arXiv subjects

Sohei Ashida

Publications and source records attributed to Sohei Ashida.

13 recordsLinked to original sources

arXiv2023

Lower bound for solutions to the secular equation in a direct sum of the Sobolev spaces of divided regions

In this paper a lower bound for solutions to the secular equation of the Schr\"odinger equation with basis functions discontinuous on boundaries of divided regions is given. If the functions do not have the discontinuity, the bound reduces to that for the usual Rayleigh-Ritz method. Difference from the usual Rayleigh-Ritz method is bounded by the degree of discontinuity. The result can be regarded as a theoretical basis of the augmented plane wave method for the band structure calculations in solid state physics. The result would be useful also for other electronic eigenvalue problems in which the behavior of the eigenfunction near nuclei is very different from that in the interstitial region, because it allows us to use different basis functions in different regions in contrast with the usual Rayleigh-Ritz method in which we need to use basis functions with inappropriate behaviors in wide regions. The proof is based only on new general results about the Sobolev space, in particular, equivalence of two semi-norms concerned with discontinuity of the functions in a direct sum of the Sobolev spaces. Therefore, the result does not depend on specific forms of basis functions at all. The main ingredient of the proof is an estimate of elements of the orthogonal complement of $H^1_0(\Omega)$ by their boundary values based on a characterization of them as weak solutions to an elliptic equation on a region $\Omega$.

math.AP↗

arXiv2023

Critical points of the discretized Hartree-Fock functional of connected molecules preserving structures of molecular fragments

In this paper a method to obtain a critical point of the discretized Hartree-Fock functional from an approximate critical point is given. The method is based on Newton's method on the Grassmann manifold. We apply Newton's method regarding the discretized Hartree-Fock functional as a function of a density matrix. The density matrix is an orthogonal projection in the linear space corresponding to the discretization onto a subspace whose dimension is equal to the number of electrons. The set of all such matrices are regarded as a Grassmann manifold. We develop a differential calculus on the Grassmann manifold introducing a new retraction (a mapping from the tangent bundle to the manifold itself) that enables us to calculate all derivatives. In order to obtain reasonable estimates, we assume that the basis functions of the discretization are localized functions in a certain sense. As an application we construct a critical point of a molecule composed connecting several molecules using critical points of the Hartree-Fock functional corresponding to the molecules as the basis functions under several assumptions. By the error estimate of Newton's method we can see that the electronic structures of the molecular fragments are preserved.

math-ph↗

arXiv2022

Convergence of SCF sequences for the Hartree-Fock equation

The Hartree-Fock equation is a fundamental equation in many-electron problems. It is of practical importance in quantum chemistry to find solutions to the Hartree-Fock equation. The self-consistent field (SCF) method is a standard numerical calculation method to solve the Hartree-Fock equation. In this paper we prove that the sequence of the functions obtained in the SCF procedure is composed of a sequence of pairs of functions that converges after multiplication by appropriate unitary matrices, which strongly ensures the validity of the SCF method. A sufficient condition for the limit to be a solution to the Hartree-Fock equation after multiplication by a unitary matrix is given, and the convergence of the corresponding density operators is also proved. The method is based mainly on the proof of approach of the sequence to a critical set of a functional, compactness of the critical set, and the proof of the \L ojasiewicz inequality for another functional near critical points.

math.AP↗

arXiv2021

Upper bounds of local electronic densities in molecules

The eigenfunctions of electronic Hamiltonians determine the stable structures and dynamics of molecules through the local distributions of their densities. In this paper an a priori upper bound for such local distributions of the densities is given. The bound means that concentration of electrons is prohibited due to the repulsion between the electrons. A relation between one-electron and two-electron densities resulting from the antisymmetry of the eigenfunctions plays a crucial role in the proof.

math-ph↗

arXiv2021

Methods for accurate calculations of multi-center integrals of squared Coulomb potentials for lower bounds to energy levels of molecular systems

In this paper methods for calculations of multi-center integrals of squared Coulomb potentials and Slater-type orbitals (STO) are derived. These integrals are necessary for accurate lower bounds to energy levels of molecular systems. All multi-center integrals are reduced to fundamental integrals using the Gaunt coefficients and translation of STO. When the potential is the usual Coulomb potential, using the Laplace expansion or the Neumann expansion of the potential the integrals can be calculated. However, for the squared Coulomb potentials such expansions are not known. For the fundamental one-center and two-center integrals with squared Coulomb potentials, by methods free from such expansions exact analytic expressions and expressions by one-dimensional integrals of analytic functions are derived. The methods mainly rely on the integration in ellipsoidal coordinates, the Fourier transform, Hobson's theorem and expansion of differential operators by simple ones suitable for the calculation. Numerical results by these expressions are given and compared.

