CC BY
20Sturm-Liouville problem with general inverse symmetric
potential
Ghulam Rabani Romal Rasa romal.rasa1399@gmail.com Yazd University, Iran Ghulam Hazrat Aimal Rasa aimal.rasa14@gmail.com Kabul Education University, Afghanistan
Abstract: For an inverse nonselfadjoint Sturm-Liouville problem with a symmetric potential and general boundary conditions, the uniqueness theorems are established and proven. Six eigenvalues and a spectrum are the spectral information utilized for the original reconstruction of Sturm-Liouville problems. Additionally, it is established that an inverse self-adjoint Sturm-Liouville problem with symmetric potential and nonseparated boundary conditions is unique. The unique reconstruction of Sturm-Liouville problems is accomplished by these theorems using a spectrum and two (or three) eigenvalues. The theorems apply the traditional Sturm-Liouville results of G. Borg and N. Levinson to the case of problems with general boundary conditions. With symmetric potential and general boundary conditions, schemes for the original reconstruction of Sturm-Liouville problems are provided.
Keywords: boundary conditions, the inverse eigenvalue problem, general inverse Sturm-Liouville, problem symmetric potential
1. Introduction
Let L stands for the Sturm-Liouville problem
Ly = -x) + p(x)y(x) = Ay(x) (i) u,(y) = an y(0) + a, 2 y ' (0) + a, 3 y(l) + a, 4 y(l) = 0, (2)
Where p(x) G Li(0,1) is a real function so that p(x) =p(1 - x)the virtually
everywhere (a.e.) a,j with 1 =1,2 and j =1,2,3,4 are complex constants.
The inverse Sturm-Liouville problem for L in the case of separated boundary conditions and the boundary value problem for second-order differential equations
(a13 - a14 - a22 = a22 = 0) have undergone thorough research (see [1, 2, 3, 4, 6,9, 11, 13, 14, 15, 16]). Researchers V.A.Sadovnichii, V.A.Yurko, V.A.Marchenko, O.A.Plaksina, I.M.Guseinov, and I.M.Nabiev investigated the inverse Sturm-Liouville problem with unknown coefficients in nonseparated boundary conditions (see [4, 5, 8, 9, 14, 16,]).
For the inverse reconstruction problem, L in all coefficients of which a,j with 1 =1,2 and j =1,2,3,4 are unknown, no uniqueness theorems have been proved. Special cases of problem L with boundary conditions
V (y) = auy (0) + y'(0) + a 13 y(1) = 0, (3)
V2 ( y) = a2iy(0) + y'(1) + a23 y(1) = 0, (4)
and
W,(y) = y(0) + p y(1) = 0, (5)
W2(y) = Py ' (0) + y' (1) + a y(1) = 0, (6)
previously researched. The following sorts of generic self-adjoin nonseparated border conditions (2) can be reduced to: the boundary conditions (3),(4), where a" and a23 if any numbers are real, a13 * 0 is any complex number, a21 = -a13, also the boundary conditions (5), (6), where P * 0 where is any complex number and is any real integer.
In addition to the problem's spectrum, the spectra of two other boundary value problems, a specific set of signs, a specific real number were employed in order to uniquely recreate these boundary value problems with asymmetric potential (see, e.g., [4,6]). Using symmetric potential and general boundary conditions (2), which may not be self-adjoint, we demonstrate a theorem in this study about the unique reconstruction of problem L. The only spectrum data used are the eigenvalues of three spectral problems.
2. Objectives of this research
The purpose of this research paper is discussion for the problems of homogeneous linear differential equations of the second order by Sturm-Liouville boundary conditions of the general inverse with Symmetric Potential problem.
3. Methodology
Information has been collected in the form of libraries, websites, domestic and foreign scientific articles, undergraduate and doctoral research dissertations.
