A POLYNOMIAL IDENTITY AND ITS APPLICATION TO INVERSE SPECTRAL PROBLEMS IN STIELTJES STRINGS

作者：Law, C K^*; Pivovarchik, V; Wang, W C

来源：Operators and Matrices, 2013, 7(3): 603-616.

DOI：10.7153/oam-07-33

• Stieltjes function
• subtree
• Lagrange polynomial
• continued fraction
• Dirichlet conditions
• Neumann conditions

The equation Phi = P(1)Q(2) P(2)Q(1) is studied where Phi, Q(1), Q(2) are known real polynomials while P-1 and P-2 are unknown polynomials. Condition are obtained for the solution (P-1,P-2) to exist and to be such that P(1)(-1)Q(1) and P(2)(-1)Q(2) are Stieltjes functions. This result is used to prove the existence of a tree with two complementary subtrees of Stieltjes strings such that the spectrum of the Neumann boundary value problem on the tree is exactly the set of zeros of Phi and the spectra of Dirichlet problems on the subtrees are the sets of zeros of Q(1) and Q(2) This result is generalized to the equation Phi = Sigma(q)(i=1) [Pi Pi Q(j) j not equal i], which is then applied to solve the inverse several spectra problem for trees of Stieltjes strings.

• 出版日期2013-9
• 单位中山大学