Series Potential Expansion Method (PSEM)

• Series Potential Expansion Method (PSEM) is a family of strategies that expand physical observables using a solvable reference, series truncation, and explicit coefficient extraction.
• The method is applied across diverse fields—such as fluid dynamics with beta-scaling, gravitational modeling via Legendre expansions, and vacuum polarization using Maclaurin series—to obtain analytic approximations.
• PSEM unifies disparate series techniques while requiring domain-specific resummation or truncation strategies to ensure convergence, physical accuracy, and computational efficiency.

Searching arXiv for papers on "Series Potential Expansion Method" and the cited arXiv IDs to ground the article. “Series Potential Expansion Method” (PSEM) is not used in the surveyed arXiv literature as a single, universally standardized formalism. Instead, the term appears across several technically distinct settings: Ramana’s thermodynamic perturbation theory of fluids, where the coupling-parameter series expansion is shown to be exactly equivalent to the high-temperature series expansion for pairwise-additive interactions; Mota et al.’s analytical gravity model for irregular celestial bodies, where the gravitational potential of a homogeneous polyhedral body is expanded in Legendre contributions over tetrahedral elements; Martines’s exact Maclaurin expansion of the Källén–Sabry vacuum polarization potential; Tong’s equation-of-motion expansion of double-time Green’s functions; and Inyang et al.’s series solutions of radial Schrödinger equations for heavy-meson spectroscopy (Ramana, 2014, Mota et al., 25 Jul 2025, Martines, 2020, Tong, 2015, Inyang et al., 2022, Inyang et al., 2023). The available literature therefore suggests that PSEM denotes a family of series-construction strategies organized around a solvable reference, a truncation order, and explicit coefficient extraction, rather than one canonical algorithm.

1. Terminological scope and disciplinary usage

The surveyed uses of PSEM are heterogeneous in both object and purpose. In each case, however, the target quantity is rewritten as a structured series whose coefficients can be computed from a reference formulation, an analytic kernel decomposition, or a polynomialized potential.

Domain	Expanded quantity	Characteristic feature
Thermodynamic perturbation theory of fluids	Helmholtz free energy, radial distribution function, direct correlation function	Exact equivalence CPSE↔HTSE and $\beta$-scaling for hard-sphere reference
Gravity field of irregular bodies	Gravitational potential and acceleration of a homogeneous polyhedron	Legendre expansion over tetrahedra with analytic gradients
Vacuum polarization	Källén–Sabry potential	Exact Maclaurin series from $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$
Double-time Green’s functions	Retarded Green’s function for $H=H_0+\lambda H_1$	EOM recursion with continued-fraction resummation
Heavy-meson spectroscopy	Radial Schrödinger solutions and bound-state masses	Frobenius-type series plus termination/quantization condition

This distribution suggests a methodological family resemblance: PSEM is repeatedly used where one isolates a tractable core problem, introduces an expansion parameter or local series variable, and reconstructs the full observable to finite order. The details, convergence properties, and physical interpretation are domain-specific.

2. Ramana’s fluid-theory formulation

In Ramana’s formulation for fluids, one considers a one-component system with pair potential

$u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$

and introduces a coupling parameter $\lambda\in[0,1]$ through

$$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$

The exact Helmholtz free energy is

$F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$

with

$$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$

Expanding either $\langle V_{\mathrm{per}}\rangle_\lambda$ or $F(\lambda)$ about $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$0 yields the coupling-parameter series expansion (CPSE) (Ramana, 2014).

The central result is the order-by-order equivalence of this CPSE to the high-temperature series expansion (HTSE, or Zwanzig expansion),

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$1

where $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$2 are cumulants evaluated in the reference ensemble. Ramana shows that the CPSE term

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$3

satisfies

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$4

so every CPSE term coincides exactly with the corresponding HTSE term for pairwise-additive interactions (Ramana, 2014).

The same framework extends to structural quantities. Ramana expands

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$5

and shows that the derivatives $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$6 and $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$7 are built from the same cumulant structure. When the reference system is a hard-sphere fluid of diameter $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$8, the reference averages $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$9 are temperature-independent, so the perturbative coefficients obey explicit $H=H_0+\lambda H_1$0-power laws:

$H=H_0+\lambda H_1$1

This is the paper’s “$H=H_0+\lambda H_1$2-scaling” result.

