Search

We propose a counterpart of the classical Rollnik-class of potentials for fractional and massive relativistic Laplacians, and describe this space in terms of appropriate Riesz potentials. These definitions rely on precise resolvent estimates. We show that Coulomb-type potentials are elements of fractional Rollnik-class up to but not including the critical singularity of the Hardy potential. For the operators with fractional exponent $\alpha = 1$ there exists no fractional Rollnik potential, however, in low dimensions we make sense of these classes as limiting cases by using $\Gamma$-convergence. In a second part of the paper we derive detailed results on the self-adjointness and spectral properties of relativistic Schr\"odinger operators obtained under perturbations by fractional Rollnik potentials. We also define an extended fractional Rollnik-class which is the maximal space for the Hilbert-Schmidt property of the related Birman-Schwinger operators.

In a previous work of 2014 on a quantum system governed by the repulsive Hamiltonian, the author proved uniqueness for short-range interactions described by a scattering operator consisting of regular and singular parts. In this paper, the singular part is assumed to have much stronger singularities and the same uniqueness theorem is proved. By applying the time-dependent method invented by Enss and Weder in 1995, the high-velocity limit for a wider class of the scattering operator with stronger singularities also uniquely determines the interactions of a multi-dimensional system.

In this paper, we study the linear and nonlinear Schr\"odinger equations with a time-decaying harmonic oscillator and inverse-square potential. This model retains a form of scale invariance, and utilizing this property, we demonstrate the asymptotic completeness of wave operators and Strichartz estimates for linear propagators.

Different from the usual harmonic oscillator, the time-decaying harmonic oscillator accelerates particles and generates scattering states. We study one of the multidimensional inverse scatterings in this two-body quantum system perturbed by short-range potential functions that have a bounded part and a locally singular part. Applying the Enss--Weder time-dependent method, we prove that the scattering operator determines the potential functions uniquely.

Sub-quadratic repulsive potentials accelerate quantum particles and can relax the decay rate in the $x$ of the external potentials $V$ that guarantee the existence of the quantum wave operators. In the case where the sub-quadratic potential is $- |x|^{\alpha} $ with $0< \alpha < 2$ and the external potential satisfies $|V(x) | \leq C (1+|x|) ^{-(1- \alpha /2) - \varepsilon} $ with $\varepsilon>0$, Bony et. al [3] determined the existence and completeness of the wave operators, and Itakura [12, 13, 14] then obtained their results using stationary scattering theory for more generalized external potentials. Based on their results, we naturally expect the following. If the decay power of the external potential $V$ is less than ${ -(1- \alpha /2) } $, V is included in the short-range class. If the decay power is greater than or equal to ${ -(1- \alpha /2) } $, $V$ is included in the long-range class. In this study, we first prove the new propagation estimates for the time propagator that can be applied to scattering theory. Second, we prove that the wave operators do not exist if the power is greater than or equal to $-(1- \alpha /2)$ and that the threshold expectation of ${ -(1- \alpha /2) } $ is true using the new propagation estimates.

We consider non-local Schr\"odinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity, and derive conditions ruling embedded eigenvalues out. Our goal in this paper is to advance techniques based on virial theorems, Mourre estimates, and an extended version of the Birman-Schwinger principle, previously developed for classical Schr\"odinger operators but thus far not used for non-local operators. We also present a number of specific cases by choosing particular classes of kinetic and potential terms, and discuss existence/non-existence of at-edge eigenvalues in a basic model case in function of the coupling parameter.

It is known in scattering theory that the minimal velocity bound plays a conclusive role in proving the asymptotic completeness of the wave operators. In this study, we prove the minimal velocity bound and other important estimates for the two-body Schroedinger-type operator with fractional powers. We assume that the pairwise potential functions belong to broad classes that include long-range decay and Coulomb-type local singularities. Our estimates are expected to be applied to prove the asymptotic completeness for the fractional Schroedinger-type operators in various (not only short-range but also long-range and N-body) situations.

We develop scattering theory for non-local Schr\"odinger operators defined by functions of the Laplacian that include its fractional power $(-\Delta)^\rho$ with $0<\rho\leqslant1$. In particular, our function belongs to a wider class than the set of Bernstein functions. By showing the existence and non-existence of the wave operators, we clarify the threshold between the short and long-range decay conditions for perturbational potentials.

Controlled time-decaying harmonic oscillator changes the threshold of decay order of the potential functions in order to exist the physical wave operators. This threshold was first reported by Ishida and Kawamoto \cite{IK} for the non-critical case. In this paper we deal with the critical case. As for the critical case, the situation changes drastically, and much more rigorous analysis is required than that of non-critical case. We study this critical behavior and clarify the threshold by the power of the log growth of the potential functions. Consequently, this result reveals the asymptotics for quantum dynamics of the critical case.

We study one of the multidimensional inverse scattering problems for quantum systems governed by the Stark Hamiltonians. By applying the time-dependent method developed by Enss and Weder in 1995, we prove that the high-velocity limit of the scattering operator determines uniquely the short-range interaction potentials. Moreover, we prove that, when a long-range interaction potential is given, the high-velocity limit of the Dollard-type modified scattering operator determines uniquely the short-range part of the interactions. We allow the potential functions to belong to very broad classes. These results are improvements on the previous results obtained by Adachi and Maehara in 2007 and Adachi, Fujiwara, and Ishida in 2013.

Controlled time-decaying harmonic potentials decelerate the velocity of the charged particle but the particle never be trapped by this harmonic potentials. This physical phenomena changes threshold between the short range class of potential and long-range class of potential in the sense of the existence of physical wave operators. In this paper, we reveal such a threshold is $1/(1-\lambda)$ for some $0\leq \lambda<1/2$ , which is determined by the mass of the particle and a coefficient of harmonic potential.

We consider quantum systems described by the fractional powers of the negative Laplacian and the interaction potentials. When a slowly decaying potential function is given, we prove the nonexistence of the wave operators, under the assumption that the Dollard-type modified wave operators exist and that they are asymptotically complete. This nonexistence indicates the borderline between short-range and long-range behavior.

Enss (1983) proved a propagation estimate for the usual free Schroedinger operator that turned out later to be very useful for inverse scattering in the work of Enss--Weder (1995). Since then, this method has been called the Enss--Weder time-dependent method. We study the same type of propagation estimate for the fractional powers of the negative Laplacian and, as with the Enss--Weder method, we apply our estimate to inverse scattering. We find that the high-velocity limit of the scattering operator uniquely determines the short-range interactions.