Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics

F. Erman; O. T. Turgut

Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics

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Pith's one-line read This paper proves that for the reflectionless sech-squared potential the continuum energy eigenfunctions are incomplete by exactly the single bound-state projector, and adding that bound state restores the full completeness relation.

desk verdict A correct, clean, explicitly verified completeness proof for the reflectionless potential; a useful pedagogical addition, not a deep new theorem. read the letter →

arxiv 2411.14941 v1 pith:II4LRFSE submitted 2024-11-22 quant-ph math-phmath.MP

F. Erman , O. T. Turgut This is my paper · ORCID

classification quant-phmath-phmath.MP MSC 81Q1034L4035P10 PACS 03.65.-w03.65.Ge02.30.Hq

keywords reflectionlesspotentialcompletenessrelationenergyeigenfunctionsboundstatescontinuumscatteringsech-squaredcreationandannihilationoperatorseven-oddparitydecomposition

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

This paper establishes, by direct integration, the completeness relation for the reflectionless one-dimensional potential $V(x)=-(\hbar^2\kappa^2/m)\,\mathrm{sech}^2(\kappa x)$. It shows that the normalized continuum eigenstates alone integrate to $\delta(x-y)-\frac{\kappa}{2}\,\mathrm{sech}(\kappa x)\,\mathrm{sech}(\kappa y)$, so they are incomplete by exactly the projector onto the single bound state. Adding $\psi_0(x)=\sqrt{\kappa/2}\,\mathrm{sech}(\kappa x)$ restores the full $\delta(x-y)$ completeness relation. Because the deficiency is rank-one, the bound-state wavefunction can be read off from the continuum states alone, giving a miniature inverse problem in closed form. The same completeness is re-derived using even- and odd-parity eigenstates.

What carries the argument

The machinery is an algebraic factorization of the Hamiltonian using the operators $a=(P-i\hbar\kappa\tanh(\kappa X))/\sqrt{2m}$ and $a^\dagger=(P+i\hbar\kappa\tanh(\kappa X))/\sqrt{2m}$, a direct analog of harmonic-oscillator creation and annihilation operators. The identity $aa^\dagger=H_0+\hbar^2\kappa^2/(2m)$ maps each free-particle plane wave $|k\rangle$ to a continuum eigenstate $a^\dagger|k\rangle$ of $H$ with the same energy, yielding the explicit normalized wavefunctions used in the integrals. The completeness proof then reduces to evaluating Fourier and contour integrals, and the key simplification is that the continuum contribution collapses to a delta function minus a rank-one sech-sech kernel; that kernel is identified with the bound-state projector.

What would settle it

Compute both sides of Eq. (4.9) numerically against smooth compactly supported test functions; any mismatch larger than quadrature error between the continuum-state integral and $\delta(x-y)-\frac{\kappa}{2}\,\mathrm{sech}(\kappa x)\,\mathrm{sech}(\kappa y)$ would show that the normalization or the completeness identity is wrong.

Watch

Extended reading notes

Core claim

The central claim is the identity Eq. (4.9): after normalizing the continuum eigenfunctions as $\psi_k(x)=e^{ikx}(k+i\kappa\tanh\kappa x)/(\sqrt{2\pi}(\kappa+ik))$, their integral over all real $k$ equals $\delta(x-y)-\frac{\kappa}{2}\,\mathrm{sech}(\kappa x)\,\mathrm{sech}(\kappa y)$. Thus the continuum states alone do not resolve the identity; the missing term is precisely the projector onto the bound state $\psi_0(x)=\sqrt{\kappa/2}\,\mathrm{sech}(\kappa x)$. Combining the two gives the full completeness relation $\psi_0^*(x)\psi_0(y)+\int_{-\infty}^{\infty}\psi_k^*(x)\psi_k(y)\,dk=\delta(x-y)$. Because the deficiency is rank-one, the bound-state wavefunction can be recovered from the continuum scattering states alone, and the paper also obtains the same completeness identity by splitting the eigenstates into even and odd parity classes.

