The absence of positive energy bound states for a class of nonlocal potentials

Abstract

We generalize in this paper a theorem of Titchmarsh for the positivity of Fourier sine integrals. We apply then the theorem to derive simple conditions for the absence of positive energy bound states (bound states embedded in the continuum) for the radial Schrödinger equation with nonlocal potentials which are superposition of a local potential and separable potentials.

Generalization of Bochner's theorem for functions of the positive type

“…We are now going to generalize the above theorem by replacing, as we did similarly in [1], the exponential function in (16) by the appropriate solution of the differential equation (3), namely the Jost solution, which satisfies [3,4,5] Here, since we consider the half-axis r ∈ [0, ∞), we must restrict the support of α(t) in (16) to be also in [0, ∞). We shall study the case of the full axis x ∈ (−∞, ∞)…”

“…In ref. [1], we gave an application of our theorem 2 to secure the absence of positive energy bound states (bound states embedded in the continuum) in the radial Schrödinger equation for a class of nonlocal potentials. We give now an application of the Bochner's Theorem to the Fourier integral representation of the phase-shift.…”

Essential spectrum of a model operator associated with a three-particle system on a lattice

“…were studied in [10], [11], where they were regarded as models associated with a system of several particles interacting via nonlocal pair potentials. For example, one of the nonlocal potentials is given by the Gauss potential, whose kernel has the form…”

“…They are also systematically used along with Faddeev equations for three-particle systems. The main feature of these equations [11] is that the particle-wave tmatrix retains its simple form and can easily be continued, which is the most important characteristic in nuclear physics and in Faddeev equations.…”

Существенный Спектр Одного Модельного Оператора, Ассоциированного С Системой Трех Частиц На Решетке

“…Они также систематически используются вместе с уравнениями Фаддеева для систем трех частиц. Основная характеристика этих уравнений состоит в том [11], что частично-волновая t-матрица имеет ту же простую форму и может быть продолжена простым способом: наиболее важная характеристика в ядерной физике и в уравнениях Фаддеева.…”