High Order Finite Difference Schemes for the Transparent Boundary Conditions and Their Applications in the 1D Schr\"odinger-Poisson Problem

Haiyan Jiang; Meili Guo; Tiao Lu; Wenqi Yao

REVIEW 3 major objections 4 minor 23 references

High Order Finite Difference Schemes for the Transparent Boundary Conditions and Their Applications in the 1D Schr\"odinger-Poisson Problem

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that a compact fourth-order interior scheme combined with its discrete fourth-order transparent boundary conditions gives a globally fourth-order discretization of the 1D Schrödinger problem, and that an analytic variant…

desk verdict Useful fourth-order TBC construction, but the uniqueness theorems have a real gap at E=V0 that lands inside your own energy integral. read the letter →

arxiv 2411.13175 v1 pith:H3MATBBN submitted 2024-11-20 math.NA cs.NA

Meili Guo , Haiyan Jiang , Tiao Lu , Wenqi Yao This is my paper · ORCID

classification math.NAcs.NA MSC 34L4065L1065L20

keywords 1DSchrödinger–Poissonproblemtransparentboundaryconditioncompactfinitedifferenceschemediscreteresonanttunnelingdiodefourth-orderconvergencequantumtransport

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Simulating nanoscale semiconductor devices with the 1D Schrödinger–Poisson model requires artificial boundary conditions that let electron waves leave the computational box without reflection. The paper introduces two boundary-condition discretizations: D4TBCs, fourth-order discrete transparent boundary conditions that match the accuracy of the compact interior scheme and avoid spurious oscillations when the potential vanishes, and aDTBCs, analytic discretizations that introduce no boundary error at all. It proves that the interior scheme closed with D4TBCs is globally fourth order, proves uniqueness for both discrete Schrödinger problems under a grid-dependent condition, and demonstrates fourth-order convergence and realistic current–voltage curves for a resistor and two resonant tunneling diodes.

What carries the argument

The load-bearing objects are the two boundary-condition families. The D4TBCs are fourth-order discrete transparent boundary conditions obtained by inserting the ansatz $\psi_j=\alpha^j$ on the left and $\psi_j=\beta^j$ on the right into the compact fourth-order stencil, solving the resulting discrete dispersion relations for $\alpha=e^{i\tilde k_1\Delta x}$ and $\beta=e^{i\tilde k_2\Delta x}$, and using these roots in the one-sided relations (38)–(39). The aDTBCs are analytic discrete transparent boundary conditions obtained by expanding $\psi(\pm j\Delta x;k)$ in infinite Taylor series around the boundary and using the exact dispersion relation $k_1=\sqrt{2m_*(E-V_0)}/\hbar$; this yields relations (55)–(56) and their multi-point generalizations (57)–(58). The interior engine is the compact fourth-order approximation (21) for $u_{xx}=f$, together with the fourth-order one-sided derivative formula (23) used to discretize the transparent boundary conditions (7)–(8).

What would settle it

Run the paper's zero-potential test with $\hbar=m_*=1$, $E=0.5$, $L=10$, and a coarse grid $\Delta x=2.5$, so that $t=6/\Delta x^2=0.96<2E=1$, solving (24) with the D4TBCs for a unit-amplitude incoming wave. If the numerical wave reproduces $e^{ikx}$ and the homogeneous version of the discrete system has only the zero solution, then the condition $t>2(E-V)$ is not necessary; if not, that hypothesis is doing essential work.

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Extended reading notes

Core claim

The paper's central claim is that transparent boundary conditions should be discretized together with the interior stencil, not as separate low-order add-ons. Theorem 1 states that the compact fourth-order interior scheme (24) equipped with the D4TBCs forms a globally fourth-order discretization of the 1D Schrödinger problem. The D4TBCs are constructed so that, when the potential is constant outside the device, the exterior discrete waves $\alpha^{\pm j}$ and $\beta^{j}$ are unimodular, which prevents the spurious oscillations seen when the boundary conditions are discretized naively. The aDTBCs go further: because the wave function is analytic near the boundaries, the paper writes the exterior relations as exact Taylor series, so the boundary introduces no discretization error and the overall order becomes whatever order the interior scheme delivers, even arbitrarily high order.

