arXiv:2606.01718

Asymptotic Recovery in Fourier Spectral Methods for the Schrödinger Equation with Point Singularities

Yanjie Li, Sihong Shao

math.NAmath.SPquant-phSubmitted Jun 1, 2026• Announced Jun 2, 2026

Abstract

This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials $V \in H^{s}$, where $s > \max\{d/2-2,-1\}$ and $d$ denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order $2s+2$ for eigenvalues and order $s+1$ for eigenfunctions in the $H^1$ norm. More importantly, the $H^1$ error with respect to the projected eigenfunction converges with a higher order $s+1+b$, where $b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0$ for arbitrarily small $\varepsilon>0$, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders $2s+2+2b$ for eigenvalues and $s+1+b$ for eigenfunctions in the $H^1$ norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in $H^{s+4}$ together with $d+1$ asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.

Comments: 43 pages, 4 figures, 3 tables

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arXiv:2606.01718

Asymptotic Recovery in Fourier Spectral Methods for the Schrödinger Equation with Point Singularities

Yanjie Li, Sihong Shao

math.NAmath.SPquant-phSubmitted Jun 1, 2026• Announced Jun 2, 2026

arXiv HTML PDF

Abstract

This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials

V \in H^{s}

, where

s > \max\{d/2-2,-1\}

and

d

denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order

2s+2

for eigenvalues and order

s+1

for eigenfunctions in the

H^1

norm. More importantly, the

H^1

error with respect to the projected eigenfunction converges with a higher order

s+1+b

, where

b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0

for arbitrarily small

\varepsilon>0

, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders

2s+2+2b

for eigenvalues and

s+1+b

for eigenfunctions in the

H^1

norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in

H^{s+4}

together with

d+1

asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.

Comments: 43 pages, 4 figures, 3 tables

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This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials

V \in H^{s}

, where

s > \max\{d/2-2,-1\}

and

d

2s+2

for eigenvalues and order

s+1

for eigenfunctions in the

H^1

norm. More importantly, the

H^1

error with respect to the projected eigenfunction converges with a higher order

s+1+b

, where

b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0

for arbitrarily small

\varepsilon>0

2s+2+2b

for eigenvalues and

s+1+b

for eigenfunctions in the

H^1

H^{s+4}

together with

d+1

asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.

1/4

-order convergence rate in the time step for many-body Yukawa interactions, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in

H^2(\mathbb R^{3N})

, the natural domain of the Hamiltonian, and our numerical experiments are consistent with the theoretical estimates. To identify the sharp obstruction behind this rate, we prove a short-time lower bound in the one-body setting of order

t^{5/4}

for the one-step error, which rules out any uniform global estimate of order better than

1/4

in general. This agreement with the optimal

1/4

rate in the Coulomb case is particularly interesting, as Yukawa potential is short-ranged compared to Coulomb potential. For the many-body upper bound, one of the new technical ingredients is the explicit polynomial-in-system-size Sobolev estimates of many-body Yukawa systems. These estimates are crucial for obtaining fully a priori bounds that depend only on the norms of the initial states, rather than on the solution at time

t

. For the one-body lower bound, we leverage a new analysis argument based on Fourier analysis and Kato smoothing.

The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians

[-{\rm d}^2/{\rm d} q^2 + V(q)]^\pm

on the half-line

\{q>0\}

, with a Dirichlet (

-

) or Neumann (

+

) condition at

q=0

. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential

V(q)=q^N

N \to +\infty

vs its (solvable)

N=\infty

limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-

N

properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants

\det( -{\rm d}^2/{\rm d} q^2 + q^4 + v q^2)^\pm

are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic

v \to +\infty

properties of Airy type. The third study addresses the potentials

V(q)=q^N+v q^{N/2-1}

of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with

v

as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to

N=2

); these results probably reflect the presence of supersymmetric potentials in the family above.

In the framework of distributionally generalized quantum theory, the object

Hψ

is defined as a distribution. The mathematical significance is a mild generalization for the theory of para- and pseudo-differential operators (as well as a generalization of the weak eigenvalue problem), where the

ψ

-do symbol (which is not a proper linear operator in this generalized case) can have its coefficient functions take on singular distributional values. Here, a distribution is said to be singular if it is not L

^p(\mathbb{R}^d)

for any

p\geq 1

. Physically, the significance is a mathematically rigorous method, which does not rely upon renormalization or regularization of any kind, while producing bound state energy results in agreement with the literature. In addition, another benefit is that the method does not rely upon self-adjoint extensions of the Laplace operator. This is important when the theory is applied to non-Schrodinger systems, as is the case for the Dirac equation and a necessary property of any finite rigorous version of quantum field theory. The distributional interpretation resolves the need to evaluate a wave function at a point where it fails to be defined. For

d=2

, this occurs as

K_o(a|x|)δ(x)

, where

K_o

is the zeroth order MacDonald function. Finally, there is also the identification of a missing anomalous length scale, owing to the scale invariance of the formal symbol(ic) Hamiltonian, as well as the common identity for the logarithmic function, with

a,\,b\in\mathbb{R}^+

\log(ab)=\log(a)+\log(b)

, which loses unitlessness in its arguments. Consequently, the energy or point spectrum is generalized as a family (set indexed by the continuum) of would-be spectral values, called the C-spectrum.

L^2

bound sate solution) being singular at the compact point support of the Dirac delta function (measure). The problem is solved here in a completely mathematically rigorous manner with no recourse to renormalization nor regularization. The method involves a distributionally generalized version of the Schrodinger theory as developed by the author, which regards the formal symbol "

Hψ

" as an element of the space of distributions, the topological dual vector space to the space of smooth functions with compact support. Two main facts come to light. The first is the bound state energy of such a system can be calculated in a well-posed context, the value of which agrees with both the mathematical and theoretical physics literature. The second is that there is then a rigorous distributional version of the Hellmann-Feynman theorem.