Finiteness of the discrete spectrum in the quantum n-particle problem

Discrete spectrum of many body Schrödinger operators with non-constant magnetic fields I

“…Schrόdinger operators with short-range scalar potentials (for example Zhislin [19]). Some different phenomena are expected to occur in non-constant magnetic fields.…”

On the discrete spectrum of a given SO(2) symmetry of multiparticle systems in potential and homogeneous magnetic fields

“…Since the support of the function Cu is contained in the ball [r I < b~, it follows that for sufficiently large rn, and, consequently, and so they are disjoint for different D2 [7]. Similarly to [5] Relations (3.9), (3.10), (3.12), (3.4), and (3.7) prove the validity of (3.1).…”

“…To avoid rewriting a significant number of calculations from [2] and [4] we perform the proof of Theorems 2.1 and 2.2 in the simplest case, where Q(/92) < 0, 7 < 2. We shall prove that in this case we have: Then it is easy to obtain by analogy with [5] Under the assumptions for potentials mentioned above one arrives at the inequality for any function ~(r) with support in a bounded domain [6]:…”

Zero-Threshold Resolvent Asymptotics of Three-Body Schrödinger Operators

“…When the essential spectrum of the three-particle Hamiltonian is the positive real axis, and when at least two of its two-body Subhamiltonians have a resonance at the threshold zero, the discrete spectrum of the three-body Schrödinger operator is infinite, even if the interactions are very short-range. This phenomenon is striking if one compares it with the results on the finiteness of eigenvalues of two-body Schrödinger operators or N -body Schrödinger operators whose bottom of essential spectrum is only reached by the spectrum of two-cluster Subhamiltonians ( [6,17]). Since then, many works, both in mathematical and physical literature, are devoted to this subject (see, for example, [1,2,3,9,10,13,15,16,19,21,23]).…”