Inverse resonance scattering for massless Dirac operators on the real line

Definition.

P = P γ is a set of all functions q ∈ L 2 ( R ) such that the convex hull of supp q equals [ 0 , γ ] for some γ > 0 . The set P is equipped with the metric ρ P given by (1.3) ρ P ( q 1 , q 2 ) = ‖ q 1 − q 2 ‖ L 2 ( 0 , γ ) , q 1 , q 2 ∈ P .

Note that P is a metric space, which is not complete. Recall known results about Dirac operators H, see e.g., [35]. The spectrum of H satisfies σ ( H ) = σ a c ( H ) = R . We introduce the 2 × 2 matrix-valued Jost solutions f ± ( x , k ) = ( f 11 ± f 12 ± f 21 ± f 22 ± ) ( x , k ) of the Dirac equation (1.4) ( f ± ) ′ ( x , k ) = Q ( x ) f ± ( x , k ) + i k σ 3 f ± ( x , k ) , ( x , k ) ∈ R × C , under the standard condition for compactly supported potentials: f + ( x , k ) = e i k x σ 3 , ∀ x ⩾ γ , f − ( x , k ) = e i k x σ 3 , ∀ x ⩽ 0 . The equation (1.4) has exactly one linear independent matrix-valued solution, then there exists a unique 2 × 2 transition matrix A ( k ) such that f + ( x , k ) = f − ( x , k ) A ( k ) , x ∈ R . The transition matrix A has the form (1.5) A = a b ∗ b a ∗ , a a ∗ − b b ∗ = 1 , where we have used the notation g ∗ ( k ) = g ( k ‾ ) ‾ , k ∈ C . It is known that a and b are entire, a ( k ) ≠ 0 for any k ∈ C ‾ + and a has zeros in C − , which are called resonances. Note that the resonances are also zeros of Fredholm determinants and poles of the resolvent ( H − k ) − 1 (see e.g. [19]). Due to (1.5) the zeros of b and a do not coincide. Let H o = − i σ 3 d d x be the free Dirac operator on L 2 ( R , C 2 ) . The scattering matrix S for the pair H, H o is given by (1.6) S ( k ) = 1 a ( k ) 1 − b ( k ‾ ) ‾ b ( k ) 1 , k ∈ R . Here 1 / a is a transmission coefficient and r + = − b ‾ / a (or r − = b / a ) is a right (or left) reflection coefficient. The matrix-valued function S admits a meromorphic continuation from R onto C , since a and b are entire. Poles of S are resonances and zeros of the reflection coefficients r ± coincide with zeros of b or b ∗ .

1.2. Main goal

Our main goal is to solve the inverse problem for Dirac operators with compactly supported potentials in terms of resonances. We sometimes write a ( · , q ) , b ( · , q ) , … instead of a ( · ) , b ( · ) , … , when several potentials are being dealt with. We show that the resonances of operator H do not uniquely determine the potential, and then we need additional data. We consider inverse problems for the following mappings:

We need to underline that in order to solve the inverse problem ii) in terms of resonances, we need to solve the problem i). An inverse problem is to determine the potential by some data, and it consists of at least the four parts:

The data considered in our paper can be divided into two sets, and we will briefly describe the results obtained for each of them.

Firstly, we consider the inverse problem in terms of a function b (or the zeros of b, and also the reflection coefficients r ± ). For these data, we obtain the following results:

Secondly, we consider the inverse problem in terms of the function a (or in terms of the resonances). We obtain the following results:

The coefficients a and b are important to study the NLS equation. Namely, log | a ( k ) | and arg b ( k ) , k ∈ R , are action and angle variables for the NLS equation (see e.g. p. 230 in [11]). For q ∈ P we obtain the representations of action-angle variables in terms of resonances and zeros of b.

We solve the inverse problem for the Dirac operators with potentials, which have some symmetries in Section 6. We show that the resonances uniquely determine a real-valued even or real-valued odd potential up to a sign.

Note also that Dirac operators can be rewritten as canonical systems (see, e.g., p. 389 in [15]). Due to this relation, we describe the class of canonical systems, which are unitary equivalent to the Dirac operators in Section 7. Then we introduce the scattering matrix and solve the inverse resonance problem for such canonical systems.

