Extension of Ambarzumyan's Theorem to General Boundary Conditions

Abstract

We extend the classical Ambarzumyan's theorem for the Sturm-Liouville equation (which is concerned only with Neumann boundary conditions) to the general boundary conditions, by imposing an additional condition on the potential function. Our result supplements the Pöschel-Trubowitz inverse spectral theory. We also have parallel results for vectorial Sturm-Liouville systems. 

A vectorial inverse nodal problem

“…Recall that in the scalar case, it was shown in [7] that the smoothness of the potential function depends only on the nodal data, regardless of the boundary conditions. In the present case, the answer is also affirmative for the general boundary condition (5). In fact, the result is more complete.…”

Ambarzumyan’s theorem for the quasi-periodic boundary conditions

“…For the first time, we obtain Ambarzumyan's theorem for the operator L t (q) with t ∈ [0, 2π), generated by quasi-periodic boundary conditions (2). The result established below show that the potential q can be determined from one spectrum and there is not any additional condition on q such as (3) for the operator L t (q) with t = π (see also [4,7]). The result of this paper is the following.…”

“…then q = 0 a.e. The present work was stimulated by the papers [4,8]. For the first time, we obtain Ambarzumyan's theorem for the operator L t (q) with t ∈ [0, 2π), generated by quasi-periodic boundary conditions (2).…”

A Class of Matrix-valued Schrödinger Operators with Prescribed Finite-band Spectra

“…Next, assuming −∞ ≤ a < b ≤ ∞, we consider the spaces 18) for some c ∈ R and z ∈ C. (Here (φ, ψ) C n = n j=1 φ j ψ j denotes the standard scalar product in C n , abbreviating χ ∈ C n by χ = (χ 1 , · · · , χ n ) t .) Both dimensions of the spaces in (2.18), dim C (N (z, ∞)) and dim C (N (z, −∞)), are constant for z ∈ C ± = {ζ ∈ C | ±Im(ζ) > 0} (see, e.g., [55]).…”

“…Inverse monodromy problems have recently been discussed in [4], [5], [6], [15], [68], [69], [96], [101], and the literature cited therein. More specific inverse spectral problems, such as compactness of the isospectral set of periodic Schrödinger operators [13], special isospectral matrix-valued Schrödinger operators, and Borg-type uniqueness theorems (for periodic coefficients as well as eigenvalue problems on compact intervals) were recently studied in [17], [18], [19], [21], [23], [26], [51], [52], [68], [69], [101], [102], [103], [104]. Moreover, direct spectral theory in the particular case of periodic Schrödinger operators (i.e., Floquet theory and the like) has been studied in [12], [14], [23], [25], [26], [34], [53], [59], [60], [86], [102], [103], [110]- [112], with many more pertinent references to be found therein.…”