4. Literature review
Differential equations have been developed for nearly 300 years, and the relationship between evolutions are functions and derivatives of functions, so its history naturally dates back to the discovery of the derivative by the English scientist Isaac Newton (1642-1772) and Gottfried Wilhelm Leibniz (Germany (1646-1716)) began. Newton worked on differential equations, including first-order differential equations, into forms. Jacob proposed the Bernoulli differential equation in (1674), but failed to prove it until Euler proved it in (1705).
In the linear differential equations of the boundary problem, Sturm-Lowville first worked, the Sturm-Lowville theory in mathematics and its applications, the classical Sturm-Lowville theory, named after Jacques François Sturm (1803-1855) and Joseph
Lowville (1809-1882), the theory of linear differential equations is the second real order of form. In 1969, the Russian scientist Nymark wrote in his book Linear Differential Operators about the Green function to solve differential equations with boundary problem conditions.
The inverse Sturm-Liouville problem for L in the case of separated boundary conditions and the boundary value problem for second-order differential equations
(a13 - a14 - a22 = a22 = o) have undergone thorough research research (see [1, 2, 3, 4, 6,9, 11, 13, 14, 15, 16].)
Researchers V.A. Sadovnichii, V.A. Yurko, V.A. Marchenko, O.A. Plaksina, I.M. Guseinov, and I.M. Nabiev investigated the inverse Sturm-Liouville problem with unknown coefficients in nonseparated boundary conditions (see [4, 5, 8, 9, 14, 16,]).
5. Borg's Uniqueness Theorems Generalization
For the inverse Sturm-Liouville problem's one and two with q(x) G Li(0,1) solution in 1946, Borg established a number of uniqueness theorems [6, p. 69]. Two of them mentioned the spectrum issues listed below.
1. ly - -y"(x) + p(x)y(x) - Ay(x), y(0) - 0, y(1) - 0, p(x) - p(1 - x) a.e.
2. ly --y "( x) + p(x)y( x) -Ay(x), y " (0) - 0, y '(1) - 0, p( x) - p(1 - x) a.e.
Borg proved the following theorems (in Borg's notation) for these problems [6, p. 69].
Theorem 1. The spectrum of Problem (1) is the sole determinant of the function
p( x) if p( x) - p( x -1) a.e.
Theorem 2. If p( x) - p( x -1) a.e., then the spectrum of Problem (2) determines the
function p( x) in (1) uniquely.
To the problem of general boundary conditions, we generalize these theorems in this work (2).
In the terms that follow, we'll refer to a problem of type L as e L . This problem has a separated set of equation coefficients and boundary form parameters. We assume that if a symbol designates an object from Problem L, then the identical symbol with
the tilde L designates the same object from Problem L throughout the work.
Let M denote the matrix coefficients ik of the boundary conditions' (2), i. e. ,
m -
2 3 ai4
V a21 a22 a23 a24 7
and let be its minors composed of 1 th and j th columns
A =
ai j
a2i a2 j
i, j = 1,2,3,4.
Boldface letters are used to denote vectors. Transposition is denoted by the symbol T. Rows with this superscript serve as column vector representations. We use the notation rank M to represent the rank of the matrix M.
Problems are also present in space of L (1) and (2) we consider the following Problems
ly = - y " (x) + P( x)y (x) = Ay( x), u u( y) = y(0) -*(A) y '(0) = 0, U 2J( y) = y(i) = 0,
and
ly = - y" (x) + p( x)y (x) = Ay( x), U i,i( y) = y '(0) -*(A) y(0) = 0,
u 2,i (y) = y' (1) = 0,
In Problems k(A is a polynomial of the form:
k(1) = a12 + (i - a13)a+(a14 - a2)a2 + a2a3 + a4a4.
