Wellposedness of the Inverse Problem for Dirac Operator

Abstract

We study the wellposedness of the inverse problem for Dirac operator. We consider two different problems (unperturbed and perturbed problem) for Dirac operator, and then we prove that if the spectral characteristics of these problems are close to each other, then the difference between their potential functions is sufficiently small.

1. Introduction

Inverse problems are studied for certain special classes of ordinary differential operators. Typically, in inverse eigenvalue problems, one measures the frequencies of a vibrating system and tries to infer some physical properties of the system. An early important result in this direction, which gave vital impetus for the further development of inverse problem theory, was obtained in [1–5].

The Dirac equation is a modern presentation of the relativistic quantum mechanics of electrons, intended to make new mathematical results accessible to a wider audience. It treats in some depth the relativistic invariance of a quantum theory, self-adjointness and spectral theory, qualitative features of relativistic bound and scattering states, and the external field problem in quantum electrodynamics, without neglecting the interpretational difficulties and limitations of the theory.

Inverse problems for Dirac system had been investigated by Moses [6], Prats and Toll [7], Verde [8], Gasymov and Levitan [9], and Panakhov [10, 11]. It is well known [12] that two spectra uniquely determine the matrix-valued potential function. In particular, in [13], eigenfunction expansions for one-dimensional Dirac operators describing the motion of a particle in quantum mechanics are investigated. Recently, Dirac operators have been extensively studied [14–19].

Mizutani showed the wellposedness problem of the Sturm-Liouville operator according to norming constants and eigenvalues [20]. The purpose of this paper is to give the wellposedness problem for Dirac operator by using Mizutani's method.

Let 𝐿 denote a matrix operator 𝑝𝐿≡¹¹(𝑥)𝑝₁₂(𝑥)𝑝₂₁(𝑥)𝑝₂₂(𝑥),𝑝₁₂(𝑥)=𝑝₂₁(𝑥), (1) where the 𝑝_𝑖𝑘(𝑥)(𝑖,𝑘=1,2) are real functions which are defined and continuous on the interval [0,𝜋]. Further, let 𝜑(𝑥,𝜆) denote a two-component vector-function 𝜑()=𝜑𝑥,𝜆¹()𝜑𝑥,𝜆₂(𝑥,𝜆). (2) Then the equation 𝐵𝑑𝑑𝑥+𝐿−𝜆𝐼𝜑=0, (3) where 𝜆 is a parameter and 𝐵≡01−10,𝐼≡1001, (4) is equivalent to the system of two simultaneous first-order ordinary differential equations 𝑑𝜑₂𝑑𝑥+𝑝₁₁(𝑥)𝜑₁+𝑝₁₂(𝑥)𝜑₂=𝜆𝜑₁,−𝑑𝜑₁𝑑𝑥+𝑝₂₁(𝑥)𝜑₁+𝑝₂₂(𝑥)𝜑₂=𝜆𝜑₂. (5)

For the case in which 𝑝₁₂(𝑥)=𝑝₂₁(𝑥)=0, 𝑝₁₁(𝑥)=𝑉(𝑥)+𝑚, 𝑝₂₂(𝑥)=𝑉(𝑥)−𝑚, where 𝑉(𝑥) is a potential and 𝑚 is the mass of a particle, the system (5) is known in relativistic quantum theory as a stationary one-dimensional Dirac system [5].

Consider the following eigenvalue problem for Dirac operator: 𝐵𝑦^′(𝑥)𝑦+𝑄𝑦=𝜆𝑦,0≤𝑥≤𝜋,₁(0)cos𝛼+𝑦₂(0)𝑦sin𝛼=0,₁(𝜋)=0, (6)(7)(8) where 𝑄(𝑥)=^{𝑝(𝑥)𝑞(𝑥)}_{𝑞(𝑥)−𝑝(𝑥)}, 𝑦(𝑥)=^𝑦1_𝑦^(𝑥)_2(𝑥), 𝑝(𝑥) and 𝑞(𝑥)∈𝐶¹[0,𝜋] are real-valued functions, 0≤𝛼≤𝜋, and 𝜆 is spectral parameter.

We denote by 𝜑()=𝜑𝑥,𝜆¹()𝜑𝑥,𝜆₂()=()𝑒𝑥,𝜆sin𝜆𝑥−𝛼+𝑂^𝜏𝑥𝜆()𝑒cos𝜆𝑥−𝛼+𝑂^𝜏𝑥𝜆 (9)

the solution of (6) satisfying the initial conditions 𝜑₁(0,𝜆)=−sin𝛼,𝜑₂(0,𝜆)=cos𝛼, (10) where 𝜏=|Im𝜆|.

