Generalised Feshbach–Villars Transformation

• Generalised Feshbach–Villars Transformation is a framework that recasts second-order relativistic wave equations into first-order Hamiltonians for improved spectral analysis.
• It employs multicomponent field representations and Coulomb–Sturmian basis expansions to handle arbitrary potentials and curved space–time geometries efficiently.
• The method leverages matrix continued fractions and rigorously defined boundary conditions to achieve high-precision numerical and analytical eigenvalue solutions.

The Generalised Feshbach–Villars Transformation (GFVT) is an advanced mathematical formalism that extends the classic Feshbach–Villars linearization of the Klein–Gordon equation to more general quantum-relativistic systems, including particles in curved space–time, with arbitrary coupling structures, and in the presence of external potentials. GFVT is central in reformulating higher-order relativistic wave equations—naturally second order in time—into first-order, Hamiltonian-like forms using multicomponent field representations. This approach broadens the applicability of Feshbach–Villars methods, facilitating both numerical and analytic studies of relativistic bound and resonance spectra, including field-theoretic situations involving nontrivial space–time geometries and boundary conditions.

1. Algebraic Structure and the Generalised Transformation

In the original Feshbach–Villars (FV) scheme for spin-0 fields, the wave equation is recast as a first-order system by splitting the field into two components, typically denoted ϕ and χ, forming a two-component spinor-like wavefunction: $$|\psi\rangle = \begin{pmatrix} \phi \ \chi \end{pmatrix}$$ The time evolution is governed by a matrix Hamiltonian,

$$i\hbar \frac{\partial}{\partial t} |\psi\rangle = H_{\mathrm{FV}} |\psi\rangle$$

where the Hamiltonian generically includes kinetic, mass, and potential terms multiplied by constant 2×2 matrices, e.g.: $H_{\mathrm{FV}} = A \frac{p^2}{2m} + B m c^2 + C V$ with $A, B, C$ representing channel couplings.

GFVT generalizes this structure to accommodate arbitrary field content, curved space–time, and complex interactions. The transformation allows multi-component fields (beyond two components if needed), and redistributes potential and kinetic energy terms among diagonal and off-diagonal blocks of the extended Hamiltonian, leading to a block-matrix (possibly block-tridiagonal) structure. The defining relation,

$$\psi = \varphi_1 + \varphi_2, \quad i\tilde{\mathcal{D}}\psi = \mathcal{N}(\varphi_1 - \varphi_2)$$

involves a nontrivial operator $\tilde{\mathcal{D}}$ incorporating, for curved background: $$\mathcal{Y} = \frac{1}{2} \{ \partial_i, \sqrt{-g}\, (g^{0i}/g^{00}) \}$$ thus introducing direct metric-coupling effects to the kinetic sector. At the algebraic level, GFVT unifies modifications due to external fields, geometry, and unconventional kinetic couplings, preserving the “first-order-in-time” feature even in the most general backgrounds (Bouzenada et al., 2023, Bouzenada et al., 2023, Bouzenada et al., 13 Apr 2024, Garah et al., 15 Sep 2025).

2. Basis Representation: Coulomb–Sturmian Functions and Separable Expansion

To efficiently represent and solve the resulting multi-component Hamiltonian, the GFVT formalism typically employs a discrete basis, most effectively the Coulomb–Sturmian (CS) functions: $$\langle r | n \rangle = \sqrt{\frac{n!}{(n+2l+1)!}}\, e^{-br} (2br)^{l+1} L_n^{2l+1}(2br)$$ Biorthonormality and analytic expressions for operator matrix elements make CS functions well-suited for representing both long-range (Coulombic, confining) and short-range potentials.

In the CS basis, key Hamiltonian sectors, such as the kinetic term and Coulomb (or analogous long-range) potentials, become tridiagonal or block-tridiagonal. The central object for solution is the operator: $J = E - \frac{p^2}{2m} - v_C$ For GFVT, $J$ inherits the block-matrix character from the transformation. Generic short-range components $v^{(s)}$ are approximated by low-rank separable expansions,

$$v^{(s)} \approx \sum_{i,j=1}^N |u_i\rangle v_{ij} \langle u_j|$$

so that the full eigenvalue problem reduces to an algebraic condition involving finite (block-)matrices (Brown et al., 2015, Motamedi et al., 2019).

