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R. O. Hryniv

Publications and source records attributed to R. O. Hryniv.

7 recordsLinked to original sources
arXiv2012

Norm resolvent convergence of singularly scaled Schrödinger operators and δ'-potentials

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as εgoes to 0, of a family S_εof one-dimensional Schrödinger operators on the line of the form S_ε:= -D^2 + ε^{-2} V(x/ε). Under certain conditions the family of potentials converges in the sense of distributions to the first derivative of the Dirac delta-function, and then the limit of S_εmight be considered as a "physically motivated" interpretation of the one-dimensional Schrödinger operator with potential δ'.

math.SP↗
arXiv2010

On norm resolvent convergence of Schrödinger operators with $δ'$-like potentials

We address the problem on the right definition of the Schroedinger operator with potential $δ'$, where $δ$ is the Dirac delta-function. Namely, we prove the uniform resolvent convergence of a family of Schroedinger operators with regularized short-range potentials $ε^{-2}V(x/ε)$ tending to $δ'$ in the distributional sense as $ε\to 0$. In 1986, P. Seba claimed that the limit coincides with the direct sum of free Schroedinger operators on the semi-axes with the Dirichlet boundary condition at the origin, which implies that in dimension one there is no non-trivial Hamiltonians with potential $δ'$. Our results demonstrate that, although the above statement is true for many V, for the so-called resonant V the limit operator is defined by the non-trivial interface condition at the origin determined by some spectral characteristics of V. In this resonant case, we show that there is a partial transmission of the wave package for the limiting Hamiltonian.

math.SP↗
arXiv2009

Inverse scattering for Schrödinger operators with Miura potentials, I. Unique Riccati representatives and ZS-AKNS systems

This is the first in a series of papers on scattering theory for one-dimensional Schrödinger operators with highly singular potentials $q\in H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive Schrödinger operators that admit a Riccati representation $q=u'+u^2$ for a unique $u\in L^1(R)\cap L^2(R)$. Such potentials have a well-defined reflection coefficient $r(k)$ that satisfies $|r(k)|<1$ and determines $u$ uniquely. We show that the scattering map $S:u\mapsto r$ is real-analytic with real-analytic inverse. To do so, we exploit a natural complexification of the scattering map associated with the ZS-AKNS system. In subsequent papers, we will consider larger classes of potentials including singular potentials with bound states.

math.SP↗
arXiv2009

Inverse scattering on the line for Schrödinger operators with Miura potentials, II. Different Riccati representatives

This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form $q=u'+u^2$ for some $u\in L^2(R)$. We consider potentials for which there exist `left' and `right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev--Marchenko potentials in $L^1(R,(1+|x|)dx)$ generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients $r$ and justify the algorithm reconstructing $q$ from $r$.

math.SP↗
arXiv2004

Inverse spectral problems for Sturm-Liouville operators with singular potentials, IV. Potentials in the Sobolev space scale

We solve the inverse spectral problems for the class of Sturm--Liouville operators with singular real-valued potentials from the Sobolev space W^{s-1}_2(0,1), s\in[0,1]. The potential is recovered from two spectra or from the spectrum and norming constants. Necessary and sufficient conditions on the spectral data to correspond to the potential in W^{s-1}_2(0,1) are established.

math.FA↗
arXiv2002

Inverse spectral problems for Sturm-Liouville operators with singular potentials

The inverse spectral problem is solved for the class of Sturm-Liouville operators with singular real-valued potentials from the space $W^{-1}_2(0,1)$. The potential is recovered via the eigenvalues and the corresponding norming constants. The reconstruction algorithm is presented and its stability proved. Also, the set of all possible spectral data is explicitly described and the isospectral sets are characterized.

math.SP↗