An Extension of The First Eigen-type Ambarzumyan theorem
aa r X i v : . [ m a t h . SP ] F e b An Extension of The First Eigen-type Ambarzumyan theorem
Alp Arslan Kıra¸c a a Department of Mathematics, Faculty of Arts and Sciences, Pamukkale University, 20070,Denizli, Turkey
ARTICLE HISTORY
Compiled February 12, 2019
ABSTRACT
An extension of the first eigenvalue-type Ambarzumyan’s theorem are provided forthe arbitrary self-adjoint Sturm-Liouville differential operators. The result makes acontribution to the P¨oschel-Trubowitz inverse spectral theory as well.
KEYWORDS
Sturm-Liouville differential operators; Ambarzumyan’s theorem; inverse spectraltheory
1. Introduction
In 1929, Ambarzumyan [1] proved that if { ( nπ ) : n ∈ N ∪ { }} is the spectrum of theboundary value problem − y ′′ ( x ) + q ( x ) y ( x ) = λy ( x ) , y ′ (0) = y ′ (1) = 0 (1)with real potential q ∈ L (0 , q = 0 a.e. Clearly, if q = 0 a.e., then theeigenvalues λ n = ( nπ ) , n ∈ N ∪ { } .We note that in Ambarzumyan’s theorem the whole spectrum is specified. But thenFreiling and Yurko [2] proved that it is enough to specify only the first eigenvalue. Moreprecisely, they proved the following first eigenvalue-type of Ambarzumyan theorem: Theorem 1.1. If λ = R q ( x ) dx , then q ( x ) = λ a.e. Consider the boundary value problems L ( q ) generated in the space L (0 ,
1) by thefollowing differential equation − y ′′ ( x ) + q ( x ) y ( x ) = λy ( x ) (2)with arbitrary self-adjoint boundary conditions, where q ∈ L (0 ,
1) is a real-valuedfunction. The operator L ( q ) is self-adjoint and its spectrum is discrete, real andbounded from below. We suppose that the eigenvalues of the operator L ( q ) consistof the sequence { λ n } n ≥ ( λ n ≤ λ n +1 , lim n →∞ λ n = + ∞ ) (counting with multiplicities)and { y n } n ≥ denotes the corresponding normalized eigenfunctions of the operator L ( q ). CONTACT Alp Arslan Kıra¸c. Email: [email protected] et us consider another known and fixed operator ˜ L := L (˜ q ) of the same domainwith L := L ( q ) but with different potential ˜ q ∈ L (0 , { ˜ λ n } n ≥ and { ˜ y n } n ≥ eigenvalues and normalized eigenfunctions of the operator ˜ L , respectively.In [3], Yurko provided the following generalization of Theorem 1.1 on wide classesof self-adjoint differential operators. Here ( ., . ) denotes inner product in L (0 ,
1) andˆ q := q − ˜ q . Theorem 1.2.
Let λ − ˜ λ = (ˆ q ˜ y , ˜ y ) , where ˜ y is a normalized eigenfunction of ˜ L related to the first eigenvalue ˜ λ . Then q ( x ) = ˜ q ( x ) + λ − ˜ λ a.e. on (0 , . Note that the proof of this theorem is based on the well-known variational principleof the smallest eigenvalue.Recently Ashrafyan [4] proved the following another generalization of the firsteigenvalue-type of Ambarzumyan theorem for Sturm-Liouville problems with arbitraryself-adjoint boundary conditions.
Theorem 1.3.
Let λ − ˜ λ = ess inf ˆ q or λ − ˜ λ = ess sup ˆ q, where ˜ λ is the first eigenvalue ˜ L . Then q ( x ) = ˜ q ( x ) + λ − ˜ λ a.e. on (0 , . As in the proof of Theorem 1.2, the above uniqueness theorem is also provided byusing the property of the smallest eigenvalue. That is, the proof of Theorem 1.3 isbased on the Sturm oscillation theorem that the first eigenfunction has no zeros oninterval (0 ,
2. Main results
The main result of this paper is as follows. Note that, to obtain the following extensiontheorem, it is enough to have information about the arbitrary eigenvalue λ n insteadof the first eigenvalue only. Theorem 2.1.
Let, for some n , λ n − ˜ λ n = (ˆ q ˜ y n , ˜ y n ) and λ n − ˜ λ n = ess inf ˆ q or λ n − ˜ λ n = ess sup ˆ q, where ˜ y n is a normalized eigenfunction of ˜ L related to the eigenvalue ˜ λ n . Then q ( x ) =˜ q ( x ) + λ n − ˜ λ n a.e. on (0 , . Proof.
