Double-Exponential Sinc-Collocation Method

• Double-Exponential Sinc-Collocation Method is a numerical technique combining Sinc interpolation with a double-exponential variable transformation to solve differential and eigenvalue problems with near spectral accuracy.
• It transforms unbounded or singular problems into symmetric generalized eigenvalue problems, effectively handling anharmonic oscillators and singular Sturm-Liouville systems.
• The method provides rigorous error estimates and optimal mesh strategies, ensuring efficient computation and high accuracy across various quantum and boundary value applications.

The double-exponential Sinc-collocation method (DESCM) is a numerical technique for solving differential equations—especially eigenvalue problems of quantum mechanics and Sturm-Liouville systems—that combines Sinc interpolation with a double-exponential variable transformation to achieve nearly spectral convergence rates. The method efficiently approximates solutions on unbounded or singular domains by exploiting the rapid decay properties induced by the transformation, resulting in symmetric generalized eigenvalue problems with well-characterized error bounds. DESCM has been extensively applied to anharmonic oscillators, rational and Coulombic potentials, and singular boundary value problems, frequently outperforming single-exponential Sinc and other spectral methods in accuracy and computational efficiency (Gaudreau et al., 2014, Gaudreau et al., 2014, Gaudreau et al., 2015, Gaudreau et al., 2016, Cassidy et al., 2015).

1. Sinc Basis Functions and Cardinal Expansion

The foundation of DESCM is the Sinc function

$$\mathrm{sinc}(z) = \begin{cases} \frac{\sin(\pi z)}{\pi z}, & z \neq 0\ 1, & z = 0 \end{cases}$$

and the shifted Sinc basis

$$S_{j,h}(x) = \mathrm{sinc}\left(\frac{x-jh}{h}\right)$$

for mesh size $h > 0$ and $j \in \mathbb{Z}$. These basis functions satisfy the cardinal property $S_{j,h}(kh) = \delta_{j,k}$, enabling discrete orthogonality.

For a function $v(x)$ over $\mathbb{R}$, the (truncated) Sinc expansion is

$C_N[v,h](x) = \sum_{j=-N}^{N} v(jh) S_{j,h}(x)$

which interpolates $v$ at nodes $x = jh$, $j \in \{-N,\ldots,N\}$ (Gaudreau et al., 2014, Gaudreau et al., 2016).

2. Double-Exponential (DE) Variable Transformation

DESCM achieves accelerated convergence by applying a conformal map $\phi: \mathbb{R} \to D$ such that the transformed unknown function $v(u) = \sqrt{\phi'(u)}\,\psi(\phi(u))$ decays double-exponentially as $|u| \to \infty$. Canonical choices include:

Target domain	DE map $\phi(u)$	Jacobian $\phi'(u)$
$(-\infty, \infty)$	$\sinh(u)$	$\cosh(u)$
$(0, \infty)$	$\log(e^{\sinh u} + 1)$	$\frac{e^{\sinh u}\cosh u}{e^{\sinh u} + 1}$
$(0, 1)$	$\frac{1}{2} (\tanh(\sinh u)+1)$	$\frac{1}{2}\mathrm{sech}^2(\sinh u)\cosh u$

For problems such as the Schrödinger equation with single-exponential tails, the transformation enforces double-exponential decay on $v(u)$, resulting in strong analytic properties and rapid convergence of the Sinc expansion (Gaudreau et al., 2014, Cassidy et al., 2015, Gaudreau et al., 2016).

3. Formulation: Collocation and the Generalized Eigenvalue Problem

After applying the DE transform, e.g. $x = \phi(u)$, the unknown function is represented as $v(u) \approx \sum_{j=-N}^{N} v_j S_{j,h}(u)$, where $v_j = v(jh)$. Collocating the transformed differential equation at the Sinc grid points $u_k = kh$ yields a generalized eigenvalue problem: $$\sum_{j=-N}^{N} \left[ -\frac{1}{h^2} \delta^{(2)}_{k,j} + \widetilde{V}(kh) \delta^{(0)}_{k,j} \right] v_j = E \sum_{j=-N}^{N} (\phi'(kh))^2 \delta^{(0)}_{k,j} v_j$$ where the transformed potential $\widetilde{V}(u)$ is given by

$$\widetilde{V}(u) = -\sqrt{\phi'(u)} \frac{d}{du} \left[ \frac{1}{\phi'(u)} \frac{d}{du} \sqrt{\phi'(u)} \right] + (\phi'(u))^2 V(\phi(u))$$

(Gaudreau et al., 2014, Gaudreau et al., 2016).

