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arxiv: 1906.10123 · v1 · pith:OFF632RDnew · submitted 2019-06-23 · 🪐 quant-ph

A Schr\"odinger potential involving x^frac{2}{3} and centrifugal-barrier terms conditionally integrable in terms of the confluent hypergeometric functions

V.A. Manukyan , T.A. Ishkhanyan , A.M. Ishkhanyan This is my paper

Pith reviewed 2026-05-25 17:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Schrödinger equationHermite functionsconditionally integrable potentialbound statesbi-confluent Heunpower-law potentialcentrifugal barrier
0
0 comments X

The pith

A Schrödinger potential with an x to the two-thirds term and fixed centrifugal barrier admits exact solutions in non-integer-order Hermite functions.

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

The paper shows that the one-dimensional Schrödinger equation can be solved exactly for a potential combining an attractive term proportional to x to the two-thirds with a repulsive centrifugal barrier term proportional to one over x squared. This exact solution exists only when the centrifugal strength is fixed to a particular value that places the potential inside one of the five bi-confluent Heun families. The general solution is assembled from fundamental pieces that are each an irreducible linear combination of two Hermite functions of non-integer order evaluated at a scaled and shifted argument. The resulting potential forms an infinitely extended confining well on the positive semi-axis and supports infinitely many bound states. A reader cares because exact closed-form solutions remain scarce for fractional-power potentials and provide precise benchmarks for numerical or approximate treatments of singular quantum wells.

Core claim

The solution of the one-dimensional Schrödinger equation for a potential involving an attractive x to the two-thirds and a repulsive centrifugal-barrier term proportional to x to the minus two is presented in terms of the non-integer-order Hermite functions. The potential belongs to one of the five bi-confluent Heun families. This is a conditionally integrable potential in that the strength of the centrifugal-barrier term is fixed. The general solution of the problem is composed using fundamental solutions each of which presents an irreducible linear combination of two Hermite functions of a scaled and shifted argument. The potential presents an infinitely extended confining well defined on

What carries the argument

Non-integer-order Hermite functions of a scaled and shifted argument, assembled into irreducible linear combinations that satisfy the Schrödinger equation under the fixed centrifugal strength.

If this is right

  • The potential supports infinitely many bound states on the positive semi-axis.
  • Exact solutions exist only for one fixed value of the centrifugal term strength.
  • The wave functions are built as linear combinations of two non-integer-order Hermite functions.
  • The potential forms an infinitely extended confining well defined only for positive x.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same conditional-integrability tactic may locate solvable cases inside other Heun families for different fractional powers.
  • The model supplies a concrete test bed for checking how well variational or WKB approximations perform on singular fractional potentials.
  • Extension to time-dependent driving or to effective radial problems in higher dimensions could be checked by substituting the same Hermite building blocks.

Load-bearing premise

The centrifugal barrier strength must be set to one specific value so that the Schrödinger equation reduces to a form solvable by Hermite functions.

What would settle it

Numerical integration of the Schrödinger equation for the potential with the required centrifugal strength should produce bound-state energies and wave functions that match the zeros and linear combinations of the constructed Hermite functions.

Figures

Figures reproduced from arXiv: 1906.10123 by A.M. Ishkhanyan, T.A. Ishkhanyan, V.A. Manukyan.

Figure 2
Figure 2. Figure 2: Solid curves - approximation (11), filled circles - numerical result. Even without appealing to analytical calculations, it is understood that the first term of this equation is small. To see this, we note that the roots of the spectrum equation (10) are close to positive half-integers (see Fig.2). The smallest root 0 a 1/2 of the exact equation (9) (not shown in the Fig.2) does not produce a bound state … view at source ↗
Figure 3
Figure 3. Figure 3: Relative error of the semiclassical result [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Energy eigenvalues for potential (2) ( 1V 1, 0 V  0 , m   1) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Normalized wave functions for the first three bound states ( 1V 1, 0 V  0 , m   1). æ æ æ æ æ æ æ æææ ææ æææ ææ æææ ææ æææææ æææ 5 10 15 20 25 30 n 0.00005 0.00010 0.00015 Relative Error En 5 10 15 x 1 2 3 4 5 6 7 E 2 4 6 8 10 x -0.4 -0.2 0.2 0.4 0.6  x [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Driving optical field configuration. -4 -2 2 4 t -2 -1 1 2 U,dt [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Atom-molecule conversion versus 2  U0 at 0   3, 6,10, 20, 40 (from the left to the right). Filled circles – numerical result, solid line – a root of Eq. (24). 0 5 10 15 20 0.0 l 0.1 0.2 0.3 0.4 0.5 p¶ [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original abstract