Ramana illustrates the scaling law with the square-well fluid,

$H=H_0+\lambda H_1$3

For square-well widths $H=H_0+\lambda H_1$4, $H=H_0+\lambda H_1$5 and $H=H_0+\lambda H_1$6, the reported result is that plots of $H=H_0+\lambda H_1$7 at different temperatures collapse onto a single master curve, and the simulation data for $H=H_0+\lambda H_1$8, $H=H_0+\lambda H_1$9, and $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$0 coincide with the scaled CPSE predictions (Ramana, 2014). In this fluid-mechanical meaning, PSEM is therefore not a merely heuristic truncation scheme; it is a reorganization of the exact HTSE for pairwise-additive potentials.

3. PSEM in gravitational modeling of irregular celestial bodies

Mota et al. use PSEM in a different sense: an analytical approximation to the gravitational field of a homogeneous, irregularly shaped body decomposed into tetrahedral elements. A body $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$1 of constant density $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$2 is partitioned into tetrahedra $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$3, and for a field point $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$4 and source point $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$5 one writes

$u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$6

with $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$7, $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$8, and $u(r)=u_{\mathrm{ref}}(r)+u_{\mathrm{per}}(r),$9. For $\lambda\in[0,1]$0, Kellogg’s generating function gives

$\lambda\in[0,1]$1

hence

$\lambda\in[0,1]$2

The tetrahedral contribution is then truncated at order $\lambda\in[0,1]$3 and integrated term by term (Mota et al., 25 Jul 2025).

After mapping each tetrahedron to a reference right-tetrahedron and using the closed-form monomial integral

$\lambda\in[0,1]$4

the method assembles the full potential as

$\lambda\in[0,1]$5

where $\lambda\in[0,1]$6 is the Kepler term and $\lambda\in[0,1]$7 is the non-central perturbation. Because the resulting expression is analytic, the acceleration follows directly from

$\lambda\in[0,1]$8

Mota et al. emphasize that the integrands remain homogeneous polynomials and can therefore be handled symbolically (Mota et al., 25 Jul 2025).

The paper’s computational claim is not higher intrinsic accuracy than the classical polyhedral approach, but markedly lower execution time at sufficiently high order. For the evaluation of $\lambda\in[0,1]$9 on a uniform grid of $$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$0 points outside the body on a Pentium 3.6 GHz, the reported CPU times are: for asteroid (87) Sylvia, 31 min 34.53 s for the classical polyhedral method, 1 min 52.60 s for TC20, and 0 min 16.98 s for PSEM order 14; for Bennu, 42 min 02.00 s, 0 min 30.18 s, and 0 min 03.19 s for PSEM order 11; for Itokawa, 475 min 36 s, 27 min 29 s, and 0 min 32 s for PSEM order 9; and for Apophis, 166 min 54.5 s, 7 min 43.8 s, and 0 min 15.55 s for PSEM order 10 (Mota et al., 25 Jul 2025).

Accuracy is expressed through

$$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$1

The reported benchmark is that outside the Brillouin sphere, PSEM orders 11 or 12 achieve $$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$2 for all four asteroids; the paper further lists $$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$3 for Sylvia at order 12, $$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$4 for Bennu at order 11, $$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$5 for Itokawa already at order 9, and $$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$6 for Apophis at order 10 (Mota et al., 25 Jul 2025). The same analytic representation is then used for equilibrium points, linear stability via the Hessian of the effective potential, zero-velocity surfaces, and halo-orbit trajectory planning; near Apophis, the reported trajectory deviation from the Tsoulis model is less than 100 m at $$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$7 km, with integration time 2.7 s versus 20 min (Mota et al., 25 Jul 2025).

