A quasi-orthogonal method based on the inverse operator for Schr{ö}dinger eigenvalue problems

Abstract

Computing many eigenpairs of the Schr{ö}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a quasi-orthogonal evolution model utilizing inverse operators and develop a corresponding discrete numerical scheme. Instead of forcing explicit orthogonalization, the proposed framework confines the numerical approximations within a quasi-Stiefel set, ensuring the iterates maintain full column rank without requiring $\left\langle U, U \right\rangle=I_N$. Moreover, the method naturally absorbs orthogonality errors and asymptotically converges to the exact eigenfunctions, even when initialized with non-orthogonal random data. The scheme guarantees monotonic dissipation of the target energy functional, with exponential convergence rates rigorously established for the discrete energy, gradient, and eigenfunction approximations. Furthermore, infinite-dimensional analysis proves that the admissible time step size is independent of the spatial discretization. This property overcomes the mesh-dependent stability constraints typical of conventional explicit or semi-implicit schemes, permitting larger time increments to accelerate global convergence. Numerical experiments validate the theoretical findings.