Load-bearing premise

The central claim rests on the assumption that the continuum states built from free-particle states by the algebraic raising operator are all the scattering states there are, with no further generalized eigenfunctions or boundary terms to include.

Editorial extensions

If this is right

For the single-bound-state reflectionless well, no spectral decomposition can omit the bound state; the continuum states contribute exactly the delta function minus the bound-state projector.
The bound-state wavefunction is completely determined by the continuum scattering states through the rank-one deficiency, and the paper gives the explicit closed form.
The even/odd parity decomposition yields the same completeness identity, so parity-resolved eigenstates do not change the spectral content.
Correct normalization of scattering states is essential: only with the normalization (4.2) does the integral over $k$ produce the exact missing projector.

Reading between the lines

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

For reflectionless potentials with $N>1$ bound states, the same reasoning should leave a finite-rank deficiency equal to the sum of bound-state projectors, so a generalized version could reconstruct all bound states from continuum data; the paper only treats $N=1$.
The explicit sech-sech defect in Eq. (4.9) could serve as a diagnostic for numerical spectral methods: a calculation on the line that ignores bound states will show exactly this kind of nonlocal missing term.
Because the algebraic construction is formal at the level of unbounded operators, a fully rigorous spectral decomposition would need to justify the intertwining relations on operator domains, a point the paper flags rather than settles.
The same read-the-missing-bound-state-from-the-continuum step used here for the sech-squared well could be tested on other exactly solvable single-bound-state potentials, where analogous rank-one deficiencies are known.

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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

Short take: the paper does exactly what it says. It verifies completeness for the sech^2 potential by direct integral evaluation, and the check is correct. The genuinely new content is the identity (4.9) and its even/odd version (5.12): the normalized continuum states alone give δ(x−y) minus a rank-one projector, so adding the bound state completes the set. That explicit verification was missing from Lekner's treatment, and the derivation is elementary and clean.

The ladder-operator review in Section 2 is standard SUSY QM and fine as review; the valuable part is the careful normalization and the residue-theorem evaluation of the completeness integral. The appendix's Fourier transform of sech^2 is self-contained and useful. The bound-state-from-continuum extraction in Section 6 is a simple but nice illustration of the inverse problem; it works because the missing piece is rank one. Citations to Patil, Brownstein, Lekner, and Jaffe are appropriate and the paper is honest about what is new.

Soft spots: the operator-domain issues are acknowledged in footnote 2 and are not load-bearing. Completeness is proven by direct computation with explicitly defined states, so the proof does not depend on the spectral theorem or on rigorously showing that a† exhausts the continuous spectrum. If a reader demands a proof that the Hamiltonian has no singular continuous spectrum, that is not supplied; but the explicit identity (4.9) plus the bound state gives the resolution of the identity on the orthogonal complement of ψ0, which is what a completeness statement needs. So I see no real hole, only a boundary of the paper's ambitions.

One can quibble that this is a worked example rather than a new theorem, but that is exactly the point: it adds the reflectionless potential to the short list of potentials with explicitly verified completeness. That is useful for teaching and for reference.

Who this is for: instructors, students, and researchers who want an explicit completeness identity for this potential. It deserves a serious referee and would fit a pedagogical journal like Am. J. Phys. I would send it to review without hesitation.

Referee Report

0 major / 5 minor

Summary. The paper studies the one-dimensional reflectionless (sech^2) potential with N=1 and provides an explicit proof that the bound state together with the continuum states forms a complete set. The continuum eigenfunctions are constructed via a supersymmetric/ladder-operator method and normalized using the free-particle delta normalization. The central result is Eq. (4.9), which shows by direct residue and distributional integration that the continuum states alone give δ(x−y) minus the rank-one projector onto the bound state ψ0(x)=√(κ/2)sech(κx). Adding the bound state restores the full completeness relation. The paper also gives an equivalent parity-decomposed proof and shows that the bound state can be recovered from the continuum contribution alone.

