Solutions to open problems of Yang concerning inverse nodal problems

Abstract

Yang [X. F. Yang, A new inverse nodal problem, Journal of Differential Equations 169 (2001), 633-653] considered a new inverse nodal problem for the Sturm-Liouville operator L(q, α, β) in L 2 [0, 1]: an s-dense subset of the nodal set in (0, b) (for any fixed b ∈ ( 1 2 , 1]) determines the potential q and boundary data α, β. (1) Since the s-dense condition is stronger than the dense condition, X. F. Yang proposed an open problem "It is open if the boundary parameter α can be determined by a dense subset of the nodal set in (0, b) but not necessarily by an s-dense subset of the nodal set in (0, b)." Cheng et al. have solved this problem and shown that a dense subset of the nodal set in (0, b) completely determines the potential q and boundary data α, β. (2) Another interesting open question: "It remains open if the result holds true for b ∈ (0, 1 2]" is also proposed by X. F. Yang. In this paper we provide a counterexample to claim that the result does not hold true for b ∈ (0, 1 2 ), and a uniqueness theorem for b = 1 2 .

Inverse problems for Sturm–Liouville operators on a compact equilateral graph by partial nodal data

“…2 The so-called partial inverse nodal problem is to recover the potential up to its mean value from the nodal information on some subinterval under certain conditions. In particular, partial inverse nodal problems for the classical Sturm-Liouville operator can be found in previous studies [3][4][5][6][7] and other works. Inverse spectral theory of differential operators were foundin previous works.…”

“…In this section, we shall study the inverse nodal problem for B from partial nodal data on arbitrary intervals having the central vertex. In this case, the eigenvalues { λ 1 n } together with the corresponding twin‐dense nodal subsets cannot be enough to recover the potentials (see Yang 7 ). Therefore, we need additional nodal information on q l ( x ) for all to deal with the uniqueness theorems.…”

“…[18][19][20][21][22][23][24] We note that the area of inverse nodal problems for differential operators remains incredibly active up to this day. Since we cannot possibly describe the recent developments in details in this paper, we suggest that the interested readers refer to, for instance, previous works, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein, which lead them into a variety of directions. Some aspects of inverse spectral and nodal problems for graphs were studied in the papers.…”

Partial inverse nodal problems for differential pencils on a star‐shaped graph

“…In this section, we shall study the inverse nodal problems for differential pencils on the star‐shaped graph from partial nodal data on arbitrary intervals. In this case, the eigenvalues together with the corresponding twin‐dense nodal subsets might not be enough to recover the potentials (for details, see Yang 21 ). Therefore, we need additional nodal information on q l 0 ( x ) and q l 1 ( x ) for all to deal with the uniqueness theorems.…”

“…It is worth mentioning that Theorem 2.2 in Guo and Wei 5 shows us that not only the potential up to its mean value and coefficients of boundary conditions can be uniquely determined by a twin‐dense subset on ( a , b ) with 1/2∈( a , b ) under some conditions but also the length b − a of subinterval ( a , b ) can be arbitrarily small. Because we cannot possibly describe the recent developments in details in this paper, one may refer to other studies, 1,3‐23 and the references therein, which lead the interested reader into a variety of directions. Meanwhile, the inverse nodal problem for differential pencils was also studied in Buterin and Shieh and Guo and Wei, 24‐26 respectively.…”