4. Exact power-series expansion in vacuum polarization

In Martines’s treatment of the Källén–Sabry vacuum polarization potential, PSEM denotes a constructive analytic procedure for converting an integral representation into an exact Maclaurin series. The starting form is

$$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$8

where $$V_\lambda(\{r_i\})=V_{\mathrm{ref}}(\{r_i\})+\lambda V_{\mathrm{per}}(\{r_i\}).$$9 is given by a double integral with exponential kernel. The method proceeds by rewriting $F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$0, after a change of variables such as $F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$1, as a finite sum

$F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$2

where $F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$3 are elementary polynomials and $F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$4 are modified Bessel functions or Bickley–Naylor–type integrals. One then derives exact power-series expansions of the $F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$5, multiplies by the $F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$6, and collects like powers of $F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$7 (Martines, 2020).

For the Källén–Sabry case specifically,

$F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$8

and the final result takes the compact form

$F(\lambda)\equiv-k_BT\ln Z_N(\lambda),$9

Consequently,

$$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$0

The coefficients are obtained by discrete convolution of the polynomial coefficients of the $$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$1 with the series coefficients of the $$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$2 (Martines, 2020).

The convergence statement is explicit: the Maclaurin expansion converges for $$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$3, equivalently $$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$4, because the nearest singularity in the $$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$5-plane lies at $$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$6. For $$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$7, Martines gives an asymptotic expansion,

$$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$8

or alternatively a rapidly convergent one-dimensional integral representation for numerical work (Martines, 2020). The same three-step construction is stated to extend to the Uehling potential, Wichmann–Kroll, and higher-order vacuum-polarization corrections.

5. Equation-of-motion series expansion of double-time Green’s functions

Tong’s framework applies a formally similar expansion logic to retarded double-time Green’s functions rather than to an interaction potential. One splits the Hamiltonian as

$$F(1)=F_{\mathrm{ref}}+\int_0^1 d\lambda\,\langle V_{\mathrm{per}}\rangle_\lambda.$$9

with $\langle V_{\mathrm{per}}\rangle_\lambda$0 exactly solvable and $\langle V_{\mathrm{per}}\rangle_\lambda$1 treated perturbatively. The retarded Green’s function

$\langle V_{\mathrm{per}}\rangle_\lambda$2

satisfies the equation of motion

$\langle V_{\mathrm{per}}\rangle_\lambda$3

Expanding in powers of $\langle V_{\mathrm{per}}\rangle_\lambda$4,

$\langle V_{\mathrm{per}}\rangle_\lambda$5

one obtains recursive relations for the $\langle V_{\mathrm{per}}\rangle_\lambda$6 (Tong, 2015).

Finite-order truncation creates a central pathology: the truncated series develops unphysical high-order poles and violates causality. Tong therefore introduces continued-fraction resummation,

$\langle V_{\mathrm{per}}\rangle_\lambda$7

and fixes the continued-fraction parameters by matching the truncated series in $\langle V_{\mathrm{per}}\rangle_\lambda$8. If $\langle V_{\mathrm{per}}\rangle_\lambda$9 and $F(\lambda)$0 are real, the representation has only simple real poles and restores the correct analytical structure (Tong, 2015).

A second pathology is zero-temperature divergence. A direct Taylor expansion of the Lehmann representation produces factors $F(\lambda)$1 in $F(\lambda)$2, so Tong replaces order-by-order averages by averages taken with respect to the full Hamiltonian $F(\lambda)$3. In the self-consistent version, the renormalized zeroth-order Green’s function is defined using full-$F(\lambda)$4 expectation values, and the needed averages are obtained only once from the continued-fraction-resummed full Green’s function through the spectral theorem. The stated consequence is that all spurious $F(\lambda)$5 divergences drop out (Tong, 2015).

The Anderson impurity model is the paper’s worked example. Tong develops both weak-coupling expansion to order $F(\lambda)$6 around a Hartree–Fock–shifted $F(\lambda)$7 and strong-coupling expansion to order $F(\lambda)$8 around the atomic limit, using Hubbard operators in the latter case. The reported numerical outcome is that bare self-energy resummation can violate sum rules and even give negative spectral weight, bare continued-fraction resummation restores causality but develops unphysical $F(\lambda)$9 Hubbard-peak shifts as $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$00, whereas the self-consistent continued-fraction scheme is free of both pathologies and yields Hubbard peaks at $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$01 with correct weight redistribution for $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$02 (Tong, 2015).