Significance. If the result holds, this is a useful pedagogical and reference contribution: it is one of the few exactly solvable one-dimensional systems with both bound and continuum spectra for which completeness is verified explicitly. The key identity (4.9) is a self-contained distributional computation for explicitly defined states, and the appendix checks the delta normalization of the continuum states. The algebraic intertwining argument in Section 2 is formal, as the authors acknowledge in footnote 2, but the explicit completeness verification does not depend on a rigorous domain theory: once the normalized states are written down, Eq. (4.9) directly provides the resolution of the identity on the orthogonal complement of the bound state. The recovery of the bound state from the continuum is a clean illustration of the inverse problem idea. The paper does not present a new spectral theorem, but that is not claimed; the value lies in the explicit, checkable verification.

minor comments (5)

[Appendix, Eq. (7.5)] The Fourier relation is stated as I1 = -ik I2, but with the conventions F(f)(k)=∫f(x)e^{-ikx}dx and I1(k)=∫sech^2(κx)e^{ikx}dx, the correct relation is I1(k) = -(ik/κ) I2(k). The final normalization (7.7) is nevertheless correct because it follows independently from the operator calculation in Eq. (4.1), but the appendix should be corrected for internal consistency.
[Title and abstract] The treatment in Section 2 is explicitly restricted to N=1, but the title and abstract refer to 'the reflectionless potential' in the singular; the N=1 restriction should be stated in the title or abstract to avoid over-generalizing the claim.
[Section 5, Eqs. (5.4)-(5.9)] The calculation of the momentum matrix element ⟨ψ^o_{k'}|P|ψ^e_k⟩ is not used in the completeness proof and interrupts the main argument; consider moving it to an appendix or condensing it to a brief remark.
[Appendix, Eq. (7.4)] The displayed expression '2πik iκ 2' should be typeset as 2πik/(iκ^2) (or an equivalent form); the current typesetting is ambiguous.
[Section 6] The statement that the bound state wave function is 'uniquely determined' from Eq. (6.1) should be qualified by noting that the overall phase is arbitrary.

Circularity Check

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No circularity: the completeness identity is derived by direct integral evaluation from explicit eigenfunctions, not assumed or fitted.

full rationale

The paper's central result, Eq. (4.9), is obtained by substituting the explicitly normalized continuum states (4.2) into the integral and evaluating the resulting trigonometric integrals by standard contour methods. This is a self-contained distributional identity: the left side is computed directly, and the right side is the delta function minus the rank-one sech kernel. The bound state wavefunction in Eq. (2.7) is found independently by solving the first-order condition a|0>=0, and later re-derived from the missing projector term in Eq. (6.1), which is an independent consistency check rather than a circular input. The algebraic intertwining argument in Section 2 is used to motivate the form of the continuum states, but the completeness claim itself does not rest on that argument's domain rigor: once the states are written down, one can check H psi_k = (hbar^2 k^2/2m) psi_k directly, and the appendix checks delta-normalization. The acknowledged formal domain issues in footnote 2 are limitations of presentation, not circular reasoning. There are no self-citations used as load-bearing evidence, and no fitted parameter is renamed as a prediction. The derivation is therefore self-contained against the explicit eigenfunctions and general spectral-theoretic benchmarks, and no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free parameters (κ is the potential's scale input), no new physical entities, and no fitted constants. The central claims rest only on standard functional analysis (spectral theorem), the standard SUSY quantum mechanics factorization, and standard contour integration and distribution theory.