Load-bearing premise

The uniqueness and non-oscillation theorems require a grid-dependent quantity $t$, proportional to the inverse square of the grid spacing, to exceed $2(E-V)$ at both contacts for every energy $E$; the resonant-tunneling diode simulations use energies up to $0.8$ eV with $\Delta x=0.5$ nm, where this condition is violated for part of the range, so the proved regime is a subset of the tested regime.

Editorial extensions

If this is right

A Schrödinger solver using the compact fourth-order interior scheme and D4TBCs converges as $O(\Delta x^4)$ throughout the computational box, including the boundary region.
Because aDTBCs introduce no boundary error, the global accuracy of a Schrödinger–Poisson calculation is limited only by the interior finite-difference scheme.
With aDTBCs, the overall accuracy can be raised to arbitrarily high order by widening the interior stencil, whereas DTBCs would require solving algebraic equations of degree at least six, which cannot be done explicitly.
The two discrete Schrödinger–Poisson models have unique solutions under the stated condition, so the iterative device-simulation algorithm is well posed at the discrete level.
The fourth-order schemes reproduce benchmark resistor and resonant-tunneling-diode current–voltage curves, including negative differential resistance, on meshes as coarse as 0.5 nm.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if aDTBCs are truly error-free, then for smooth potentials the practical convergence rate of Schrödinger–Poisson codes should track the interior stencil order exactly; a clean test would be to compare measured convergence orders for sixth- and eighth-order interior stencils against the predicted rates.
Beyond the paper: the violation of the uniqueness condition at high energies in the resonant-tunneling diode runs suggests a concrete numerical experiment, namely checking whether spurious oscillations or non-uniqueness appear in a zero-potential test as $\Delta x$ is increased until $t\leq 2E$, which would map the boundary of the proved regime.
Beyond the paper: the aDTBC construction relies only on analyticity of the wave function at the boundary and the constant-exterior-potential dispersion relation, so the same Taylor-series idea should transfer to time-dependent Schrödinger equations or to higher-dimensional settings where the exterior solution is known in closed form, though the paper does not explore these cases.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper deserves a serious look. The D4TBCs are a genuine fourth-order extension of Arnold's discrete transparent boundary conditions, and the aDTBC framework — using exact plane-wave relations at ghost points — is elegant and does let you push the boundary closure to arbitrary order without solving high-degree algebraic equations. The derivation of the D4TBCs is careful, Theorem 1 (global fourth-order accuracy) is supported by the Taylor analysis, and the numerical work is honest: convergence rates are reported, NEGF benchmarks are used as reference, and the data are on figshare.

The main soft spot is not the one the reader flagged. The condition t > max{2(E−V0), 2(E−VNx)} holds comfortably in the RTD experiments; the central issue is the equality case E = V0. With V ≡ 0 and E = 0 you get α = β = 1, the D4TBCs reduce to ψ−1 = ψ0 and ψNx+1 = ψNx, and the homogeneous problem has nonzero constant solutions. The proof of Theorem 2 fails exactly there: equation (53) hinges on Im(α^{-1}) and Im(β^{-1}) being nonzero, and at α = 1 they vanish. Since E = 0 is the endpoint of the energy integral (11) used in the experiments, this is not a purely academic edge case. Theorem 3 inherits the problem, and its proof is omitted, so the aDTBC uniqueness claim is doubly unsupported.

Other, smaller concerns: the abstract says aDTBCs introduce no discretization error, but Fig. 1 shows spurious oscillations (smaller than C4TBCs, but visible), so that claim should be read narrowly as "no error from the boundary closure itself." And the claim about improving accuracy to arbitrarily high order is conditional on having an interior scheme of matching order; the paper only demonstrates fourth order. The self-citation to Arnold is appropriate and properly contextualized.

Who is this for? Numerical analysts working on transparent boundary conditions for quantum transport. It is a solid, citable contribution in that niche, and the flaws are repairable. I would send it to peer review with a request for major revision: state and prove a corrected uniqueness theorem excluding E = V0 (or handle the degenerate case), write out the aDTBC uniqueness proof, and soften the no-error wording. Not a desk reject.