Theorem 3. If Problems L and L have a nonempty discrete spectrum; the spectra
of problems L and L, (1) and (i), Li and Li coincide with algebraic multiplicities taken into account, and rank M=2 , then these boundary value problems themselves
coincide, i.e., p(x) = ~(x) a.e. and the matrices M = (a'j )24 and M = (a'J)24
Proof. When we refer to Problem (1) Borg's uniqueness theory P1 for the inverse Sturm-Liouville problem with symmetric potential [6, p. 69], we observe that
P(x) = ~(x) a• e. (7)
Let us show that, for the vectors N =(Ai2Ai3Ai4As2'A42A34) and
N =(4^4J composed of the minors of the matrices (a,J)2x4 and (a,J)2x4 respectively, we have:
N = N (8)
Let yi( x,^) and y2( x,^) be linearly independent solutions to Equation (1) that meet the conditions
y(0,A) = i, yi(0,A) = 0, y2(0,A) = 0, y2(0,A) = i. (9)
The roots of the entire function are the Problem L eigenvalues. ([17, pp. 33-36], [19, p. 29].
A(A) = ai2 + a34 + (n,a) + a42y[(n,a) + a^(n,a) + a^(n,a), (10)
and the roots of the entire function are Problems Li eigenvalues.
Ai (A) = -^(A) yi (n, A) + y2 (n, A)
If A(A) *0 (i.e., the spectrum of the boundary value problem is discrete), Hadamard's Theorem states that the function A(A) can be reconstructed from its zeros up to a factor C * 0. Therefore, the functions A(A) and A(A) are related by the identity
m=c a(a) (ii)
Where C is a nonzero constant.
If AW -0 (i.e., each A is an eigenvalue of Problem L), subsequently, the condition that the eigenvalues of Problems L and L coincide also implies (11) whence A(A) = 0. Similarly, we have:
A1(*)
= C
Ai(*)
Where Ci is a nonzero constant. The following asymptotic relations hold:
y (x, A) = cos VXx + ^^ f (x) sin VXx + 0(-1),
y[ (x, A) = -VA sin VAx + f (x) cos VAx + 0(-^),
y (x, A) = sin yfAx - — f (x) cosVAx + 0(—
"J A A Ayj A
y'2 (x, A) = cos VAx + ^^ f (x) cosVAx + 0(-1),
1 r
f (x) = -l q(t)dt
Where 2 and for sufficiently large Ag ir .
These connections imply that the functions yi(^,A) and y2(n,A) in the decomposition of the function Ai (A) are linearly independent. Therefore,
A12 = A12, A13 = A13, A42 = A42, A34 = A34, (12)
The functions >i(^'A) = y2(^,A), and 1 in the decomposition of A(A) are linearly
independent as well (the relation >i(^'A) = y2(^,A) holds if and only if p( x) = p( x . These observations, together with (10) and (11), implies
= C
A1 2 + A34
A1 2 + A34 A 4 + A3 2 = C
Ai4 + A32 (13a)
A
= C
A42
4-3 = C
A13
(13b)
At least one of the numbers Al2 + As4' A32 + Al4 A42and Al3 is different from zero. Alternatively, we would have A(^ = 0 contradiction to the theorem's presumption that Problems L and L have discrete spectrum. This finding, along with (12), (13a) and (13b) suggests
If C =1 then the result we have:
A = A A = A A = A A = A A = A A = A
a12 a12 , A13 A13, a14 a14, a32 a32, a42 a42, a34 a34,
Whence we obtain (8).
From (8) it follows that the matrices (a'J )2 4 and (A)2 4 coincide up to a linear transformation of the rows (see [4, p. 32]). Combining this with (7), it becomes clear that the boundary value problems L and LA are related.
Under certain conditions, the following theorem (stronger than Theorem 1) holds
true.
6. Levinson's Uniqueness Theorem Generalizations
In 1949, Levinson considered the following Sturm-Liouville problem Lo with symmetric potential [12]
Problem L0 :
Ly = - y"( x) + p( x) y( x) = Ay( x), y'(0) - h y (0) = 0, y'(n) + h y(n) = 0, h e IR.
Levinson provided the following theorem for this problem. Theorem. If p(x) =p(x The spectrum of Problem L then determined uniquely the function p(x) and the number of h .
This section expands on this theorem to encompass nonseparated boundary conditions.