We show the spectral characteristics of the problems (6)–(8) by {𝜆_𝑛,𝜌_𝑛}(𝑛∈ℤ), where {𝜆_𝑛} is the spectrum (set of eigenvalues) and {𝜌_𝑛} is the norming constants of this problem. These spectral characteristics satisfy the following asymptotic expressions, respectively [9]: 𝜆_𝑛𝛼=𝑛−𝜋+𝛼₁𝑛1+𝑂𝑛²,𝜌_𝑛𝑐=𝜋+¹𝑛1+𝑂𝑛², (11) where 𝛼, 𝛼₁, and 𝑐₁ are real numbers.

Now let us consider the second eigenvalue problem 𝐵𝑦^′+𝑄(𝑥)𝑦=𝜇𝑦,0≤𝑥≤𝜋, (12) with the same boundary conditions (7), (8), where 𝑝𝑄(𝑥)=^(𝑥)𝑞^(𝑥)𝑞_(𝑥)−𝑝_(𝑥), 𝑝(𝑥) and 𝑞(𝑥)∈𝐶¹[0,𝜋] are real-valued functions and 𝜇 is spectral parameter. We denote the spectral characteristics of the problems (12), (7), and (8) by {𝜇_𝑛,𝜎_𝑛}(𝑛∈ℤ), where {𝜇_𝑛} is the spectrum and {𝜎_𝑛} is the norming constants of this problem. These spectral characteristics satisfy the following asymptotic expressions, respectively: 𝜇_𝑛𝛼=𝑛−^′𝜋+𝛼^′₁𝑛1+𝑂𝑛²,𝜎_𝑛𝑐=𝜋+^′1𝑛1+𝑂𝑛², (13) where 𝛼^′, 𝛼^′₁, and 𝑐^′₁ are real constants.

Theorem 1. Let 𝐸 be a linear topological space and 𝐸₁,𝐸₂⊂𝐸, where 𝐸₁ and 𝐸₂ are linear topological subspaces of 𝐸. The transformation operator 𝑋=𝑋_𝐴,𝐵 mapping 𝐸₁ to 𝐸₂ can be defined as follows [9]: 𝑋𝜑(𝑥,𝜆)=𝜑(𝑥,𝜆)=𝜑(𝑥,𝜆)+^𝑥₀𝐾(𝑥,𝑠)𝜑(𝑠,𝜆)𝑑𝑠, (14) where the kernel 𝐾(𝑥,𝑠) is a solution of the equation 𝐵𝜕𝐾+𝑄(𝑥)𝐾()𝜕𝑥𝑥,𝑠=−𝜕𝐾(𝑥)𝐾()𝜕𝑠𝐵+𝑄𝑥,𝑠 (15) and also satisfies the following conditions: ()()𝑄(𝑥)(𝑥),𝐾𝐵𝐾𝑥,𝑥−𝐾𝑥,𝑥𝐵=−𝑄₂₁(𝑥,0)=𝐾₁₁(𝑥,0)=0, (16)(17) and 𝐾(𝑥,𝑠)=^{𝐾11(𝑥,𝑠)𝐾12}_𝐾^(𝑥,𝑠)_{21(𝑥,𝑠)𝐾22(𝑥,𝑠)} is continuously differentiable on [0,𝜋].
The 𝐾(𝑥,𝑠) function in (14) satisfies the following integral equation: 𝐾(𝑥,𝑠)+𝐹(𝑥,𝑠)+^𝑥₀𝐾(𝑥,𝑡)𝐹(𝑡,𝑠)𝑑𝑡=0,0≤𝑠≤𝑥≤𝜋, (18) where 𝐹(𝑥,𝑠) function is as such 𝐹(𝑥,𝑠)=^∞_𝑛=1𝜑𝑥,𝜇_𝑛𝜑^𝑇𝑠,𝜇_𝑛𝜎_𝑛−𝜑𝑥,𝜆_𝑛𝜑^𝑇𝑠,𝜆_𝑛𝜌_𝑛, (19) and 𝐹(𝑥,𝑠)=^{𝐹11(𝑥,𝑠)𝐹12}_𝐹^(𝑥,𝑠)_{21(𝑥,𝑠)𝐹22(𝑥,𝑠)} is continuously differentiable on [0,𝜋].

Now we present main theorem in this paper.