Referee Report

3 major / 4 minor

Summary. The manuscript develops two families of discrete transparent boundary conditions for the one-dimensional Schrödinger equation on a bounded interval, both paired with a compact fourth-order interior finite-difference scheme. The D4TBCs are derived from the discrete dispersion relation of the interior scheme and are shown to have fourth-order consistency and to avoid the spurious oscillations seen with the simpler C4TBCs when the potential vanishes. The aDTBCs are obtained by using the exact plane-wave form of the solution outside the device, so they introduce no additional discretization error and can in principle be combined with arbitrarily high-order interior schemes. The authors prove uniqueness for the D4TBC discrete system, state an analogous theorem for the aDTBC system without proof, formulate a fourth-order discretization of the coupled Schrödinger–Poisson problem, and validate the schemes on a resistor and two resonant-tunneling diodes, comparing against NEGF benchmarks. The central numerical claims are plausible, but the uniqueness theorem for the D4TBCs is false at the degenerate point E = V0, which is included in the energy range used in the experiments; this requires a substantive revision.

Significance. If the technical gaps are repaired, the paper would be a useful contribution to high-order quantum-transport simulation: the D4TBC construction is parameter-free and derived in-paper from the discrete dispersion relation, the fourth-order consistency proof is explicit, and the numerical experiments against NEGF benchmarks support the practical value of the schemes. The data availability statement is a further positive feature. However, the current manuscript's headline claim that uniqueness of both discrete Schrödinger systems is "rigorously proved" is undermined by a concrete counterexample at E = V0, and the omitted proof of Theorem 3 leaves the aDTBC system in the same regime unsupported. Because the energy quadrature in Section 4 samples E = 0, the issue is not merely cosmetic; it affects the actual computational procedure.

major comments (3)

[§3.2, Theorem 2] Theorem 2 is false as stated at E = V0. For V ≡ 0 and E = 0, the formulas for α± and β± on page 7 give α = β = 1, so the D4TBCs (38)–(39) reduce to ψ−1 = ψ0 and ψNx+1 = ψNx, while the interior equation (24) reduces to the discrete Laplacian. On a mesh with Nx = 2 every constant vector solves the homogeneous system, so the solution is not unique. The hypothesis t > max{2(E−V0), 2(E−VNx)} is satisfied in this example (it only requires t > 0), so the counterexample lies inside the stated hypotheses. In the proof, g0 and gNx are real and α^{-1} = β^{-1} = 1, so equation (53) becomes identically 0 and cannot imply ψ−1 = ψNx+1 = 0. The theorem should be restricted to E > V0 and E > VNx, or the degenerate case must be analyzed separately. I note that the earlier concern that t is too small is not supported: with Δx = 0.5 nm and m* = 0.067m0 one has t ≈ 27 eV, comfortably above 2(E−V0) for E ≤ 0.8 eV; the actual gap is the equality E = V0.
[§3.3, Theorem 3] Theorem 3 is stated without proof, and the omitted proof cannot be merely "similar" to that of Theorem 2 in the degenerate case. At E = V0 the aDTBCs (55)–(56) give e^{ik1Δx} = e^{ik2Δx} = 1, producing the same constant null vector as in the D4TBC counterexample above. Since the energy integral (11) is truncated to [0, 0.8] eV and evaluated with an adaptive Simpson rule, the endpoint E = 0 lies in the computational range; for vanishing contact potential the discrete system at that energy is singular. The authors should either provide a complete proof covering all admissible E, or explicitly restrict the uniqueness theorem to E > max{V0, VNx} and adjust the numerical quadrature so that no singular energy is sampled.
[§4, Algorithm 1 and density evaluation] The numerical procedure evaluates the density integral (11) on [0, 0.8] eV using the adaptive Simpson routine, which typically samples the endpoints. At E = 0 with V(0) = V(L) = 0, the discrete Schrödinger system with either D4TBCs or aDTBCs is singular, so the linear solve at that energy is not well defined. The paper does not discuss how this energy point is handled. Please state explicitly whether an open quadrature rule, a small positive cutoff, or a limiting argument is used, and confirm numerically that the reported densities and I-V curves are insensitive to that choice.

