{'fraction_non_alphanumeric': 0.0550180340069846, 'fraction_numerical': 0.019408026564378542, 'mean_word_length': 3.812121212121212, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
An improved algorithm for reconstructing reflectionless potentials

A fully algebraic approach to reconstructing one-dimensional reflectionless potentials is described. A simple and easily applicable general formula is derived, using the methods of the theory of determinants. In particular, useful properties of special determinants - the alternants - have been exploited. The main formula takes an especially simple form if one aims to reconstruct a symmetric reflectionless potential. Several examples are presented to illustrate the efficiency of the method.


I. INTRODUCTION
Formulation and solution of inverse problems is an increasingly important field of scientific research. However, compared with a well-posed (in Hadamard's sense) direct or forward problem, the corresponding inverse problem is much more difficult and, as a rule, ill-posed.
Inverse scattering problem can be considered an exception to this rule. Namely, in the simplest one-dimensional case, the inverse scattering theory provides strict mathematical criteria for the existence, uniqueness and stability of the solution. It means that in this particular case the inverse problem is well-posed as well.
The related forward problem is the solution of the simplest time-independent Schrödinger for a given potential V (x), and subjected to appropriate physical boundary conditions. Eq.
(1) can be easily solved numerically and thus, in principle, all spectral characteristics of the potential V (x) can be accurately ascertained.
The inverse problem is to determine the unknown potential starting from the known spectral characteristics. This is a serious task even in this simple case for the following reasons: 1. It is not obvious what kind of input information is actually needed to solve the problem uniquely.
2. There must be a theoretical basis (a fundamental equation) which enables to solve the problem.  Krein 4,5 and others (see, e.g., 6 , section III.7 for a brief overview). In addition, three different methods to solve the problem have been worked out, based on the integral equations by Gelfand-Levitan 3 , Marchenko 2 and Krein 5 .
Solution of the inverse scattering problem on the full line (−∞ < x < ∞) is a more challenging problem, which was first addressed by Kay 7 , Moses 8 , and few years later in a series of papers by  . These fundamental studies provide full description of the solution procedure, while the correct necessary and sufficient criteria for the uniqueness of the solution were given by Marchenko (see 13 , section III.5). These criteria apply to the following pair of integral equations 8,12,13 : which are formally very similar to the Marchenko equation on the half line (see 13 , p. 218). On the other hand, their operator-theoretical content is closer to the Gelfand-Levitan approach.
Therefore, as a kind of compromise, Eqs. (2) (where y > x) is completely specified by the so-called right scattering data, while the kernel A 2 in Eq. (3) (y < x) is determined by the left scattering data. These two sets of input data are equivalent: the left data are uniquely determined by the right ones and vice versa. In the following analysis we will rely on Eq.
(2). If one is able to solve this integral equation then the potential is given by Now, let us briefly discuss Point 3 in the above list. Unfortunately, the criteria for the uniqueness of the solution are so strict that it is nearly impossible to get all the necessary input data experimentally. However, if one sets an additional constraint that the resulting potential must be reflectionless then the inverse scattering problem can be solved much more easily. Moreover, a symmetric reflectionless potential is uniquely determined if its full spectrum of bound states is known. In the following analysis it is assumed that the potential V (x) that corresponds to Eq. (4) is reflectionless by definition.
As the general principles of building confining reflectionless potentials are long known 12,14 , a natural question arises: is there any need to revisit the topic? A motivation comes from Point 4 stated above: if the total number of bound states is large then the technical side of the procedure becomes important. This in turn motivates development of more efficient algorithms. In this paper, a new analytic algorithm is derived, which enables to easily calculate a reflectionless potential with an arbitrary number (N) of bound states.
The paper is organized as follows. In Sec. II, the general principles are briefly described, which form the overall basis for the approach. Secs. III and IV make an excursion to the theory of determinants, the benefits of which are described and illustrated in Sec. V. Finally, Sec. VI concludes the work.