2. Main Results

Theorem 2. If 𝑀≡^∞_𝑛=1𝜎_𝑛−𝛼_𝑛+𝜇_𝑛−𝜆_𝑛 (20) is sufficiently small, then we have max_{0≤𝑥≤𝜋}𝑝(𝑥)−𝑝(𝑥)≤𝐶^′𝑀,max_{0≤𝑥≤𝜋}𝑞(𝑥)−𝑞(𝑥)≤𝐶^′′𝑀, (21) where 𝐶^′>0 and 𝐶^′′>0 are constants.

Proof. Let us solve the Gelfand-Levitan equation (18). We start from 𝐹(𝑠,𝑡) and construct the iterated kernels 𝐹^(𝑛)(𝑠,𝑡;𝑥)(𝑛=1,2,3,…) as follows: 𝐹⁽¹⁾()(),𝐹𝑠,𝑡;𝑥=𝐹𝑠,𝑡_(𝑛+1)(𝑠,𝑡;𝑥)=^𝑥₀𝐹(𝑠,𝑢)𝐹^(𝑛)(𝑢,𝑡;𝑥)𝑑𝑢, (22) for 𝑛≥1.
We put 𝑆(𝑠,𝑡;𝑥)=∑^∞_𝑛=1(−1)^𝑛𝐹^(𝑛)(𝑠,𝑡;𝑥) and assume that ^𝜋₀|𝐹(𝑠,𝑡)|²𝑑𝑠𝑑𝑡<1. (23)
Then it is easily seen that 𝐾(𝑥,𝑠)=𝑆(𝑥,𝑠;𝑥),for0≤𝑠≤𝑥≤𝜋. (24) From (19), we have 𝐹(𝑥,𝑠)=^∞_𝑛=1𝜑𝑥,𝜇_𝑛𝜑^𝑇𝑠,𝜇_𝑛𝜎_𝑛−𝜑𝑥,𝜆_𝑛𝜑^𝑇𝑠,𝜆_𝑛𝜌_𝑛=^∞_𝑛=11𝜎_𝑛𝜑₁𝑥,𝜇_𝑛𝜑₂𝑥,𝜇_𝑛𝜑₁𝑠,𝜇_𝑛𝜑₂𝑠,𝜇_𝑛−1𝜌_𝑛𝜑₁𝑥,𝜆_𝑛𝜑₂𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛𝜑₂𝑠,𝜆_𝑛=^∞_𝑛=1𝜑₁𝑥,𝜇_𝑛𝜑₁𝑠,𝜇_𝑛𝜎_𝑛−𝜑₁𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛𝜌_𝑛×𝜑₁𝑥,𝜇_𝑛𝜑₂𝑠,𝜇_𝑛𝜎_𝑛−𝜑₁𝑥,𝜆_𝑛𝜑₂𝑠,𝜆_𝑛𝜌_𝑛𝜑₂𝑥,𝜇_𝑛𝜑₁𝑠,𝜇_𝑛𝜎_𝑛−𝜑₂𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛𝜌_𝑛×𝜑₂𝑥,𝜇_𝑛𝜑₂𝑠,𝜇_𝑛𝜎_𝑛−𝜑₂𝑥,𝜆_𝑛𝜑₂𝑠,𝜆_𝑛𝜌_𝑛. (25) From (25), it is easy to see that 𝐹₁₁(𝑥,𝑠)=^∞_𝑛=1𝜑₁𝑥,𝜇_𝑛𝜑₁𝑠,𝜇_𝑛𝜎_𝑛−𝜑₁𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛𝜌_𝑛,𝐹₁₂(𝑥,𝑠)=^∞_𝑛=1𝜑₁𝑥,𝜇_𝑛𝜑₂𝑠,𝜇_𝑛𝜎_𝑛−𝜑₁𝑥,𝜆_𝑛𝜑₂𝑠,𝜆_𝑛𝜌_𝑛,𝐹₂₁(𝑥,𝑠)=^∞_𝑛=1𝜑₂𝑥,𝜇_𝑛𝜑₁𝑠,𝜇_𝑛𝜎_𝑛−𝜑₂𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛𝜌_𝑛,𝐹₂₂(𝑥,𝑠)=^∞_𝑛=1𝜑₂𝑥,𝜇_𝑛𝜑₂𝑠,𝜇_𝑛𝜎_𝑛−𝜑₂𝑥,𝜆_𝑛𝜑₂𝑠,𝜆_𝑛𝜌_𝑛. (26)(27)(28)(29)
