Reviews in Mathematical Physics

We study a mathematically rigorous derivation of a quantum mechanical Hamiltonian in a general framework. We derive such a Hamiltonian by taking a scaling limit for a generalization of the Nelson model, which is an abstract interaction model between particles and a Bose field with some internal degrees of freedom. Applying it to a model for the field of the nuclear force with isospins, we obtain a Schrödinger Hamiltonian with a matrix-valued potential, the one pion exchange potential, describing an effective interaction between nucleons.

We give a two-parameter variational formula for the grand-canonical pressure of the Pair Boson Hamiltonian model. By using the Approximating Hamiltonian Method we provide a rigorous proof of this variational principle.

In this paper, first, the general projective Riccati equation method (PREM) is applied to derive variable separation solutions of (2 + 1)-dimensional systems. By further studying, we find that these variable separation solutions obtained by PREM, which seem independent, actually depend on each other. A common formula with some arbitrary functions is obtained to describe suitable physical quantities for some (2 + 1)-dimensional models such as the generalized Nizhnik–Novikov–Veselov system, Broer–Kaup–Kupershmidt equation, dispersive long wave system, Boiti–Leon–Pempinelli model, generalized Burgers model, generalized Ablowitz–Kaup–Newell–Segur system and Maccari equation. The universal formula in Tang, Lou, and Zhang [2] can be simplified to the common formula in the present paper. Second, this method is successfully generalized to (1 + 1)-dimensional systems, such as coupled integrable dispersionless equations, shallow water wave equation, Boiti system and negative KdV model, and is able to obtain another common formula to describe suitable physical fields or potentials of these (1 + 1)-dimensional models, which is similar to the one in (2 + 1)-dimensional systems. Finally, based on the common formula for (2 + 1)-dimensional systems and by selecting appropriate multivalued functions, elastic and inelastic interactions among special dromion, special peakon, foldon and semi-foldon are investigated. Furthermore, the explicit phase shifts for all the local excitations offered by the common formula have been given, and are applied to these novel interactions in detail.

The addendum refers mainly to Sec. 5 of the paper (Friedrichs model on the positive half line). The "Schwartz space framework" is omitted because it is dispensable for the results. Improvements of the proofs are indicated. A supplement presents in the special case G₀ := ℂ\(-∞,0] a surprising implication: the scattering matrix has only simple poles and its "main part" is a linear combination of all Gamov vectors.