This thesis aims to develop an advanced numerical solver capable of efficiently computing the resonant states of quantum mechanical two-body and three-body problems, thereby expanding our understanding of these complex systems. The quantum three-body problems feature at least two dimensions, which necessitates substantial computational efforts. Therefore, in order to tackle these challenging computations, we need to seek assistance from supercomputers. By harnessing the capabilities of high-performance computing, we can significantly reduce the amount of time spent waiting for programs to run for hours.

In this thesis, we first introduce some basic knowledge about quantum few-body problems and resonant states, showing how the physical problem gives rise to a mathematical problem, the quadratic eigenvalue problem (QEP). Building upon the physical background, our journey in developing the methodology begins with two fundamental components: discretization and eigensolver. The pseudo-spectral methods are introduced to represent the quadratic eigenvalue problem as a matrix problem, by which we can solve the problem numerically through some eigensolvers. We describe a classical approach called linearization for solving QEPs, which transforms the quadratic problem into a generalized eigenvalue problem. Following the linear transformation, we apply the Jacobi-Davidson QZ (JDQZ) method, an iterative eigensolver, to solve the linearized problem. Alternatively, we could also use the Jacobi-Davidson (JD) method to approximate the quadratic eigenvalue problem's eigenpairs directly. In this thesis, we provide an outline of the Jacobi-Davidson process for solving both linear and quadratic problems. Two routes for solving QEPs are utilized and compared: linearization combined with the Jacobi-Davidson QZ method, and the quadratic Jacobi-Davidson method. Through our research and analysis, we demonstrate that the Jacobi-Davidson algorithm exhibits superior computational efficiency when adapted to solve QEPs directly.

Another significant objective of this thesis is to parallelize the eigensolver on supercomputers. We implement a hybrid distributed/shared memory parallelization of the Jacobi-Davidson algorithm to solve quadratic eigenvalue problems that arise from one-dimensional three-body problems. We leverage the tensor structure inherent in the three-body problem to optimize computational efficiency. Specifically, we implement an efficient tensor product scheme for the application of the stiffness and damping operators, which are realized as dense matrix-matrix products. By incorporating a preconditioner that also preserves the tensor structure, we enhance the performance of our Jacobi-Davidson algorithm in computing three-body resonance poles within an acceptable speed.