Letting $H$ and $S$ denote, respectively, the stiffness and mass matrices, the discrete problem becomes

$H \mathbf{v} = E S \mathbf{v}$

with $H$ symmetric and $S$ positive-definite diagonal. This form admits standard dense eigensolvers and supports efficient computation.

4. Optimal Mesh Size and Principle of Minimal Sensitivity

The convergence and accuracy of DESCM depend critically on the choice of collocation mesh size $h$. For functions decaying as $|v(u)| \leq A \exp(-B e^{\gamma |u|})$ and analytic in a strip $|\operatorname{Im} u| < d$, the optimal mesh is

$h_{\mathrm{opt}} = \frac{W(\alpha N)}{\gamma N}$

where $W$ is the Lambert–W function and $\alpha$ is determined by asymptotic decay constants, potential degree, and domain strip-width (Gaudreau et al., 2014, Cassidy et al., 2015, Gaudreau et al., 2016, Gaudreau et al., 2014).

The principle of minimal sensitivity refines $h$ numerically by minimizing the trace of the symmetrized matrix $K = S^{-1/2} H S^{-1/2}$, further improving accuracy, particularly for multi-well and complex potentials.

5. Convergence Theory and Error Estimates

DESCM achieves nearly spectral convergence rates, with rigorous error bounds established. For eigenvalue approximations $E_N$ to simple eigenvalues $E$, one has

$$|E_N - E| \leq C N^{5/2} (\ln N)^2 \exp\left(-\kappa \frac{N}{\ln N}\right)$$

for some $C, \kappa > 0$ depending on the analytic strip and decay properties. For initial value problems, the error for solution vectors $\hat{y}(t)$ obeys

$$\max_{1 \leq i \leq m} \sup_{t \geq 0} |y_i(t) - \hat{y}_i^{(l)}(t)| \leq (C + \widehat{C} \|B_{lm}^{-1}\|_\infty) \,\operatorname{arsinh}(d n/\mu) \ln(n) \exp\left( - \frac{d n}{\operatorname{arsinh}(d n/\mu)} \right)$$

showing $O(e^{-c n/\ln n})$ convergence (Gaudreau et al., 2014, Gaudreau et al., 2016, Okayama et al., 2023, Yamamoto, 3 Jan 2026).

6. Practical Implementation and Algorithmic Steps

The typical DESCM workflow entails:

• Selecting the appropriate DE transformation $\phi(u)$ for the domain and potential.
• Estimating decay parameters ($B$, $\gamma$) and analytic strip width $d$.
• Calculating the optimal mesh size $h$ via the analytic or minimal-sensitivity principle.
• Computing Sinc grid points, transformed potential values, and matrix entries ($\delta^{(0)}$, $\delta^{(2)}$ derivatives).
• Assembling and solving the symmetric generalized eigenvalue problem.
• Post-processing eigenvalues and, for IVPs, reconstructing solutions via Sinc expansions (Gaudreau et al., 2014, Cassidy et al., 2015, Gaudreau et al., 2014, Okayama et al., 2023, Yamamoto, 3 Jan 2026).

For systems with parity symmetry, resulting matrices are centrosymmetric, enabling substantial savings by splitting the computation into two smaller problems and requiring storage and computation of only $1/(N+1)$ of all matrix entries (Gaudreau et al., 2015).

7. Applications and Representative Results

DESCM has produced highly accurate results for:

• Single-well and multi-well anharmonic oscillators: $V(x) = x^2 + \beta x^4$, $V(x) = 4x^2 - 6x^4 + x^6$ (Gaudreau et al., 2014). Errors decay as $e^{-a N / \ln N}$, with 12-digit accuracy often attained with $N \approx 30-50$.
• Rational potentials of the form $V(x) = p(x) / q(x)$ (Gaudreau et al., 2016).
• Anharmonic Coulombic potentials using generalized DE maps and scaling factors for stability (Cassidy et al., 2015).
• Singular Sturm-Liouville problems, e.g. Bessel-type equations, with $10^{-14}$–level errors for moderate $N$ (Gaudreau et al., 2014).
• Initial-value ODEs, where Gauss-Seidel-type fixed-point iteration for the nonlinear DESCM system yields rapid reduction of error—two orders of magnitude per sweep with contraction factor $O((\ln N)^2 / N)$ (Yamamoto, 3 Jan 2026).

The method is routinely implemented in Julia, MATLAB, and Fortran. For all classes of problems above, DESCM outperforms single-exponential Sinc and many alternative spectral methods due to the double-exponential mapping and mesh optimization strategies.