An inverse nodal problem for two-parameter Sturm-Liouville systems Bruce A. Watson joint work with Paul Binding School of Mathematics University of the Witwatersrand Johannesburg South Africa Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Introduction For the two-parameter system j + ( λ a j + µ b j + q j ) y j = 0 , [ 0 , 1 ] , y ′′ on (1) of Sturm-Liouville equations, linked by the eigen-parameters ( λ, µ ) , with (for simplicity of presentation) boundary conditions y j ( 0 ) 0 , (2) = y j ( 1 ) 0 , j = 1 , 2 , (3) = j ∈ AC and q j ∈ L 1 for j = 1 , 2 , we show where a j , b j ∈ C 1 , a ′ j , b ′ that a single sequence of pairs of nodal points, one of the pair for each eigenfunction y jk , j = 1 , 2 , k = 1 , 2 , ..., suffices to determine q j , j = 1 , 2 , uniquely. Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Klein’s oscillation theorem The oscillation theory for the eigenfunctions of the system (1, 2, 3) has be well studied, and yields a natural indexing of the eigenvalues. Theorem Assume a 1 ( t 1 ) b 1 ( t 1 ) � � ( t 1 , t 2 ) ∈ [ 0 , 1 ] 2 , 0 for all δ ( t 1 , t 2 ) := � � (4) < � , a 2 ( t 2 ) b 2 ( t 2 ) � � � and that n = ( n 1 , n 2 ) is a pair of non-negative integers. Then there exists a unique eigenvalue pair ( λ n , µ n ) of (1) - (3) so that the corresponding eigenfunctions y j have, respectively, n j zeros in ( 0 , 1 ) for j = 1 , 2 . Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Definiteness - 1 From the definiteness assumption (4), there exist constants c , c > 0 so that t 1 , t 2 ∈ [ 0 , 1 ] . c ≤ δ ( t 1 , t 2 ) ≤ c , for all It follows from the work of Faierman that we may assume 0 t 1 , t 2 ∈ [ 0 , 1 ] , a 1 ( t 1 ) , b 2 ( t 2 ) , a 2 ( t 2 ) , b 1 ( t 1 ) , for all (5) < after a nonsingular change of λ, µ axes (although for simplicity we retain the original λ, µ notation). Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Definiteness - 2 Lemma There exist m 1 , m 2 ∈ N , such that � 1 � 1 √ b 1 √ a 1 > m 1 0 0 > , � 1 � 1 √ b 2 √ a 2 m 2 0 0 where m 1 , m 2 are even. Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Definiteness - 3 Proof. From (4), a 1 ( t 1 ) b 2 ( t 2 ) > a 2 ( t 2 ) b 1 ( t 1 ) , for all t 1 , t 2 ∈ [ 0 , 1 ] . Since a 1 , a 2 , b 1 , b 2 are all positive, we may take square roots of both sides of the above inequality to yield � a 1 ( t 1 ) � b 2 ( t 2 ) > � a 2 ( t 2 ) � b 1 ( t 1 ) . Integrating this inequality with respect to t 1 and then with respect to t 2 gives � 1 � 1 � 1 � 1 � a 1 ( t 1 ) dt 1 � b 2 ( t 2 ) dt 2 > � a 2 ( t 2 ) dt 2 � b 1 ( t 1 ) dt 1 . 0 0 0 0 Hence � 1 � 1 √ b 1 √ a 1 0 0 > � 1 � 1 √ b 2 √ a 2 0 0 from which the lemma follows directly. Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Eigencurves Several authors, including Richardson, have studied two-parameter problems via eigencurve methods. Standard results on parametric dependence then allow us to consider the n th eigenvalue µ jn of the j th problem in (1)-(3) as a continuous function of λ . The graph of this function is called the n th eigencurve for problem j . The eigenvalue pair ( λ n , µ n ) of Theorem 1 is at the intersection of the n j th eigencurves for problems j = 1 and j = 2 . Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Eigencurve asymptotics Richardson gave eigencurve asymptotics. Turyn considered the “asymptotic directions", ( γ, 1 ) , of an eigencurve, i.e. if λ/µ n ( λ ) → γ as λ → ∞ along the eigencurve. Binding and Browne, Faierman and Rynne also considered eigenvalue asymptotics but most give them to an order less than we need, or only for more restricted coefficients, or are for ( λ n , µ n ) with one of the n j fixed, corresponding to eigenvalue pairs along a fixed eigencurve. Faierman gave asymptotics of the type we need, but with implicit coefficients varying over an interval, whereas we require explicit coefficients. Our approach depends instead on using (4) to select a one-parameter subset of eigenvalue pairs ( λ k , µ k ) with a special asymptotic direction ( γ, 1 ) . Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Asymptotic direction - 1 The following corollary to the lemma gives a key step towards finding an asymptotic direction along which eigenfunction and eigenvalue approximations can easily be found, as opposed to along eigencurves. We write � j = 1 , 2 , f j ( τ )[ t ] := τ a j ( t ) + b j ( t ) , (6) Corollary If m 1 , m 2 are as in Lemma 2, then there exists γ > 0 for which � 1 � 1 m 2 f 1 ( γ ) = m 1 f 2 ( γ ) . 0 0 Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Asymptotic direction - 2 Proof. Observe from Lemma 2 that � 1 � 1 � 1 � 1 f 1 ( 0 ) = m 2 f 2 ( 0 ) . � � m 2 b 1 < m 1 b 2 = m 1 0 0 0 0 Also from Lemma 2 it follows that � 1 � 1 � 1 � 1 f 1 ( γ ) f 2 ( γ ) √ a 1 > m 1 √ a 2 = m 1 lim m 2 lim √ γ = m 2 √ γ . γ →∞ γ →∞ 0 0 0 0 Let � 1 g ( γ ) = [ m 2 f 1 ( γ ) − m 1 f 2 ( γ )] . 0 Then g ( 0 ) < 0 while for sufficiently large γ > 0 , g ( γ ) > 0 . Consequently, since the f j are continuous, there exists γ > 0 for which g ( γ ) = 0 . Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
One-parameter family For k = 0 , 1 , 2 , . . . let ω ( k ) = ( ω 1 ( k ) , ω 2 ( k )) where k + 1 � � − 1 , j = 1 , 2 . ω j ( k ) := m j 2 We consider the subset of ( λ n , µ n ) defined by (7) λ k := λ ω ( k ) , (8) µ k := µ ω ( k ) . It will be seen that the leading terms in the asymptotics of λ k , µ k are given by ¯ µ k where λ k , ¯ � 2 � 2 m 1 ( k + 1 m 2 ( k + 1 � � 2 ) π 2 ) π (9) µ k ¯ = � 1 = � 1 , 0 f 1 ( γ ) 0 f 2 ( γ ) ¯ (10) λ k = γ ¯ µ k . Thus ( λ k , µ k ) have asymptotic direction ( γ, 1 ) . Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Notation For ( λ, µ ) = ( λ k , µ k ) denote y j by y k j , we write ( λ k a j + µ k b j ) 1 / 4 , σ jk = µ k b j ) 1 / 4 = (¯ µ k ) 1 / 4 � (¯ λ k a j + ¯ f j ( γ ) , σ jk ¯ = � t ρ j ( t , λ, µ ) � λ a j + µ b j , = 0 � t σ 2 ρ k j ( t ) ρ j ( t , λ k , µ k ) = = jk , 0 � t σ 2 ρ k ρ j ( t , ¯ j ( t ) ¯ = λ k , ¯ µ k ) = ¯ jk . 0 Observed that there exists a constant c > 0 , depending only on a 1 , b 1 , a 2 , b 2 , such that √ min k ∈ N , j = 1 , 2 . σ jk ≥ c k , for all t ∈ [ 0 , 1 ] ¯ Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Solution asymptotics - 1 Theorem For ( λ, µ ) ∈ S n as n → ∞ , where S n is a family of non-empty subsets of S with inf { σ 2 j ( t ) | t ∈ [ 0 , 1 ] , ( λ, µ ) ∈ S n } ≥ n , for all n ∈ N , if y j ( t ) is the solution of (1) with y j ( 0 ) = 0 and y ′ j ( 0 ) = 1 , � 1 � t 1 � q ∗ j − q j �� sin ρ j + sin [ ρ j ( t ) − ρ j ] sin ρ j d τ + O y j ( t ) = σ j ( 0 ) σ j σ 2 n 2 0 j � σ j ( t ) σ ′ j cos ρ j − sin ρ j y ′ ( t ) = σ 3 σ j ( 0 ) j � 1 � t q ∗ j − q j �� cos [ ρ j ( t ) − ρ j ( τ )] sin j ρ d τ + O + . σ 2 n 2 0 j Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Solution asymptotics - 2 Here � 2 σ ′′ � σ ′ j j − 2 q ∗ (11) = , j σ j σ j q j q ∗ j − q j , (12) ¯ = � 1 q j ¯ Q j ¯ (13) = f j ( γ ) , 0 for ( λ, µ ) ∈ S := { ( λ, µ ) | λ, µ ∈ R , λ, µ > 0 } , (14) then sup � q ∗ � L 1 < ∞ . (15) ( λ,µ ) ∈ S Note that q ∗ ( t ) depends on both λ and µ . Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Nodal points Let 0 < x k j ( 1 ) < x k j ( 2 ) < · · · < x k j ( ω j ( k )) < 1 denote the nodal points (zeros in ( 0 , 1 ) ) of the eigenfunction y k j corresponding to the eigenvalue ( λ k , µ k ) . Define t k j ( n ) , ξ k j ( n ) and ζ k j ( n ) via the equations ρ k j ( t k j ( n )) n π, = n − 3 � � ρ k j ( ξ k j ( n )) = π, 4 n − 1 � � ρ k j ( ζ k j ( n )) = π. 4 Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
Eigenvalue and nodal asymptotics - 1 Theorem In the notation of (9), (10) we have λ k + O ( 1 ) , ¯ λ k = µ k + O ( 1 ) , µ k = ¯ as k → ∞ . The nodal points of y k j are given by � 1 � x k j ( n ) = t k n = 1 , . . . , ω j ( k ) . j ( n ) + O , k 2 Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system
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