Firstly, we calculate formula of (26). Adding and subtracting 𝜑₁(𝑥,𝜆_𝑛)𝜑₁(𝑠,𝜆_𝑛)/𝜎_𝑛 to the right side of (26), then we obtain 𝐹₁₁(𝑥,𝑠)=^∞_𝑛=1𝜑₁𝑥,𝜇_𝑛𝜑₁𝑠,𝜇_𝑛𝜎_𝑛−𝜑₁𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛𝜌_𝑛+^∞_𝑛=1𝜑₁𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛𝜎_𝑛−𝜑₁𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛𝜎_𝑛=^∞_𝑛=1𝜌_𝑛−𝜎_𝑛𝜌_𝑛𝜎_𝑛𝜑₁𝑥,𝜆_𝑛𝜑₁𝑠,𝜆_𝑛+1𝜎_𝑛^𝜇𝑛_𝜆𝑛𝜑₁(𝑥,𝜆)𝜑₁(𝑠,𝜆)^.𝑑𝜆, (30) where ^.=𝜕/𝜕𝜆. If the absolute value is taken in the formula of (30) and (9) is used and 𝑠=𝑥 is written, the following is obtained: 𝐹₁₁(𝑥,𝑥)≤^∞_𝑛=1𝜌_𝑛−𝜎_𝑛𝜌_𝑛𝜎_𝑛𝑐₁+1𝜎_𝑛^𝜇𝑛_𝜆𝑛𝑐₂≤𝑑𝜆_∞𝑛=1𝜌_𝑛−𝜎_𝑛𝑐₃+𝑐₄^𝜇𝑛_𝜆𝑛𝑑𝜆≤𝐶₁^∞_𝑛=1𝜎_𝑛−𝜌_𝑛+𝜇_𝑛−𝜆_𝑛, (31) where 𝑐₁, 𝑐₂, 𝑐₃, 𝑐₄, and 𝐶₁ are constants and positive numbers. From (20), we have 𝐹₁₁(𝑥,𝑥)≤𝐶₁𝑀. (32) From (24) and (32) for 𝑛=1, we have 𝐹⁽²⁾₁₁(𝑠,𝑡;𝑥)=^𝑥₀𝐹₁₁(𝑠,𝑢)𝐹⁽¹⁾₁₁()𝐹𝑢,𝑡;𝑥𝑑𝑢₍₂₎₁₁(≤𝐶𝑠,𝑡;𝑥)¹𝑀𝜋𝜋², (33) similarly for 𝑛=2𝐹⁽³⁾₁₁(𝑠,𝑡;𝑥)=^𝑥₀𝐹₁₁(𝑠,𝑢)𝐹⁽²⁾₁₁()𝐹𝑢,𝑡;𝑥𝑑𝑢₍₃₎₁₁(≤𝐶𝑠,𝑡;𝑥)¹𝑀𝜋³𝜋,⋮ (34) consequently, 𝐹^(𝑛)₁₁(𝑠,𝑡;𝑥)=^𝑥₀𝐹₁₁(𝑠,𝑢)𝐹^(𝑛−1)₁₁()𝐹𝑢,𝑡;𝑥𝑑𝑢_(𝑛)11(≤𝐶𝑠,𝑡;𝑥)¹𝑀𝜋^𝑛𝜋. (35) From (24), we get the following: 𝐾₁₁(𝑥,𝑠)≤^∞_𝑛=1(−1)^𝑛𝐹^(𝑛)₁₁(𝑠,𝑡;𝑥). (36) From (35) and (36), we have 𝐾₁₁(𝑥,𝑥)≤^∞_𝑛=1(−1)^𝑛𝐶₁𝑀𝜋^𝑛𝜋=^∞_𝑛=1𝐶₁𝑀𝜋^𝑛𝜋. (37) If 𝐶₁𝑀𝜋 is small, for example, 𝐶₁𝑀𝜋<1/2, then we have 𝐾₁₁(𝑥,𝑥)≤^∞_𝑛=1𝐶₁𝑀𝜋^𝑛𝜋=𝐶₁𝑀+𝜋𝐶₁𝑀²+𝜋²𝐶₁𝑀³+⋯≤𝐶₁𝑀11+2+12²+⋯≤2𝐶₁𝑀. (38) If the operations for 𝐹₁₁(𝑥,𝑠) are repeated, we get the following: 𝐹₁₂(𝑥,𝑠)≤^∞_𝑛=1𝜌_𝑛−𝜎_𝑛𝜌_𝑛𝜎_𝑛𝑐₅+1𝜎_𝑛^𝜇𝑛_𝜆𝑛𝑐₆,≤𝑑𝜆_∞𝑛=1𝜌_𝑛−𝜎_𝑛𝑐₇+𝑐₈^𝜇𝑛_𝜆𝑛𝑑𝜆,≤𝐶₂^∞_𝑛=1𝜎_𝑛−𝜌_𝑛+𝜇_𝑛−𝜆_𝑛, (39) where 𝑐₅, 𝑐₆, 𝑐₇, 𝑐₈, and 𝐶₂ are constants and positive numbers.
