. Levitan, The inverse problem for the Dirac system (in Russian), Dokl. Akad. Nauk SSSR167(1966), No. 5, 967–970.MR0194650

[7] T. N. Harutyunyan, The eigenvalues function of the family of Sturm–Liouville and Dirac operators(in Russian), dissertation, Yerevan State University, 2010.

[8] T. N. Harutyunyan, Isospectral Dirac operators (in Russian), Izv. Nats. Akad. Nauk Ar- menii Mat.29(1994), No. 2, 3–14; in English: J. Contemp. Math. Anal.29(1994), No. 2, 1–10.

MR1438873

[9] T. N. Harutyunyan, H. Azizyan, On the eigenvalues of boundary value problem for canonical Dirac system (in Russian),Mathematics in Higher School2(2006), No. 4, 45–54.

[10] M. Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans.

Amer. Math. Soc.353(2001), No. 10, 4155–4171.MR1837225;url

[11] E. L. Isaacson, H. P. McKeen, E. Trubowitz, The inverse Sturm–Liouville problem. II, Comm. Pure Appl. Math.37(1984), No. 1, 1–11.MR728263;url

[12] E. L. Isaacson, E. Trubowitz, The inverse Sturm–Liouville problem. I,Comm. Pure Appl.

Math.36(1983), No. 6, 767–783.MR720593;url

[13] B. M. Levitan, I. S. Sargsyan, Sturm–Liouville and Dirac operators (in Russian), Moscow, Nauka, 1988.MR958344

[14] V. A. Marchenko, Sturm–Liouville operators and its applications (in Russian), Naukova Dumka, Kiev, 1977.MR0481179

[15] V. A. Mar ˇcenko, Concerning the theory of a differential operator of the second order (in Russian),Doklady Akad. Nauk SSSR. (N.S.)72(1950), 457–460.MR0036916

[16] M. A. Naimark, Linear differential operators (in Russian), Nauka, Moscow, 1969.

MR0353061

[17] J. Pöschel, E. Trubowitz,Inverse spectral theory, Academic Press, Inc., Boston, MA, 1987.

MR894477

[18] B. A. Watson, Inverse spectral problems for weighted Dirac systems, Inverse Problems 15(1999), No. 3, 793–805.MR1694351;url

[19] Z. Wei, G. Wei, The uniqueness of inverse problem for the Dirac operators with partial information,Chin. Ann. Math. Ser. B36(2015), No. 2, 253–266.MR3305707;url