assumptions (4)

standard math The spectral theorem for self-adjoint operators ensures the generalized eigenfunctions form a complete set (Eq. 3.5).
Invoked in Section 3 as the general framework; the paper then verifies it explicitly for this potential.
domain assumption The Hamiltonian H = -ℏ²/2m d²/dx² - (ℏ²κ²/m) sech²(κx) is essentially self-adjoint on a suitable domain.
Needed for the spectral decomposition; the paper defers to Ref. [24] and states in footnote 2 that domain issues are not treated.
domain assumption The intertwining relations a and a† map eigenstates of H0 to eigenstates of H, and the continuous spectra coincide except for the bound state (Eqs. 2.10-2.11).
This is the standard SUSY QM argument used to construct the continuum states; it is stated formally without full operator-domain proof.
standard math The Fourier transform of tanh(κx) exists as a tempered distribution and satisfies F(tanh') = ik F(tanh) (Eq. 7.5).
Used in the appendix to compute the orthonormality integral; the paper notes the integral converges only in the principal value sense.

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Pith. "Pith review of Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics." pith.science (2026). https://pith.science/paper/II4LRFSE

@misc{pith2026241114941,
  author       = {Pith},
  title        = {Pith review of: Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/II4LRFSE}},
  note         = {Machine review of arXiv:2411.14941}
}

read the original abstract

There are few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.

Figures

Figures reproduced from arXiv: 2411.14941 by the authors.

**Figure 1.** The reflectionless potential V (x) = −(~ 2κ 2/m) sech2 κx (blue curve) and the bound state wave function ψ0(x) (red curve) and the corresponding energy E0 is shown for κ = 1 in the units ~ = 2m = 1. Notice that the wave function ψ0 has a different unit and is only shown for comparison. function and it has no nodes, as expected. The corresponding bound state energy E0 = −~ 2κ 2/2m can be found by substituting this so… view at source ↗

**Figure 2.** The choice of the closed contour C for the integral I1. where the last term is the result of the coinciding limit of the vertical lines along the imaginary axis so that we end up with integrals over clockwise oriented N small circles. After taking the limit of the above equation as N → ∞ (which is equivalent to RN → ∞) and applying Jordan’s lemma, the second integral goes to zero so that we find I1 = 2πik iκ2 X∞ n=0… view at source ↗

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Pith tools

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

[1] [1]

L. D. Faddeev, O. A. Yakubovski ˘ ı,Lectures on Quantum Mechanics for Mathematics Students , American Mathematical Society (2000), p. 80

work page 2000

[2] [2]

Bohm, Quantum Mechanics, Foundations and Applications , third edition, Springer (2001), pp

A. Bohm, Quantum Mechanics, Foundations and Applications , third edition, Springer (2001), pp. 8-18

work page 2001

[3] [3]

Appel, Mathematics for Physics and Physicists , Princeton University Press, Princeton (2007), pp

W. Appel, Mathematics for Physics and Physicists , Princeton University Press, Princeton (2007), pp. 379-385

work page 2007

[4] [4]

D. J. Griﬃths, D. F. Schroeter, Introduction to Quantum Mechanics , third edition (2018), p. 51 for completeness relation, and Problem 2.52 for the discussion of the re ﬂectionless potential

work page 2018

[5] [5]

Gasiorowicz, Quantum Physics , John Wiley and Sons, (2007), p

S. Gasiorowicz, Quantum Physics , John Wiley and Sons, (2007), p. 112

work page 2007

[6] [6]

Completeness of the Energy Eigenfunctions for the One-dimensionalδ-Function Potential

S. H. Patil, “Completeness of the Energy Eigenfunctions for the One-dimensionalδ-Function Potential”, Am. J. Phys. 68, 712-714 (2000)

work page 2000

[7] [7]

Completeness of the Coulomb Wave Functions in Qua ntum Mechanics

N. Mukunda, “Completeness of the Coulomb Wave Functions in Qua ntum Mechanics”, Am. J. Phys. 46, 910-913 (1978)

work page 1978

[8] [8]

Calculation of a Bound State Wavefunction Us ing Free State Wavefunctions Only

K. R. Brownstein, “Calculation of a Bound State Wavefunction Us ing Free State Wavefunctions Only”, Am. J. Phys. 43, 173–176 (1975)

work page 1975