Reconstruction of the Sturm-Liouville operator on a $p$-star graph with nodal data

Inverse spectral problems for the Sturm–Liouville operator on a d-star graph

“…, d, (2.2) at the pendant vertices 0, and y 1 (π , λ) = y 2 (π , λ) = · · · = y d (π , λ), ( [18,19], Pivovarchik gave the asymptotic expansion of eigenvalues and showed that there are d sequences of eigenvalues which one sequence is simple while the others might not be. In [5], Cheng presented the sharper estimates of eigenvalues for the problem A. …”

“…There exists an extensive literature devoted to inverse spectral problems for ordinary differential operators on a finite interval; we mention only the literature [2,4,[6][7][8][9][10][11][12]16,[20][21][22]. Recently, the spectral problems of quantum graphs have become a rapidly-developing field of mathematics and mathematical physics, and spectral properties of quantum graphs and different inverse problems have been studied in both forward [13,17,24] and inverse [1,3,5,14,18,19,23,[25][26][27][28], etc.…”

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

“…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.…”

“…The inverse spectral problem for differential operators on graphs have been completely studied in pevious works, 27‐29,34,36,39‐42 and the references therein. We note that Currie and Watson 6 , or Yurko, 43 or Cheng 4 studied the inverse nodal problem for Sturm–Liouville operators on graphs from the dense nodal set on the whole interval, respectively. C.‐F.…”

Reconstruction of the Sturm-Liouville operators on a graph with δ′ s couplings

“…Recently, the spectral problems of quantum graphs have become a rapidly-developing field of mathematics and mathematical physics, and spectral properties of quantum graphs and different inverse problems have been studied in both forward [20,21,22,31,39] and inverse [4,23,32,38,40,41,42], etc. Nowadays there are only a number of papers devoted to inverse nodal problems for differential operators on graphs (for example, refer to [8,11,41,42]). …”