Definition.

B = B γ is the metric space of all entire functions u such that u ˆ ∈ P γ equipped with the metric ρ B given by (2.1) ρ B ( u 1 , u 2 ) = ‖ u ˆ 1 − u ˆ 2 ‖ L 2 ( 0 , γ ) , u 1 , u 2 ∈ B .

Note that the metric space B is not complete. We recall needed facts about entire functions. An entire function f ( z ) is said to be of exponential type if there exist constants τ , C > 0 such that | f ( z ) | ⩽ C e τ | z | , z ∈ C . We introduce the Cartwright class of entire functions E C a r t by

Definition.

E C a r t is a class of entire functions f of exponential type such that ∫ R log ( 1 + | f ( k ) | ) d k 1 + k 2 < ∞ , τ + ( f ) = 0 , τ − ( f ) = 2 γ , where τ ± ( f ) = lim sup t → + ∞ log | f ( ± i t ) | t .

We shortly describe needed properties of functions from the Cartwright class. Let f ∈ E C a r t and let ν be the multiplicity of zero of f ( k ) at k = 0 . Let ( k n ) n ⩾ 1 be zeros of f in C ∖ { 0 } counted with multiplicity and labeled by (2.2) 0 < | k 1 | ⩽ | k 2 | ⩽ ⋯ . Then f has the Hadamard factorization (see, e.g., pp. 127–130 in [34]) (2.3) f ( k ) = C k ν e i γ k lim r → + ∞ ∏ | k n | ⩽ r ( 1 − k k n ) , C = f ( ν ) ( 0 ) ν ! , k ∈ C , where the product converges uniformly on compact subsets of C and ∑ n ⩾ 1 | Im k n | | k n | 2 < + ∞ , ∃ lim r → + ∞ ∑ | k n | ⩽ r 1 k n ≠ ∞ . We denote the number of zeros of function f having modulus less than r by N ( r , f ) , each zero being counted according to its multiplicity. For any function f ∈ E C a r t , the Levinson theorem holds true (see, e.g., p. 58 in [23]): (2.4) N ( r , f ) = γ π r + o ( r ) as r → + ∞ .

We present our first result.

Theorem 2.1.

The mapping q ↦ b ( · , q ) is a homeomorphism between P and B .

Remark.

1) The proof of this theorem consists of two parts. Firstly, we solve the inverse scattering problem for the Dirac operators with potentials from P , see Theorem 2.3. Secondly, we prove that there exists a one-to-one correspondence between b and r ± , and to get this result we use arguments from [27].

2) In Corollary 6.2, we give the characterization of b for even, odd or real-valued potentials.

3) We discuss the recovery procedure for this inverse problem in Section 3.

Now we introduce the class of scattering data associated with the resonances, i.e. the poles of the reflection coefficients.

Definition.

A = A γ is the set of all entire functions f such that:

f ( k ) ≠ 0 for any k ∈ C + ;

| f ( k ) | ⩾ 1 for any k ∈ R ;

| f | 2 − 1 ∈ L 1 ( R ) ;

f − 1 ∈ B γ .

We consider the inverse problem for the Dirac operator H in terms of a ∈ A . The idea here is to construct b ∈ B for given a ∈ A such that a a ∗ − b b ∗ = 1 and then we can determine a potential using Theorem 2.1. So let a ∈ A be given and let B = a a ∗ − 1 . Thus, to determine a potential we need to solve the equation b b ∗ = B . However, in general, this equation has many solutions b ∈ B . It follows that the coefficient a does not uniquely determine a potential and to obtain a unique solution b ∈ B of the equation b b ∗ = B we need additional information about b.

We consider the function B. It is easy to see that B is an exponential type function, B ( z ) ⩾ 0 for each z ∈ R and the zeros of B are symmetric with respect to the real line. Let b ∈ B be a solution of the equation b b ∗ = B . Then some zeros of B are zeros of b, and the other are zeros of b ∗ . We will show that this information is sufficient to obtain a unique solution of this equation. Namely, if the multiplicities of the zeros of b in C ± are fixed then there exists a unique solution b ∈ B of the equation b b ∗ = B . Note also that zeros on the real line are the same for all solutions.

