The mathematical analogy of the three magnon bound-state equation with other momentum-space integral equations is studied. It is shown that a variable transformation, similar to Wick’s transformation in the Bethe–Salpeter equation, leads to alternative methods of complete analytic solutions of this equation.
Recently several unphysical solutions of the three magnon bound state equation for the isotropic Heisenberg Hamiltonian have been found, and one unphysical solution for the Hamiltonian with longitudinal anisotropy has been computed. Here we complete the work for such unphysical solutions for all anisotropy from the Ising to the isotropic Heisenberg limit by directly solving the integral equation. Two types of wavefunctions are constructed. The eigenvalue of the first type satisfies a cubic equation in general, but gives only a real root in the Ising limit and a pair of complex conjugate roots in the isotropic limit. The other type has a single eigenvalue; this one, previously known numerically, is shown to have a simple analytic form.
The Bethe–Salpeter equation for three bosons with zero-range interaction is solved for the first time. For comparison the light-front equation is also solved. The input is the two-body scattering length and the outputs are the three-body binding energies, Bethe–Salpeter amplitudes and light-front wave functions. Three different regimes are analyzed: ( i ) For weak enough two-body interaction the three-body system is unbound. ( ii ) For stronger two-body interaction a three-body bound state appears. It provides an interesting example of a deeply bound Borromean system. ( iii ) For even stronger two-body interaction this state becomes unphysical with a negative mass squared. However, another physical (excited) state appears, found previously in light-front calculations. The Bethe–Salpeter approach implicitly incorporates three-body forces of relativistic origin, which are attractive and increase the binding energy.
We present a gate-based quantum computing solution of the homogeneous Bethe-Salpeter equation (hBSE) for the bound state of two massive relativistic scalar particles interacting via ladder-approximation scalar exchange. After Wick rotation to Euclidean space and O(4) S-wave partial-wave projection, the hBSE is reduced to a symmetric matrix eigenvalue problem of dimension N = 2^n. We decompose the resulting Hamiltonian into a sum of n-qubit Pauli operators and solve it with the Variational Quantum Eigensolver (VQE) using a Matrix Product State (MPS) tensor-network ansatz. For N=16 (n=4 qubits), the VQE recovers the maximum eigenvalue - which encodes the minimum coupling constant for binding - to better than 1 % mean relative error compared to classical diagonalization. Entanglement analysis of the BSE amplitude shows low, area-law-like entanglement over the tested sizes, which motivates the MPS ansatz. A critical analysis of three independent barriers - exponential Pauli overhead, approximately size-independent low entanglement, and decreasing VQE gradient scales for the tested ansatz, consistent with generic barren-plateau concerns at larger n - reveals that the specific problem studied here lies in a classically tractable regime: over the tested range it is well described by low-bond-dimension tensor networks and efficiently handled by MPS/Lanczos methods. This negative result provides, to our knowledge, the first entanglement quantification of the BSE amplitude in qubit encoding, establishes the Pauli Hamiltonian framework for future BSE variants, and identifies 2D Minkowski-space BSE, N-body bound states, and non-ladder kernels as physically motivated extensions where genuine quantum advantage may become plausible.