minor comments (4)

[§3.2, page 7] The nomenclature is potentially confusing: equations (27)–(28) are first introduced as C4TBCs, while equations (38)–(39) are later called D4TBCs. Please clarify at first use that these are two different boundary closures and explain why C4TBCs are abandoned in favor of D4TBCs.
[§3.3, page 14] The notation L = max{l, s} in the discussion of (57)–(58) refers to stencil parameters l and s introduced in (17)–(18), but this connection is not restated; a brief reminder would improve readability.
[Abstract and §3.3] The abstract's phrase "essentially non-oscillating" is stronger than what is proved: the manuscript proves uniqueness for vanishing potential and shows in Fig. 1 that oscillations disappear in that case, while the authors themselves note in §3.3 that it is not clear whether spurious oscillations still exist when V ≠ 0. Please soften the wording accordingly.
[Throughout] There are several grammatical and typographical issues, e.g., "the dispersion relation brought numerical error, so does the D4TBCs consequently" in §3.3, and "V /nequivalence0" in the same section. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the D4TBCs and aDTBCs are derived in-paper from the discrete/exact dispersion relations, and the accuracy claims are checked against external NEGF benchmarks.

full rationale

All load-bearing derivations are carried out in-paper from explicit assumptions. The D4TBCs (Eqs. 38-39) are constructed by substituting the discrete wave ansatz (29) into the interior compact scheme (24), solving the resulting characteristic equation for alpha and beta, and selecting the outgoing wave branches; the consistency order (Theorem 1 and Lemma 1) is then proven by Taylor expansion of the exact TBC together with the discrete dispersion relations (31)-(32), and the non-oscillation claim for V identically zero is supported by a uniqueness/energy argument (Theorem 2). The aDTBCs (Eqs. 55-56) are defined as the exact TBCs written at ghost points via infinite Taylor series, so their 'introduce no discretization error' status is a direct property of the definition rather than a fitted or imported prediction. No fitted parameters appear, and the numerical experiments are benchmarked against second-order NEGF results from the literature. The only flagged items are non-circular: Theorem 3 explicitly omits its proof ('the proof is omitted due to its similarity with the proof of Theorem 2,' Sect. 3.3), and Theorem 2's hypothesis t > max{2(E-V0), 2(E-VNx)} does not cover the equal-energy case E = V0, which is a correctness/rigor gap, not a circular dependency.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard numerical analysis assumptions (smoothness, constant exterior potential, Taylor expansions) plus a stated parameter condition for uniqueness. No new physical entities are postulated. One hand-chosen numerical truncation (energy cutoff) is used in the algorithm but is not part of the boundary condition derivation.

free parameters (1)

Energy integration upper bound E_max = 0.8 eV
Section 4 truncates the energy integral in (11) to [0, 0.8] eV because the Fermi factor is below 1e-10 beyond this value. It is a numerical truncation chosen by hand, not a fitted model parameter.

assumptions (5)

domain assumption The potential V is extended to the left and right of the device as constants: V(x)=V(0) for x<0 and V(x)=V(L) for x>L.
Used in Section 2 to construct plane wave solutions outside the device and to derive TBCs (7)-(8).
domain assumption The wave function ψ(x;k) is analytic near both physical boundaries.
Section 3.3 states this assumption to justify the infinite Taylor series used for the aDTBCs.
domain assumption The energy satisfies E ≥ max{V(0), V(L)} so that k1 and k2 are real.
Section 2 says E ≥ max{V(0), V(L)} is mandatory for the plane wave decomposition.
ad hoc to paper The grid and energy satisfy t > max{2(E-V0), 2(E-VNx)} for the uniqueness theorems.
Theorems 2 and 3 and the non-oscillation claims require this inequality; it may be violated for high energies or fine grids in the practical integration range.
standard math The interior compact scheme (21) is fourth-order accurate, obtained by Taylor series matching.
Section 3.1 derives the scheme using standard Taylor expansion.