From (20), we have 𝐹₁₂(𝑥,𝑠)≤𝐶₂𝑀. (40) Similarly from (24) and (40) for 𝑛=1, we have 𝐹⁽²⁾₁₂(𝑠,𝑡;𝑥)=^𝑥₀𝐹₁₂(𝑠,𝑢)𝐹⁽¹⁾₁₂()𝐹𝑢,𝑡;𝑥𝑑𝑢₍₂₎₁₂(≤𝐶𝑠,𝑡;𝑥)²𝑀𝜋²𝜋, (41) for 𝑛=2, 𝐹⁽³⁾₁₂(𝑠,𝑡;𝑥)=^𝑥₀𝐹₁₂(𝑠,𝑢)𝐹⁽²⁾₁₂()𝐹𝑢,𝑡;𝑥𝑑𝑢₍₃₎₁₂(≤𝐶𝑠,𝑡;𝑥)²𝑀𝜋³𝜋, (42) consequently, ⋮𝐹^(𝑛)₁₂(𝑠,𝑡;𝑥)=^𝑥₀𝐹₁₂(𝑠,𝑢)𝐹^(𝑛−1)₁₂()𝐹𝑢,𝑡;𝑥𝑑𝑢,_(𝑛)12(≤𝐶𝑠,𝑡;𝑥)²𝑀𝜋^𝑛𝜋. (43) From (24), we get the following: 𝐾₁₂(𝑥,𝑠)=^∞_𝑛=1(−1)^𝑛𝐹^(𝑛)₁₂(𝑠,𝑡;𝑥). (44) If we repeat the above operations, we have 𝐾₁₂(𝑥,𝑥)≤^∞_𝑛=1(−1)^𝑛𝐶₂𝑀𝜋^𝑛𝜋=^∞_𝑛=1𝐶₂𝑀𝜋^𝑛𝜋. (45) If 𝐶₂𝑀𝜋 is small, for example, 𝐶₂𝑀𝜋<1/2, then we have 𝐾₁₂(𝑥,𝑥)≤^∞_𝑛=1𝐶₂𝑀𝜋^𝑛𝜋=𝐶₂𝑀+𝜋𝐶₂𝑀²+𝜋²𝐶₂𝑀³+⋯≤𝐶₂𝑀11+2+12²+⋯≤2𝐶₂𝑀. (46) When the similar operations are repeated, the following is obtained: 𝐾₂₁(𝑥,𝑥)≤2𝐶₃𝐾𝑀,₂₂(𝑥,𝑥)≤2𝐶₄𝑀, (47) where 𝐶₃>0 and 𝐶₄>0 are constants.
On the other side, from (16) we get the following: ()()𝐾𝐵𝐾𝑥,𝑥−𝐾𝑥,𝑥𝐵=01−10¹¹𝐾₁₂𝐾₂₁𝐾₂₂−𝐾₁₁𝐾₁₂𝐾₂₁𝐾₂₂=𝐾01−10₁₂+𝐾₂₁𝐾₂₂−𝐾₁₁𝐾₂₂−𝐾₁₁−𝐾₁₂−𝐾₂₁=𝐾₁₂+𝐾₂₁𝐾₂₂−𝐾₁₁𝐾₂₂−𝐾₁₁−𝐾₁₂−𝐾₂₁=𝑝(𝑥)(𝑥)𝑞(𝑥)(𝑥)𝑞(𝑥)(𝑥)𝑝(𝑥)−𝑝(𝑥).−𝑝−𝑞−𝑞 (48) From here, we have 𝑝(𝑥)−𝑝(𝑥)=𝐾₁₂+𝐾₂₁,𝑞(𝑥)−𝑞(𝑥)=𝐾₂₂−𝐾₁₁. (49) Consequently from (38), (46), (47), and (49), we obtained the desired estimate max_{0≤𝑥≤𝜋}𝑝(𝑥)−𝑝(𝑥)≤𝐶^′𝑀,max_{0≤𝑥≤𝜋}𝑞(𝑥)−𝑞(𝑥)≤𝐶^′′𝑀, (50) where 𝐶^′>0 and 𝐶^′′>0 are constants.

3. Conclusion

The more norming constants and spectrums which are taken as spectral characteristics of Dirac operators are close to each other, the more difference of potential functions is sufficiently small.