6. Schrödinger-series implementations for heavy-meson spectroscopy

Inyang et al. use “series expansion method” for analytic solutions of the radial Schrödinger equation after a local expansion of the interaction potential. In the Hulthén–Hellmann application, the temperature-dependent potential is

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$03

which is expanded for small $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$04 up to $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$05 and rearranged as

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$06

The reduced radial wavefunction is written as

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$07

with

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$08

and the substituted equation is organized as a sum of powers $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$09, $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$10. Matching coefficients yields five algebraic relations, a three-term recurrence, and a termination condition $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$11 that quantizes the spectrum (Inyang et al., 2022).

The paper states that the resulting algebra gives a closed-form expression for the energy eigenvalues, Eq. (29), and that four special limits are recovered by setting appropriate couplings to zero: Hellmann, Yukawa, Coulomb, and Hulthén. Meson masses are then obtained from

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$12

Using the optimized parameters reported in the paper, the authors state that the present potential provides satisfying results in comparison with experimental data and the work of other researchers with a maximum error of 0.034 GeV (Inyang et al., 2022).

The related quarkonium application uses the temperature-dependent potential

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$13

expands the exponential to third order, and rewrites the result as a cubic polynomial

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$14

With the same ansatz

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$15

the coefficient-matching conditions give

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$16

and the quantization condition leads to

$I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$17

followed by the paper’s explicit Eq. (27) for $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$18 after undoing definitions (Inyang et al., 2023).

For spectroscopy, the paper reports parameter sets for charmonium and bottomonium and compares predicted masses with PDG 2018 values. The abstract states that the method provides satisfying results with a maximum error of 0.058 GeV. The detailed summary supplied here also notes, however, that the displayed charmonium table contains entries with $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$19 up to 0.117 GeV, while simultaneously stating “overall max quoted in paper 0.058 GeV via $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$20 fit” (Inyang et al., 2023). That reporting nuance is important: it indicates that the numerical assessment depends on which summary statistic is being cited.

7. Common structure, non-equivalence, and recurrent caveats

Across these usages, the surveyed literature suggests a shared procedural template: choose a tractable reference problem or local representation, introduce a formal expansion variable or series point, derive coefficients recursively or by analytic decomposition, and truncate at order $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$21 or $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$22 to reconstruct the target observable. A plausible implication is that “PSEM” functions as a cross-domain label for explicit series engineering rather than as a field-independent theorem or algorithm (Ramana, 2014, Mota et al., 25 Jul 2025, Martines, 2020, Tong, 2015).

The same survey also shows that the status of the approximation differs sharply by domain. In Ramana’s fluid theory, CPSE and HTSE are exactly equivalent for pairwise-additive interactions, so the method has a rigorous perturbative interpretation; when a hard-sphere reference is used, the resulting coefficients obey simple $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$23-power laws (Ramana, 2014). In Mota et al.’s gravity model, by contrast, the paper explicitly states that the method does not offer higher accuracy than the classical polyhedral approach, and the quoted $I_{KS}(x)=\sum_{i=1}^{17}P_i(x)f_i(x)$24 accuracy is tied to sufficiently high order and to field points outside the Brillouin sphere (Mota et al., 25 Jul 2025). In Tong’s Green’s-function framework, naive truncation violates causality and generates zero-temperature divergences, so continued-fraction resummation and self-consistency are structural necessities rather than optional refinements (Tong, 2015). In the heavy-meson papers, the method depends on a local polynomialization of the interaction and on termination of the Frobenius-type series; this suggests that the reported spectra are inseparable from the chosen truncation of the potential itself (Inyang et al., 2022, Inyang et al., 2023).

A recurrent misconception would therefore be to treat all invocations of PSEM as mathematically interchangeable. The literature instead supports a narrower conclusion: the name is shared, but the underlying objects being expanded—free energies, pair correlation functions, Newtonian kernels, vacuum-polarization integrals, Green’s functions, and bound-state wavefunctions—belong to different analytical regimes with different notions of convergence, accuracy, and physical control.