We construct the data ξ = ( ξ n ) n ⩾ 0 for any b ∈ B . Let ( z n ) n ⩾ 1 be the zeros of B = b b ∗ in C + counted without multiplicity and labeled by (2.5) 0 < | z 1 | ⩽ | z 2 | ⩽ ⋯ , Re z n < Re z n + 1 , if | z n | = | z n + 1 | , and let ( ζ n ) n ⩾ 1 be theirs multiplicities. Recall that N o = N ∪ { 0 } . Let also ν ∈ N o be the multiplicity of the zero k = 0 of b ( k ) . Then we introduce a mapping ξ : b ↦ ξ ( b ) = ( ξ n ) n ⩾ 0 by (2.6) ξ o = b ( ν ) ( 0 ) | b ( ν ) ( 0 ) | , ξ n = ξ n + ξ n − , n ⩾ 1 , where ξ n + is the multiplicity of the zero z n of b and ξ n − is the multiplicity of the zero z ‾ n of b. Note that ξ n + + ξ n − = ζ n for each n ⩾ 1 and ξ n shows how the zero z n of B is allocated between b and b ∗ and ξ o is a general phase multiplier.

Now for any given a ∈ A we construct the set Ξ ( B ) , where B = a a ∗ − 1 , such that for any ξ ∈ Ξ ( B ) there exist a solution b of the equation b b ∗ = B and ξ ( b ) = ξ . We introduce Ξ ( B ) as follows (2.7) Ξ ( B ) = T × ( × n ⩾ 1 T ζ n ) , where T = { λ ∈ C : | λ | = 1 } , T ζ n = α + α − ∈ N o 2 | α + + α − = ζ n , n ⩾ 1 . Here T is a unite circle in C and T ζ n is, in some sense, its discrete analogues. Now, we give our second main result.

Theorem 2.2.

The mapping q ↦ ( a , ξ ) given by ( 2.6 ) is a bijection between P and A × Ξ ( B ) , where a = a ( · , q ) , ξ = ξ ( b ( · , q ) ) , and B = a a ∗ − 1 .

Remark.

1) In order to construct the set Ξ ( B ) , B = a a ∗ − 1 we need to know the function a. Thus, to obtain a unique potential q ∈ P we need to choose a ∈ A , construct Ξ ( B ) , B = a a ∗ − 1 , by (2.7) and then we need to choose ξ ∈ Ξ ( B ) .

2) Corollary 6.4 gives the characterization of a for even, odd or real-valued potentials and we will show how Ξ ( B ) is reduced in these cases.

3) The inverse resonance problem was solved for the Schrödinger operators on the half-line [25] and on the line [27]. The inverse resonance problem for the Dirac operator on the half-line was considered in [33]. In the proof we use arguments from [25,33] and mainly from [27]. Note that the case of half-line is simpler than the case of the real line since we have:

∙ the scattering matrix depends on two coefficients a and b;

∙ the potential is not uniquely determined by resonances.

These features lead to significant modification of methods given in [33]. Moreover, arguments from [27] required big changes since there are differences between Dirac and Schrödinger operators. Mainly it is the second term in the asymptotic expansion of the Jost solutions of the Dirac equations, which decreases more slowly for the large spectral parameter. Roughly speaking, it means that the spectral problem for Dirac operators corresponds to the spectral problem for the Schrödinger operators with distributions.

Using Theorems 2.1 and 2.2, we give the characterization of the reflection coefficients for the compactly supported potentials. We introduce the following classes.

Definition.

R + = R γ + (or R − = R γ − ) is a set of all meromorphic functions r + (or r − ), given by r + = − b ∗ a (or r − = b a ) for some ( a , b ) ∈ A γ × B γ such that | a ( k ) | 2 − | b ( k ) | 2 = 1 , k ∈ R .