UNIVERSAL RECIPE
Suppose we are given 2N parameters for an unknown reflectionless potential V (x): the positions of N discrete energy levels E n = −Cκ 2 n and N norming constants C n (n = 1, 2, ..., N) Then it can be shown 14 that Here A is a symmetric matrix with the following elements: A (n) is obtained from A by replacing the nth column with its derivative, and A simple formula can also be obtained for the potential 12,14 : which is uniquely determined by parameters κ n and C n .
Note that the potential remains unchanged if det (A) is multiplied by a function exp (αx + β), where α and β are arbitrary constants. Consequently, we can multiply, for example, any row of the initial matrix by a function exp [κ n (x − x n )] (n = 1, 2, ..., N), where x n are new parameters, equivalent to norming constants: and thus, according to definition (8), As a result, we get another matrix which contains full information for reconstructing the potential according to Eq. (9):Ã HereB Remark: A subscript was added to denote the number of the bound states (and the rank of the matrix).
From now on, the determinants having the structure det(A) · exp (αx + β) with the elements A mn defined in Eq. (7), will be sometimes called τ -functions (as is common in soliton theory). In the next two sections it will be shown that such determinants can be easily calculated even for an arbitrarily large N.

III. GENERAL FORMULA FOR THE τ -FUNCTIONS
To evaluate a non-trivial determinant, one can use the Laplace expansion (see 15  Consequently, applying Eq. (15) to the τ -function defined by Eqs. (12)- (14), where the coefficients a 0 , a i , a ij , a ijk ,... as well as the corresponding arguments of the exponents can be easily fixed with the help of Eqs. (12)- (14). Indeed, δ il being the Kronecker's symbol.
To further simplify Eq. (16), let us group the terms into pairs, so that the arguments of the corresponding exponents differ only by sign. For example, the first pair is formed of the terms with coefficients Here we defined a new coefficient A 0 , whose subscript "0" emphasizes that the expression Analogously, we can form a pair from a 12 exp(α 12 ) and a 345...N exp(−α 12 ), where The principle is simple: for any a n 1 n 2 ...n i exp(α n 1 n 2 ...n i ) the partner is a n i+1 n i+2 ...n N exp(−α n 1 n 2 ...n i ).
In addition, as we will see below, it is convenient to define a relevant coefficient where the indices point at the terms with plus sign on the right side of the expression Looking at the structure of the matrixÃ N , it is obvious that all these plus sign terms can only originate from the expansion of det B N and they correspond to the product The terms with minus sign on the right side of Eq. (19) are related to the expansion of det C N , without any contribution from det B N .
On the basis of the above arguments, the following conclusions can be made: • All terms on the right side of Eq. (16) can be grouped into pairs. There is only one term, a 123...N exp(−α 0 ) = det C N (with partner det B N = a 0 exp(α 0 )) which is entirely formed of the elements of the matrixC N . Any other term (both partners) contains some diagonal elements of the matrixB N as well.
• Any term a n 1 n 2 ...n i exp(α n 1 n 2 ...n i ) (i = 1, 2, ...) can be obtained by replacing all elements of the rows and columns n i+1 , n i+2 , ...n N of the matrixC N with corresponding elements of the matrixB N (mostly with zeros). As a result, one gets a modified matrix C N , while a n 1 n 2 ...n i exp(α n 1 n 2 ...n i ) = det C N . The Laplace expansion (15) of this determinant for the fixed rows n i+1 , n i+2 , ...n N contains only one term!
• For any term a n 1 n 2 ...n i exp(α n 1 n 2 ...n i ) of the expansion (16) there is a partner where C N can be obtained by replacing all elements of the rows and columns n 1 , n 2 , ...n i of the matrixC N with corresponding elements of the matrixB N . The Laplace expansion of det C N for the rows n 1 , n 2 , ...n i also contains only one term:

It means, for example, that
• One should avoid re-use of the terms: an already existing pair must not be included again! It means that the members of the modified expansion (16)  As the final result of the above analysis, we get the following general formula: where exp(β n 1 n 2 ...n i ) ≡ a n 1 n 2 ...n i a n i+1 n i+2 ...n N .
It is easy to be convinced that the expansion (22) contains 2 N −1 members in total (apart from inessential factor 2). Indeed, according to Newton's binomial theorem  where we took into consideration that only half of this formal series is actually needed.

