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We consider the Sturm–Liouville operator on quantum graphs with a loop with the standard matching conditions in the internal vertex and the jump conditions at the boundary vertex. Given the potential on the loop, we try to recover the potential on the boundary edge from the subspectrum. The uniqueness theorem and a constructive algorithm for the solution of this partial inverse problem are provided. [ABSTRACT FROM AUTHOR]

The half-inverse problem is studied for the Sturm–Liouville operator with an eigenparameter dependent boundary condition on a finite interval. We develop a reconstruction procedure and prove the existence theorem for solution of the inverse problem. Our method is based on interpolation of entire functions. [ABSTRACT FROM AUTHOR]

In this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on (0 , π) {(0,\pi)} with the Robin boundary conditions and the jump conditions at the point π 2 {\frac{\pi}{2}}. We prove that the potential M ⁢ (x) {M(x)} on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential M ⁢ (x) {M(x)} is given on (0 , (1 + α) ⁢ π 4) {(0,\frac{(1+\alpha)\pi}{4})} ; (ii) the potential M ⁢ (x) {M(x)} is given on ( (1 + α) ⁢ π 4 , π) {(\frac{(1+\alpha)\pi}{4},\pi)} , where 0 < α < 1 {0<\alpha<1} , respectively. It is also shown that the potential and all the parameters can be uniquely recovered by one spectrum and some information on the eigenfunctions at some interior point. [ABSTRACT FROM AUTHOR]

In this paper, we deal with some partial inverse problems for the Sturm–Liouville operator on a star graph with Robin and/or Dirichlet boundary conditions in pendant vertices. It is shown that if all but one of the potentials are known a priori, then Horváth-type theorems hold. Our method is based on the related theory about Weyl function and the growth of entire functions. [ABSTRACT FROM AUTHOR]

In this work, we study inverse resonance problems for the Schrödinger operator on the real line with the potential supported in [0, 1]. In general, all eigenvalues and resonances cannot uniquely determine the potential. (i) It is shown that if the potential is known a priori on [0, 1 / 2], then the unique recovery of the potential on the whole interval from all eigenvalues and resonances is valid. (ii) If the potential is known a priori on [0, a], then for the case $$a>1/2$$ , infinitely many eigenvalues and resonances can be missing for the unique determination of the potential, and for the case $$a<1/2$$ , all eigenvalues and resonances plus a part of so-called sign-set can uniquely determine the potential. (iii) It is also shown that all eigenvalues and resonances, together with a set of logarithmic derivative values of eigenfunctions and wave-functions at 1 / 2, can uniquely determine the potential. [ABSTRACT FROM AUTHOR]

Three spectral problems generated by the same Sturm-Liouville equation are considered: Neumann-Dirichlet problem (the Neumann condition at the left end and the Dirichlet condition at the right end) on the whole interval [0, a], Neumann-Dirichlet problem on [ABSTRACT FROM AUTHOR]

Inverse spectral problems are considered for Dirac equations with boundary conditions depending polynomially on the spectral parameter and with a transmission condition. We give formulations of the associated inverse problems such as Titchmarsh–Weyl theorem, and prove corresponding uniqueness theorems. Using Titchmarsh–Weyl theorem, we also obtain two analogues of a theorem of Hochstadt–Lieberman and a theorem of Mochizuki–Trooshin for this kind of Dirac operators. The obtained results are generalizations of the similar results for the classical Dirac operator. [ABSTRACT FROM AUTHOR]

The inverse spectral problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b . A uniqueness theorem for the potential q of the derived Sturm–Liouville problem B ( q ) is presented from the data involving fractions of the eigenvalues of the problem B ( q ) on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl-function techniques and properties of zeros of a class of entire functions. [ABSTRACT FROM AUTHOR]

The inverse nodal problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b from zeros for eigenfunctions corresponding to the transmission eigenvalues. It is shown that the space of potential functions q which correspond to interior transmission problems characterized by Ω : = { q : q ∈ L 2 [ 0 , 1 ] } , under a certain metric, is homeomorphic to the partition set of the space of quasinodal sequences. As a consequence, the inverse nodal problem defined on the partition set of admissible sequence, is stable. [ABSTRACT FROM AUTHOR]

In this paper the Schrödinger operators endowed with Neumann or Dirichlet boundary conditions, where the potential q ∈ L 1 [0, π ] is single-well with transition point a ∈ (0, π ), are considered. Suppose { ν n } n ≥ 0 and { λ n } n ≥ 1 are the set of eigenvalues for Neumann or Dirichlet problem, respectively, and the potential q ( x ) possesses an additional condition, we present estimates of lower bound for the differences ν 1 − ν 0 and λ 2 − λ 1 , respectively. The results show that the energy transition for quantum particles jumping to the ground state is related to transition point a . [ABSTRACT FROM AUTHOR]

Dirac operator with eigenvalue-dependent boundary and jump conditions is studied. Uniqueness theorems of the inverse problems from eitherWeyl functionor thespectral data(the sets of eigenvalues and norming constantsexcept for one eigenvalue and corresponding norming constant; two sets of different eigenvaluesexcept for two eigenvalues) are proved. Finally, we investigate two applications of these theorems and obtain analogues of a theorem ofHochstadt-Liebermanand a theorem ofMochizuki-Trooshin. [ABSTRACT FROM PUBLISHER]

We consider the regular Sturm-Liouville problem [image omitted] with boundary conditions [image omitted] where ad - bc > 0, c ≠ 0. In this work, using the Hochstadt and Lieberman's method [15] we show that if q(x) is prescribed on the half interval [image omitted], then a single spectrum suffices to determine q(x) on [0, 1], the boundary condition at x = 0 and “the asymptotic” boundary condition at x = 1. [ABSTRACT FROM AUTHOR]