Below, we prove that r ˆ ± ∈ L 2 ( R ) ∩ L 1 ( R ) for any r ± ∈ R ± . Thus, we introduce the metric ρ R ( r 1 , r 2 ) = ‖ r ˆ 1 − r ˆ 2 ‖ L 2 ( R ) + ‖ r ˆ 1 − r ˆ 2 ‖ L 1 ( R ) , r 1 , r 2 ∈ R ± . Note that R ± equipped with the metric ρ R are incomplete metric spaces.

Theorem 2.3.

The mappings q ↦ r ± ( · , q ) are homeomorphisms between P and R ± .

The Paley–Wiener Theorem (see e.g. p. 30 in [23]) gives that A ⊂ E C a r t and B ⊂ E C a r t . Thus, using the properties of E C a r t , we get the following corollary.

Corollary 2.4.

i) The potential q ∈ P is uniquely determined by zeros of b ( · , q ) ∈ B and b ( ν ) ( 0 , q ) , where ν is the multiplicity of zero of b ( k , q ) at k = 0 .

ii) A potential q ∈ P is uniquely determined by ( ( k n ) n ⩾ 1 , ξ ) , where ( k n ) n ⩾ 1 are zeros of a = a ( · , q ) ∈ A and ξ ( b ( · , q ) ) ∈ Ξ ( B ) for B = a a ∗ − 1 .

iii) Let q ∈ P . Then the functions a ( · , q ) , b ( · , q ) satisfy ( 2.3 – 2.4 ).

Remark.

The asymptotics of distribution of resonances of the Dirac operators on the real line was obtained in [19] for compactly supported potentials q , q ′ ∈ L 1 ( R ) . Recall that for Schrödinger operators on the real line such results were obtained in [49].

It follows from Theorem 2.2 that for some q o ∈ P there exist distinct potentials q ∈ P such that a ( · , q o ) = a ( · , q ) . Moreover, by Corollary 2.4, the function a ∈ A is uniquely determined by its zeros. For any q o ∈ P , we introduce an iso-resonance set in P by Iso ( q o ) = { q ∈ P ∣ a ( · , q ) = a ( · , q o ) } . Using Theorem 2.2 for fixed a ( · , q o ) , we give the characterization of Iso ( q o ) in terms of  Ξ ( B ) .

Corollary 2.5.

Let q o ∈ P . Then the mapping q ↦ ξ given by ( 2.6 ) is a bijection between Iso ( q o ) and Ξ ( B ) , where ξ = ξ ( b ( · , q ) ) , a = a ( · , q o ) , and B = a a ∗ − 1 .

On the other hand, it follows from Theorem 2.1 that for each q ∈ P there exists a unique b = b ( · , q ) ∈ B . Now, we describe how b ( · , q ) changes for q ∈ Iso ( q o ) . Theorem 2.6.

Let q o ∈ P and let ( z n ) n ⩾ 1 be zeros of b ( · , q o ) in C + counted without multiplicity and labeled by ( 2.5 ). Let also ξ = ( ξ n ) n ⩾ 0 = ξ ( b ( · , q o ) ) , where ξ n = ( ξ n + ξ n − ) , n ⩾ 1 . Then q ∈ Iso ( q o ) if and only if b ( · , q ) = b ( · , q o ) P ( · ) , where (2.8) P ( k ) = α o lim r → + ∞ ∏ | z n | ⩽ r ( 1 − k z n ) α n ( 1 − k z n ‾ ) − α n , k ∈ C , for some α o ∈ T and ( α n ) n ⩾ 1 such that α n ∈ Z , − ξ n + ⩽ α n ⩽ ξ n − , n ⩾ 1 .

Remark.

It is easy to see that P is a meromorphic function such that P ∗ = 1 P , which yields that | P ( k ) | = 1 for each k ∈ R and then for each q ∈ Iso ( q o ) we obtain | b ( k , q ) | = | b ( k , q o ) | , | r ± ( k , q ) | = | r ± ( k , q o ) | , k ∈ R .