3. Matrix Continued Fraction Method

The cornerstone of the GFVT's computational machinery is the matrix continued fraction, designed to invert infinite block-tridiagonal matrices inherent in the CS-basis-represented Green's operator,

$g_{\rm GFVT}^C(E) = [E - H_{\rm GFVT}^C]^{-1}$

For an $N$-dimensional truncation, the inversion is: $$g^\mathrm{C}_{\underline{N}} = \big(J_{\underline{N}} - \delta_{iN} J_{N,N+1} C_{N+1} J_{N+1,N}\big)^{-1}$$ with the continued fraction recursion,

$$C_{n+1} = \left(J_{n+1,n+1} - J_{n+1, n+2} C_{n+2} J_{n+2, n+1}\right)^{-1}$$

All objects are block matrices whose size is determined by the GFVT's field structure.

This formalism “sums” the contribution of the infinite basis tail exactly, ensuring that even small basis truncations yield highly accurate eigenvalues, found as roots of the determinant condition: $$\big| [g_{\rm GFVT}^C]^{-1} - v_{\rm GFVT}^{(s)} \big| = 0$$ For resonant phenomena, analytic continuation of the matrix continued fraction to the complex energy plane is possible by analyzing the asymptotic behavior and deriving corresponding recursion closure conditions (Motamedi et al., 2019).

4. Boundary Conditions, Singular Matrices, and Self-Adjointness

An intrinsic feature of FV-type Hamiltonians is the appearance of a singular matrix in the kinetic term, such as $(f_3 + i f_2)$ in 1D, which satisfies $(f_3 + i f_2)^2 = 0$. The correct physical boundary conditions for GFVT must be imposed not directly on the (multi-)component wavefunction and its derivatives, but on their images under the action of these singular matrices: $(f_3 + i f_2) Y(x),\quad (f_3 + i f_2) Y_x(x)$ where $Y(x)$ is the GFVT wavefunction. The set of pseudo self-adjoint boundary conditions is fully parameterized (e.g., by four real parameters for the 1D box), and must maintain the vanishing of surface terms in the pseudo-inner product, which is required for the pseudo self-adjointness (in the Krein space sense) of the GFVT Hamiltonian (Vincenzo, 2023).

The mapping between boundary conditions in the one-component (KFG) and GFVT (two-component) representations involves applying the inverse of the transformation relations,

$$\psi = \varphi_1 + \varphi_2;\quad \frac{1}{mc^2}(i\hbar \partial_t - V)\psi = \varphi_1 - \varphi_2$$

ensuring the consistency of initial and boundary value problems.

5. Applications in Curved Space–Time and Topological Defect Backgrounds

GFVT is particularly powerful when applied to quantum systems in non-Euclidean geometries. Notable examples include:

• Rotating and Dislocated Cosmic Strings: In metrics with angular deficits and off-diagonal terms (e.g., $$ds^2 = dt^2 + 2a dt d\varphi - dr^2 - (\alpha^2 r^2 - a^2) d\varphi^2$$), GFVT introduces corrections to kinetic operators and additional terms to the Hamiltonian that depend explicitly on geometric and rotational parameters. The resulting radial equations are typically of Bessel, Whittaker, or confluent hypergeometric type, with quantization conditions explicitly shifted by topological defects (deficit angle $\alpha$, dislocation parameter $x$, rotation parameter $a$) (Bouzenada et al., 2023, Bouzenada et al., 2023, Bouzenada et al., 2023, Bouzenada et al., 2023).
• Cosmological Backgrounds: In the Bonnor–Melvin cosmological model, the transformed radial equation is reducible to an associated Legendre polynomial form, with quantization connected to the curvature and topology (cosmological constant $\Lambda$, angular deficit $\sigma$) (Bouzenada et al., 13 Apr 2024).
• Black Hole (Schwarzschild) Space–Times: In Schwarzschild and related coordinate systems (e.g., Painlevé–Gullstrand), the GFVT renders the Klein–Gordon equation into a two-component system whose radial equation admits both hydrogen-like and, upon potential modification, oscillator-like solutions. The energy eigenvalues (for s-wave in the far–field) are

$E_n^+ = m \sqrt{1 - (GM m)^2/(n+1)^2}$

and, when a relativistic harmonic oscillator is included, the spectrum becomes independent of the gravitational field except through selection rules, with wavefunctions involving confluent or biconfluent Heun functions (Garah et al., 15 Sep 2025).