IV. ALTERNANTS OF τ -FUNCTIONS
We have shown that not only det Ã N itself but also the coefficients in the expansion (22) are τ -functions. Consequently, the solution of the inverse scattering problem has been reduced to evaluating a number of determinants fixed by the parameters κ 1 < κ 2 < ... < κ n , n being an appropriate natural number. We are now going to derive a simple formula for calculating such τ -functions. First, let us set a one-to-one correspondence between each row of the determinant and a fixed parameter For example, the modified elements of the first row of Eq. (24) will be The usefulness of this trick soon becomes evident, although there seems to be only a formal change: Here we separated a factor 2q i from each row and √ κ i from each column, i.e., 2q i √ κ i = 2κ i (i = 1, 2, ..., n) from each such pair.
The result is where a new determinant was introduced. As can be seen, the factor 2 n · κ 1 κ 2 ...κ n was canceled out from Eq. (27).
We can see that the elements of the columns of D n correspond to different values of the same function, while any row is characterized by a single fixed parameter. Indeed, Eq. (28) can be expressed as ... ... ... ... where A determinant that has a structure of Eq. (29) is called alternant (see 16 which can be easily evaluated (see 17 , p. 16): An important point is that the factor V n (q 1 , q 2 , ..., q n ) can be separated from any nth-order (q n − q i ) . Analogous reasoning applied to q n−1 shows that D n also has a factor Putting it all together, we conclude that an nth-order alternate always has a factor V n (q 1 , q 2 , ..., q n ).
To continue the analysis, let us recall some useful properties of the elementary symmetric functions: Here σ k (k ≥ 2) is a sum of all possible products of exactly k variables arranged in ascending order of their indices. According to the Fundamental Theorem for symmetric polynomials (see 18 , p. 312), any such polynomial can be uniquely expressed as a polynomial in σ 1 , σ 2 , ..., σ n . This in turn is a basis for the following important theorem: Theorem 1: Let |A n | be an nth order alternant generated by the functions where the parameters a ij do not depend on x, and define S k ≡ (−1) k σ k (q 1 , q 2 , ..., q n ).
or, in a more compact form, where we took into consideration that F (q i ) = 0 if i = 1, 2, ..., n.
Consequently, D n = V n (q 1 , q 2 , ..., q n ) × |R n | , |R n | ≡ (39) σ n−1 (κ 2 , κ 3 , ..., κ n ) σ n−2 (κ 2 , κ 3 , ..., κ n ) ... Here we defined a new determinant |R n | which seems to be another alternant, so we can apply Theorem 1 to evaluate it. To be convinced that |R n | is indeed an alternant, we have to specify the generating functions. Obviously, Also, it is easy to prove that where we dropped the arguments to get a more compact formula. Let us agree that if no arguments are explicitly given then the corresponding function depends on n arguments: Taking, e.g., x = κ 1 we can check the validity of Eq. (41). Indeed, as needed according to Eq. (39). Continuing in the same manner, we get the following result: In summary, we have obtained a very simple and universal recipe for calculating determinants defined by Eq. (28): Combining Eqs. (27) and (47), we can formulate a general and important result: Theorem 2: Let κ 1 , κ 2 , .., κ n be arbitrary positive real numbers arranged in the ascending order, so that κ 1 < κ 2 < ... < κ n , and let D(κ 1 , κ 2 , .., κ n ) be a determinant, defined by Eq.
(24). Then where the product contains all possible combinations of the pairs (κ j , κ i ) satisfying the condition 1 ≤ i < j ≤ n.