The solution of the one-dimensional Schr\"odinger equation for a potential involving an attractive $x^\frac{2}{3}$ and a repulsive centrifugal-barrier $\sim x^{-2}$ terms is presented in terms of the non-integer-order Hermite functions. The potential belongs to one of the five bi-confluent Heun families. This is a conditionally integrable potential in that the strength of the centrifugal-barrier term is fixed. The general solution of the problem is composed using fundamental solutions each of which presents an irreducible linear combination of two Hermite functions of a scaled and shifted argument. The potential presents an infinitely extended confining well defined on the positive semi-axis and sustains infinitely many bound states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to solve the one-dimensional Schrödinger equation for a potential with an attractive x^{2/3} term and a repulsive centrifugal ~x^{-2} barrier, expressing the general solution as irreducible linear combinations of non-integer-order Hermite functions of a scaled and shifted argument. The potential is identified as belonging to one of the five bi-confluent Heun families under a conditional integrability condition that fixes the centrifugal strength; it is asserted to form an infinitely extended confining well on the positive semi-axis that supports infinitely many bound states.

Significance. If the central claims hold, the work contributes an explicit example of conditional integrability within the bi-confluent Heun class, linking it to Hermite-function solutions. This adds a concrete case to the catalog of exactly solvable potentials and may help delineate the boundary between solvable and non-solvable members of the Heun hierarchy.

major comments (2)
  1. [Abstract] Abstract: the description of the x^{2/3} term as 'attractive' (conventionally a negative coefficient) yields V(x) ~ -c x^{2/3} + d/x^2 (c>0) on x>0, so V(x)→-∞ as x→+∞. This directly contradicts the claims of an 'infinitely extended confining well' and 'infinitely many bound states.' The sign must be positive for confinement; the inconsistency is load-bearing for the spectral claims.
  2. [Abstract] Abstract and potential definition: the conditional integrability fixes the centrifugal coefficient, but the manuscript must explicitly state the sign and domain of the x^{2/3} coefficient (and any domain restrictions such as x>0) so that the confining character and the applicability of the Hermite-function basis can be verified.
minor comments (1)
  1. [Abstract] The abstract states that the general solution is composed of fundamental solutions that are linear combinations of two Hermite functions; the main text should include the explicit change of variable and the resulting recurrence or differential equation satisfied by the coefficients to allow direct verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the inconsistency in the abstract's terminology regarding the sign of the x^{2/3} term. We address the comments point by point below and will revise the manuscript to resolve the issues.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the description of the x^{2/3} term as 'attractive' (conventionally a negative coefficient) yields V(x) ~ -c x^{2/3} + d/x^2 (c>0) on x>0, so V(x)→-∞ as x→+∞. This directly contradicts the claims of an 'infinitely extended confining well' and 'infinitely many bound states.' The sign must be positive for confinement; the inconsistency is load-bearing for the spectral claims.

    Authors: We acknowledge the error in terminology. The potential in the manuscript is constructed with a positive coefficient for the x^{2/3} term (V(x) = A x^{2/3} + B x^{-2} with A > 0 for x > 0) to produce the claimed confining behavior as x → +∞. The word 'attractive' was applied incorrectly and will be removed from the abstract. The revised version will explicitly define the potential with the positive sign to support the spectral claims. revision: yes

  2. Referee: [Abstract] Abstract and potential definition: the conditional integrability fixes the centrifugal coefficient, but the manuscript must explicitly state the sign and domain of the x^{2/3} coefficient (and any domain restrictions such as x>0) so that the confining character and the applicability of the Hermite-function basis can be verified.

    Authors: We agree that greater explicitness is required. The revised abstract and introduction will state that the x^{2/3} coefficient is positive, the domain is restricted to x > 0, and the centrifugal strength is fixed by the conditional integrability condition. This clarification will confirm the confining character and the validity of the non-integer Hermite function solutions. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained construction from special functions

full rationale

The paper constructs the general solution of the Schrödinger equation as linear combinations of non-integer-order Hermite functions for a potential placed in the bi-confluent Heun class under a fixed centrifugal strength (the conditional integrability condition). No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional renaming. The classification and solution form are presented as direct outcomes of matching the potential to known Heun families and composing fundamental solutions, without load-bearing self-referential steps or ansatz smuggling. The abstract and description contain no evidence that the bound-state claim or integrability condition is equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the mathematical properties of confluent hypergeometric and Hermite functions (standard) and on the assumption that the barrier coefficient takes one specific value that maps the equation onto the bi-confluent Heun equation (domain assumption). No free parameters beyond that fixed coefficient are mentioned. No new entities are introduced.

axioms (2)
  • domain assumption The Schrödinger equation with the stated potential reduces to the bi-confluent Heun equation when the centrifugal coefficient is fixed to the required value.
    Stated in the abstract as the condition for conditional integrability.
  • standard math Non-integer-order Hermite functions form a fundamental set of solutions for the transformed equation.
    Invoked when composing the general solution from linear combinations of these functions.

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discussion (0)

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Reference graph

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