2.2. The NLS equation

Using analytical properties of a and b, we obtain representations of action-angle variables for the nonlinear Schrödinger equation (NLS equation) in terms of resonances and zeros of b. We recall some known facts about the NLS equation from [11]. Let ψ ( x , t ) be a solution of the defocusing NLS equation (2.9) i ψ t = − ψ x x + 2 | ψ | 2 ψ satisfying the initial condition ψ ( · , 0 ) = q ∈ L 2 ( R ) ∩ L 1 ( R ) . For such equation, there exists the Lax pair and the Dirac operator is the corresponding self-adjoint operator. Thus, considering the solution ψ ( · , t ) of (2.9) as a potential of the Dirac operator H, we get b ( · , ψ ( · , t ) ) , a ( · , ψ ( · , t ) ) and these functions have a simple dynamics: (2.10) a ( k , ψ ( · , t ) ) = a ( k , q ) , b ( k , ψ ( · , t ) ) = e 4 i t k 2 b ( k , q ) , ( k , t ) ∈ R 2 , see, e.g., p. 52 in [11]. Note that there are differences in notations. Namely, if we denote by a F and b F the coefficients of the transition matrix, which was used in [11], then we get a ( k , q ) = a F ( 2 k , q ) , b ( k , q ) = − b F ( 2 k , q ) ‾ , k ∈ C . Thus, solving the inverse problem for b given by (2.10), we obtain ψ ( x , t ) . Moreover, the NLS equation is a Hamiltonian system with the following Hamiltonian H = ∫ R ( | ψ x ( x , t ) | 2 + | ψ ( x , t ) | 4 ) d x . Such system is completely integrable and there exist action variables ϱ ( k ) and angle variables ϕ ( k , t ) , k ∈ R , given by ϱ ( k ) = 1 π log | a ( k , q ) | ∈ R , ϕ ( k , t ) = ( 4 k 2 t + arg b ( k , q ) ) mod 2 π , ( k , t ) ∈ R 2 , where q = ψ ( · , 0 ) (see, e.g., p. 230 in [11]). Note that there is the ambiguity in the definition of ϕ ( k , t ) at zeros of b ( k , q ) on the real line but it can be avoided by introducing new variables (see formula (7.9), p. 230 in [11]). Now, we give the representation of these variables in terms of resonances and zeros of b, when q ∈ P .

Theorem 2.7.

Let a = a ( · , q ) for some q ∈ P and let ( k n ) n ⩾ 1 be its zeros in C − counted with multiplicities and labeled by ( 2.2 ). Then the action variable ϱ ( k ) has the form (2.11) ϱ ( k ) = 1 π log | a ( 0 ) | + 1 π lim r → + ∞ ∑ | k n | ⩽ r log | 1 − k k n | , k ∈ R , where the series converges uniformly on compact subsets of R .

Let in addition, b = b ( · , q ) , ξ ( b ) = ( ξ n ) n ⩾ 0 , where ξ n = ( ξ n + ξ n − ) is defined by ( 2.6 ). Let ( z n ) n ⩾ 1 be zeros of B = b b ∗ in C + counted without multiplicities and labeled by ( 2.5 ) and let ( r n ) n ⩾ 1 be zeros of B in R ∖ { 0 } counted with multiplicities. Then the angle variable has the form (2.12) ϕ ( k , t ) = ( 4 k 2 t + arg ξ 0 + γ k + π I ( k ) + ∫ 0 k w ( s ) d s ) mod 2 π , k ∈ R ∖ ( r n ) n ⩾ 1 , where (2.13) I ( k ) = # { r n ∈ [ 0 , k ) , n ⩾ 1 } , k ⩾ 0 , # { r n ∈ ( k , 0 ) , n ⩾ 1 } , k < 0 , w ( k ) = ∑ n ⩾ 1 ( ξ n + − ξ n − ) Im z n | z n − k | 2 , and the series converges uniformly on compact subsets of R .

Remark.

1) It follows from (2.10) that if q ∈ P , then a ( · , ψ ( · , t ) ) and b ( · , ψ ( · , t ) ) are entire functions for any t ∈ R . Moreover, their zeros do not move when t changes.

2) If q ∈ P , then e 4 i t k 2 b ( k , q ) is not exponential type as function of k for any t ≠ 0 . Thus, by Theorem 2.1, ψ ( · , t ) is not compactly supported for any t > 0 .

2.3. Location of resonances

Now, we describe locations of resonances and zeros of b. Firstly, we discuss a forbidden domain for resonances.