V. SYMMETRIC REFLECTIONLESS POTENTIALS
The excursion to theory of determinants concluded with a surprisingly simple final result.
Indeed, as is seen from Eqs. (18), (21) and (23), all coefficients in Eq. (22) can be evaluated with the help of Eq. (48), which means that the general formula for τ -functions can be essentially simplified. For example, etc. Thus Eq. (22) transforms to and Eq. (12) can be rewritten as In general, as mentioned, the reflectionless potential is uniquely determined if 2N parameters κ n and C n (n = 1, 2, ..., N) are known. However, if one sets an additional constraint then the number of necessary input parameters is twofold reduced. In other words, a symmetric reflectionless potential is uniquely determined by its N binding energies 19 E n = −Cκ 2 n . Let us analyze this in more detail. where and consequently, Using Eqs. (17) and Eq. (23), we get a similar expression for any other combination κ i x i .
From Eqs. (10) and (55) we therefore obtain the following symmetricity conditions: If N = 1 then It can be easily shown that Eqs. (56) are indeed the symmetricity conditions for τ N and V (x). To this end, in full analogy with Eq. (54), we can write Summing these equations, we get which coincides with Eq. (53).
Remark: If N = 2 then first two equations of the system (58) are not linearly independent, since The next step is to complement Eq. (58), for example, with another condition which is a direct conclusion from Eq. (56). Thus which means that is a symmetric function. Analogously, one can prove that any other term cosh(α in Eq. (51) is a symmetric function as well. This in turn proves that the norming constants C n of a symmetric reflectionless potential are uniquely determined by the given binding energies E n .

A. Few practical examples
To illustrate the results, let us take, for example, N = 4. Then Eq. (51) reads while the symmetricity conditions, according to Eqs. (54)-(56), can be given as Summing the corresponding sides of Eq. (61) we get is a symmetric function.
Analogously, substructing the sides of the last two equations from the corresponding sides of the first two equations of (61), we get which means that cosh(α 12 + β 12 ) is a symmetric function. Continuing in a similar manner, it is easy to be convinced that cosh(α 13 + β 13 ) and cosh(α 14 + β 14 ) are symmetric functions as well.
Therefore, according to Eq. (60), and exactly as needed.
Example 2: Next, let us construct a reflectionless approximant to a symmetric rectangular potential with four energy levels (see Fig. 1). These levels can be determined from (see, e.g., 20 , Sect. II.9) where a and U 0 denote the half-width and the depth of the potential well, respectively.
Eq. (63) fixes the symmetric and (64) -the antisymmetric solutions to the Schrödinger equation. Again, it is convenient to use dimensionless units for the length and energy, taking x 0 = a = 1 and E 0 = C/a 2 = 1. In addition, let us fix U 0 a 2 C = 5.
Then the system has four discrete levels (as assumed) corresponding to κ 1 = 1.3064400089, In this case Eq. (60) cannot be further simplified, but this is not any serious problem.
Indeed, let us define the coefficients A i and B i , such that Then the corresponding potential becomes The result for the input data (65) can be seen in Fig. 1.
Example 3: Fig. 2 demonstrates two isospectral potentials corresponding to the following the four energy levels being E n = −Cκ 2 n (n = 1, 2, 3, 4) as previously. The solid curve in this figure corresponds to a Morse potential 21 where a ≡ D/C/α. In Fig. 2, the value a = 4 has been chosen, so that α = 1/4 in our dimensionless units. As in the previous example, the red dashed curve shows the symmetric reflectionless potential derived by Eq. (66) from the input parameters (67).

VI. CONCLUSION
The main result of this work is a general formula for calculating τ -functions. This important formula, Eq. (51), is a direct conclusion from Theorem 2 that has been proved with the help of well-known methods of the theory of determinants. We demonstrated that τ -functions can be expanded in terms of special determinants called alternants 16  The efficiency of the method has been explicitly demonstrated for the case N = 4, and there is no doubt that the algorithm can be successfully applied to a much higher number (in principle, to an arbitrary number) of given binding energies. The described approach can also be applied to building N-soliton solutions to the Korteweg-de Vries equation, but this would be a subject for another paper.