It follows from the first assumption that((ˆ q + ˜ λ n − λ n )˜ y n , ˜ y n ) = 0 . (3)2ince ˜ y n is a normalized eigenfunctions of the operator ˜ L corresponding to theeigenvalue ˜ λ n , we obtain that ˜ y n has at most finitely many isolated zero points in(0 , q + ˜ λ n − λ n )˜ y n ∈ L (0 , q = λ n − ˜ λ n a.e. on (0 , Theorem 2.2.
Let the assumptions of Theorem 2.1 be valid and let R q ( x ) dx = R ˜ q ( x ) dx . Then q ( x ) = ˜ q ( x ) a.e. on (0 , . Let us give an example to illustrate Theorem 2.1.Consider the Dirichlet boundary value problem − y ′′ ( x ) + q ( x ) y ( x ) = λy ( x ) , y (0) = y (1) = 0 . (4)Let ˜ q ( x ) ≡
0. Then ˜ λ n = ( nπ ) , ˜ y n = √ nπx for some n ≥ Corollary 2.3.
Let, for some n , λ n − ( nπ ) = 2 Z q ( x ) sin nπx dx (5) and λ n − ( nπ ) = ess inf q or λ n − ( nπ ) = ess sup q. Then q ( x ) = λ n − ( nπ ) a.e. on (0 , . And, Theorem 2.2 implies the following assertion. The assertion makes a contribu-tion to the P¨oschel-Trubowitz inverse spectral theory.
Corollary 2.4.
Let the assumptions of Corollary 2.3 be valid and let R q ( x ) dx = 0 .Then q ( x ) = 0 a.e. on (0 , . Remark 1.
In [5], P¨oschel and Trubowitz showed that, for the Dirichlet problem,there is an infinite dimensional set of L (0 ,
1) potentials with the same Dirichlet zerospectrum σ := { ( nπ ) : n ∈ N } as q = 0 a.e. That is, if the spectrum is σ , then thepotential is not necessarily zero. Thus Ambarzumyan’s theorem is not valid.Arguing as in the paper [6] (see p. 337), from P¨oschel and Trubowitz [5], an isospec-tral L (0 ,
1) potential can be written in the form q = u + e ( u ), where u is odd part and e is even part which can be uniquely expressed as a function of its odd part ( e = e ( u )).Let q be an isospectral potential with the Dirichlet zero spectrum σ = { ( nπ ) : n ∈ N } . Then R q ( x ) dx = 0 and from (5), we get for some n Z q ( x ) sin nπx dx = Z q ( x ) dx − Z q ( x ) cos 2 nπx dx. (6)3ence, for some n , Z q ( x ) cos 2 nπx dx = 0which implies that the n-th even Fourier coefficient a n has the following equalities a n = Z q ( x ) cos 2 nπx dx = Z e ( u ) cos 2 nπx dx = 0 . Then a n = 0 for all n . Similarly, all the odd Fourier coefficients vanish. Therefore, theeven part of an isospectral potential does not vanish. Consequently, from Corollary2.4, for the isospectral potential q (i.e. q = 0), we get, for all n ,2 Z q ( x ) sin nπx dx = 0 , that is, we get from (6), for all n , a n = Z q ( x ) cos 2 nπx dx = 0 . Thus, the present paper supplements the P¨oschel-Trubowitz inverse spectral theory.
References [1] Ambarzumian V. ¨Uber eine Frage der Eigenwerttheorie. Zeitschrift f¨ur Physik. 1929;53:690–695.[2] Freiling G, Yurko VA. Inverse SturmLiouville problems and their applications. New York:NOVA Science Publishers; 2001.[3] Yurko VA. On Ambarzumyan-type theorems. Applied Mathematics Letters. 2013;26:506–509.[4] Ashrafyan Y. A remark on Ambarzumian’s theorem. Results in Mathematics. 2018 Feb;73(1):36. Available from: https://doi.org/10.1007/s00025-018-0806-9 .[5] P¨oschel J, Trubowitz E. Inverse spectral theory. Boston: Academic Press; 1987.[6] Chern HH, Law CK, Wang HJ. Extension of Ambarzumyan’s theorem to general boundaryconditions. J Math Anal Appl. 2001;263(2):333–342..[5] P¨oschel J, Trubowitz E. Inverse spectral theory. Boston: Academic Press; 1987.[6] Chern HH, Law CK, Wang HJ. Extension of Ambarzumyan’s theorem to general boundaryconditions. J Math Anal Appl. 2001;263(2):333–342.