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Cite this review

Pith. "Pith review of High Order Finite Difference Schemes for the Transparent Boundary Conditions and Their Applications in the 1D Schr\"odinger-Poisson Problem." pith.science (2026). https://pith.science/paper/H3MATBBN

@misc{pith2026241113175,
  author       = {Pith},
  title        = {Pith review of: High Order Finite Difference Schemes for the Transparent Boundary Conditions and Their Applications in the 1D Schr\"odinger-Poisson Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H3MATBBN}},
  note         = {Machine review of arXiv:2411.13175}
}

read the original abstract

The 1D Schr\"odinger equation closed with the transparent boundary conditions(TBCs) is known as a successful model for describing quantum effects, and is usually considered with a self-consistent Poisson equation in simulating quantum devices. We introduce discrete fourth order transparent boundary conditions(D4TBCs), which have been proven to be essentially non-oscillating when the potential vanishes, and to share the same accuracy order with the finite difference scheme used to discretize the 1D Schr\"odinger equation. Furthermore, a framework of analytic discretization of TBCs(aDTBCs) is proposed, which does not introduce any discretization error, thus is accurate. With the accurate discretizations, one is able to improve the accuracy of the discretization for the 1D Schr\"odinger problem to arbitrarily high levels. As numerical tools, two globally fourth order compact finite difference schemes are proposed for the 1D Schr\"odinger-Poisson problem, involving either of the D4TBCs or the aDTBCs, respectively, and the uniqueness of solutions of both discrete Schr\"odinger problems are rigorously proved. Numerical experiments, including simulations of a resistor and two nanoscale resonant tunneling diodes, verify the accuracy order of the discretization schemes and show potential of the numerical algorithm introduced for the 1D Schr\"odinger-Poisson problem in simulating various quantum devices.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Mathematical concepts of open quantum boun dary conditions

Anton Arnold. Mathematical concepts of open quantum boun dary conditions. Transp. Theory Stat. Phys., 30:561–584, 2001. https://doi.org/10.1081/TT-100105939

work page doi:10.1081/tt-100105939 2001

[2] [2]

On a one-dimensional Schr¨ odinger-Poisson scattering model

Naoufel Ben Abdallah, Pierre Degond, and Peter A Markowic h. On a one-dimensional Schr¨ odinger-Poisson scattering model. Z. angew. Math. Phys. , 48:135–155, 1997. https://doi.org/10.1007/pl00001463. High Order FD Schemes for the TBCs of the 1D Schr¨ odinger-Poisson Problem 25

work page doi:10.1007/pl00001463 1997

[3] [3]

L. L. Chang, L. Esaki, and R. Tsu. Resonant tunneling in sem iconductor double barriers. Appl. Phys. Lett., 24:593–595, 1974. https://doi.org/10.1063/1.1655067

work page doi:10.1063/1.1655067 1974

[4] [4]

Datta, F

S. Datta, F. Assad, and M.S. Lundstrom. The silicon MOSFET from a transmission viewpoint. Su- perlattices Microstruct., 23:771–780, 1998. https://doi.org/10.1006/spmi.1997.0563

work page arXiv 1998

[5] [5]

Nanoscale device modeling: the Green’s fu nction method

Supriyo Datta. Nanoscale device modeling: the Green’s fu nction method. Superlattices Microstruct., 28:253–278, 2000. https://doi.org/10.1006/spmi.2000.0920

arXiv 2000

[6] [6]

Quantum transport i n ultrasmall electronic devices

Supriyo Datta and Michael J McLennan. Quantum transport i n ultrasmall electronic devices. Rep. Prog. Phys., 53:1003, 1990. https://doi.org/10.1088/0034-4885/53/8/001

work page doi:10.1088/0034-4885/53/8/001 1990

[7] [7]

Frensley

William R. Frensley. Wigner-function model of a resonant -tunneling semiconductor device. Phys. Rev. B, 36:1570–1580, 1987. https://doi.org/10.1103/PhysRevB.36.1570

work page doi:10.1103/physrevb.36.1570 1987

[8] [8]

K. L. Jensen and F. A. Buot. Numerical simulation of intrin sic bistability and high-frequency current oscillations in resonant tunneling structures. Phys. Rev. Lett. , 66:1078–1081, 1991. https://doi.org/10.1103/PhysRevLett.66.1078

work page doi:10.1103/physrevlett.66.1078 1991