6. Extensions to Spin-½ Fields and Curved Space–Time

GFVT generalizes to relativistic spin-½ systems by squaring the Dirac equation and deriving a generalized Feshbach–Villars system in curved backgrounds. The resulting Hamiltonian involves both the metric and spin connection: $$H = -i \gamma^0 \gamma^i D_i + m \gamma^0 - \omega_0$$ The transformation decomposes the Dirac spinor into two components (ϕ, χ), separating positive and negative energy sectors, while coupling both to gravitational and electromagnetic background fields: $$H = \sigma_z Y + m \sigma_x - \omega_{ab} \gamma^{ab}$$ This framework accommodates (1+2)– and (1+3)–dimensional spacetimes, and the interaction of spin with external curvature and field strengths is represented via explicit spin–field coupling terms (Boumali, 1 Apr 2025, Wingard et al., 26 Aug 2024).

7. Numerical and Analytical Solution Techniques

GFVT, especially when combined with CS basis expansions and the matrix continued fraction method, enables highly accurate numerical solution of spectra, as well as analytic paper of boundary and resonance phenomena. For generic systems the workflow is:

• Decompose the system into a multi-component first-order time evolution form via GFVT.
• Expand all relevant operators (kinetic, potential) in a CS basis, ensuring analytical tractability for matrix elements.
• Use a matrix continued fraction to invert the block-tridiagonal Green's operator, summing the infinite tail exactly and efficiently.
• Reduce the problem to a finite-dimensional determinant condition for the energy eigenvalues.
• Analytically continue the continued fraction if resonant (complex) energies are required.

Accuracy to $\sim$10 digits can be achieved for spectra if the block matrix structure is fully incorporated (Brown et al., 2015, Motamedi et al., 2019).

Table: Core Mathematical Elements in GFVT Applications

Element	Description	Typical Formula or Structure
GVFT Hamiltonian	Block-matrix with kinetic, potential, mass terms	$H_{\rm GFVT} = \sum_i A_i O_i$
Basis Expansion	Coulomb–Sturmian (tridiagonal/block-tridiagonal)	$$\langle r | n \rangle = \sqrt{n!/(n+2l+1)!} e^{-br}\dots$$
Green's Operator	Inverse via matrix continued fraction	$g_C^\underline{N} = [\cdots C_{N+1} \cdots]^{-1}$
Eigenvalue Condition	Determinant root for spectrum	$$\left| [g_{\rm GFVT}^C]^{-1} - v_{\rm GFVT}^{(s)} \right| = 0$$
Boundary Condition	Weighted by singular matrices in component space	$(f_3 + i f_2) Y(a), \; (f_3 + i f_2) Y_x(a)$

Summary

The Generalised Feshbach–Villars Transformation provides a robust, unified framework for the paper of relativistic wave equations (spin-0 and spin-½) in a broad class of physical settings, including external fields, arbitrary potentials, and nontrivial geometries. The methodology leverages discrete basis representations and matrix continued fractions for both computational efficiency and analytic tractability. It rigorously incorporates boundary condition subtleties arising from singular structures in the kinetic sector and ensures self-adjointness in the domain of the Hamiltonian. Applications demonstrate that GFVT is capable of capturing fine relativistic effects, producing high-precision bound and resonance spectra, and elucidating the influence of geometry and topology on quantum dynamics (Brown et al., 2015, Motamedi et al., 2019, Vincenzo, 2023, Bouzenada et al., 2023, Bouzenada et al., 2023, Bouzenada et al., 2023, Bouzenada et al., 2023, Bouzenada et al., 13 Apr 2024, Wingard et al., 26 Aug 2024, Boumali, 1 Apr 2025, Garah et al., 15 Sep 2025).