Consider the following spectral problem. Problem 3:
Ly = - y " ( x) + p( x) y( x) = Ay( x), Uii (y ) = a iy(0) - a2 y(n) + y ' (0) = 0,
U2,1 (y) = a2^.y(0) + y '+ a23y(x) = 0 all, a21, a23 e IR.
Problem 3 boundary conditions are the boundary conditions (3) and (4), where
an, a23, and a2i are any real numbers and a21 = -ai3 in (33, YURKO) showed that problem 3 can be uniquely recreated using two spectra and a series of indicators,
namely from the spectrum of Problem 3, the spectrum n ^ of the problem for equation (2) and boundary conditions y (0) + any(0) = y(fa) =0 and the sequence of signs <®n = faZn) \a2\), where is the solution of equation (2) under the
boundary conditions v(0A) = 0, v '(0Â) = -aii.
In what follows, we show that if the potential of Problem 3 is symmetric, then Problem 3 can be reconstructed from two spectra (a sequence of signs is not needed in this case).
Let 4 denote the following spectral problem. Problem 4.
Ly = - y " ( x) + p( x) y( x) = Ay( x),
ai i y(0) + y" (0) = 0, - a„ y(n) + y" (n) = 0.
Theorem 4. If p(x) = p(n-x) x) = ~(n-x) and the spectral of Problems 3 and ~ , 4 and 4 coincide with algebraic multiplicities taken into account, then these boundary
value problems themselves coincide, i.e., p(x) = p(x) ^ = ^ ^ = ^ and a23 = a23
Proof. Using problem 4 as an example, we can see that, for the inverse Sturm-Liouville problem with symmetric potential, Levinson's uniqueness theorem [12] is true.
p(x) = x) aii = ~ii. (14) The relations still need to be proved in order to verify the theorem = and a23 = a23 Problem 3 eigenvalues are the function's full roots.
A3 (A) = -a2i - a23yi (n, A) - yi(n, A) + (aii a23 + a2i a2i) y2 (n, A) + aii y2 (n, A) (15)
According to Hadamard's theorem, the function A 3(A) (which is entire of order 1/2) can be reconstructed from its zeros up to a multiplier C * 0. Therefore, the
functions A s(A) and A3(A) are related by the identity
A b(a)_c
A3(A) 3 (16)
where C is a nonzero constant. The asymptotic relations show that the functions
yi(n,A) = y2(n,A), yi(n,A), y2(n,A) and 1 are linearly independent. Therefore, if C =i then the result we have:
a2i = a2i, a23 = a23.
under certain conditions, stronger results than Theorem 4 hold true.
Let's say that the value and the function p(x) are rebuilt. Then, given
conditions (9), the linearly independent solutions yi( x) and y2( x) of equation (1) are known. As a result, we can establish the following conditions.
Condition 1. Numbers A and A satisfy equations
y2(n,A) = y2(n,A) = 0.
and inequalities
yi(n,A) * yi(n,A2) * 0. Condition 1. Numbers *2 and A satisfy equations
i yi(n,A) - aii y2(n,A) y2(n,A) i yi (T,A2) - ai i y2(T,A2) y2(T,A2) * 0.
i y (T,A3) - ai i y2(T,A3) y2(T,A3)
7 conflict
In this research, it has been determined that a Storm-Lewell self-adjoint inverse problem with symmetric potential and non-separated boundary conditions is unique. In addition, the unique reconstruction of Sturm-Liouville problems is performed by these theorems using one-spectrum and two-spectrum (or three-spectrum) eigenvalues. A unique reconstruction of Sturm-Liouville problems is performed by these theorems using a spectrum and two (or three) eigenvalues.
8. Conclusion
In this research, it has been determined that the second-order differential equations with boundary problem conditions and eigenvalue conditions on a Storm-Liouville self-adjoint inverse problem with symmetric potential and non-separated boundary conditions become unique.
In addition, a unique reconstruction of Sturm-Liouville problems using one spectrum and two spectra (or three spectra) of eigenvalues is performed. A unique reconstruction of Sturm-Liouville problems is performed by these theorems using a spectrum and two (or three) eigenvalues.
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