physics.chem-ph↗

arXiv2020

Structures of sets of solutions to the Hartree-Fock equation

The Hartree-Fock equation which is the Euler-Lagrange equation corresponding to the Hartree-Fock energy functional is used in many-electron problems. Since the Hartree-Fock equation is a system of nonlinear eigenvalue problems, the study of structures of sets of all solutions needs new methods different from that for the set of eigenfunctions of linear operators. In this paper we prove that the sets of all solutions to the Hartree-Fock equation associated with critical values of the Hartree-Fock energy functional less than the first energy threshold are unions of a finite number of compact connected real-analytic spaces. The result would also be a basis for the study of approximation methods to solve the equation.

math.AP↗

arXiv2020

Finiteness of the number of critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold

We study the Hartree-Fock equation and the Hartree-Fock energy functional universally used in many-electron problems. We prove that the set of all critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold is finite. Since the Hartree-Fock equation which is the corresponding Euler-Lagrange equation is a system of nonlinear eigenvalue problems, the spectral theory for linear operators is not applicable. The present result is obtained establishing the finiteness of the critical values associated with orbital energies less than a negative constant and combining the result with the Koopmans' well-known theorem. The main ingredients are the proof of convergence of the solutions and the analysis of the Fr\'echet second derivative of the functional at the limit point.

math-ph↗

arXiv2019

Lower bounds to eigenvalues of one-electron Hamiltonians

A method for computing lower bounds to eigenvalues of sums of lower semibounded self-adjoint operators is presented. We apply the method to one-electron Hamiltonians. To improve the lower bounds we consider symmetry of molecules and use Temple's inequality. These methods would be useful in estimating eigenvalues of electronic Hamiltonians needed for studies of properties of molecules.

math-ph↗

arXiv2018

Scattering matrices and generalized Fourier transforms in long-range $N$-body problems

We give a definition of scattering matrices based on the asymptotic behaviors of generalized eigenfunctions and show that these scattering matrices are equivalent to the ones defined by wave-operator approach in long-range $N$-body problems. We also define generalized Fourier transforms by the asymptotic behaviors of outgoing solutions to nonhomogeneous equations and show that they are equivalent to the definition using wave operators. We also prove that the adjoint operators of the generalized Fourier transforms are given by Poisson operators.

math-ph↗

arXiv2018

Propagation Estimates for Two-cluster Scattering Channels of N-body Schrödinger Operators

In this paper we prove propagation estimates for two-cluster scattering channels of N-body Schrödinger operators. These estimates are based on the estimate similar to Mourre's commutator estimate and the method of Skibsted. We also obtain propagation estimates with better indices using projections onto almost invariant subspaces close to two-cluster scattering channels. As an application of these estimates we obtain the resolvent estimate for two-cluster scattering channels and microlocal propagation estimates in three-body problems without projections. Our method clearly illustrates evolution of the solutions of the Schrödinger equation.

math-ph↗

arXiv2017

Scattering theory for multistate Schrödinger operators

We study multistate Schrödinger operators related to molecular dynamics. We consider potentials which do not necessarily decay and prove absence of the singular continuous spectrum and propagation estimates which mean the scattering at speed larger than a positive constant and decay of the state with potentials higher than considered energy at infinity. We also consider the multistate Schrödinger operators with many-body structures. We obtain the Mourre estimate and the minimal velocity estimate for the many-body operators. The lower bound of the velocity is determined by the distance between the energy and thresholds below the energy.

math-ph↗

arXiv2017

Molecular predissociation resonances below an energy level crossing

We study the resonances of $2\times 2$ systems of one dimensional Schrödinger operators which are related to the mathematical theory of molecular predissociation. We determine the precise positions of the resonances with real parts below the energy where bonding and anti-bonding potentials intersect transversally. In particular, we find that imaginary parts (widths) of the resonances are exponentially small and that the indices are determined by Agmon distances for the minimum of two potentials.

math-ph↗

arXiv2015

Born-Oppenheimer approximation for an atom in constant magnetic fields

We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. In Martinez-Sordoni \cite{MaSo2} such a case is also studied but their reduced Hamiltonian includes the vector potential terms. In this paper, using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.

math-ph↗