WKB-Based Inverse Problem Approach

• WKB-Based Inverse Problem Approach is a semiclassical method that uses direct and inverse mappings to reconstruct effective potentials and geometrical features from observed spectra and transmission data.
• It is applied in diverse fields including radioactive decay, energy-dependent potentials, and phaseless inverse scattering, leveraging turning-point analysis and action integrals.
• The method faces challenges such as non-uniqueness and sensitivity near coalescing turning points, often requiring additional constraints or prescriptions for unique reconstruction.

A WKB-Based Inverse Problem Approach denotes a family of methods in which semiclassical WKB relations are used not only to compute spectra, tunneling probabilities, and transmission coefficients from a prescribed model, but also to reconstruct or constrain the underlying potential, dielectric coefficient, decay law, geometry, or effective background from observed or known data. In the literature, this program appears in the reconstruction of energy-dependent potentials from bound states and transmission data, the recovery of spatially varying dielectric constants from phaseless backscattering, the estimation of superheavy-nucleus half-lives from neighboring nuclei, the extraction of scattering data for non-self-adjoint Dirac operators in inverse scattering theory, the inverse design of accelerating sections, and the engineering of holographic dilaton profiles from hadronic spectra (Albuquerque et al., 2024, Le et al., 26 Jun 2025, Dong et al., 2011, Fujiié et al., 2019, Ayzatsky, 2021, Contreras et al., 5 Sep 2025).

1. Semiclassical inverse logic

The common mathematical core is a direct semiclassical map from a model to observables, followed by an inverse step that uses those observables to recover geometrical or dynamical information. For wave equations of the form

$$\frac{d^2}{dx^2}\psi(x)+Q(x,E)\psi(x)=0,\qquad Q(x,E)=E-V(x,E),$$

turning points are defined by $Q(x,E)=0$. In a two-turning-point well, the bound-state energies satisfy the Bohr-Sommerfeld rule

$$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$

while for a barrier the transmission is written in terms of the Gamow formula

$$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$

These direct formulas can be inverted to obtain the turning-point separation $L(E)=x_1(E)-x_0(E)$. For wells,

$L_1(E)=\frac{\partial}{\partial E}I(E),$

and for barriers,

$$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$

In radioactive decay, the corresponding WKB quantity is the barrier penetrability

$$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$

These formulas organize the inverse problem around turning points, action integrals, and semiclassical widths rather than around a unique closed-form potential (Albuquerque et al., 2024, Dong et al., 2011).

This framework does not, in general, reconstruct a unique potential from the data. The inversion typically fixes only the separation of turning points as a function of energy. A further prescription—such as symmetry, a boundary condition, or the choice of one turning-point function—is required to build an explicit representative of the WKB-equivalent family. This suggests that WKB-based inversion is often best understood as recovery of an equivalence class rather than of a unique model.

Domain	Inferred quantity	WKB ingredient
Radioactive decay	Half-life or penetrability	Barrier-action integral
Energy-dependent potentials	Turning-point width $L(E)$	Bohr-Sommerfeld and Gamow inversion
Phaseless scattering	Dielectric coefficient $c(\mathbf{x})$	WKB phase ansatz
Dirac/NLS inverse scattering	Eigenvalues, reflection, norming constants	Exact WKB and Bohr-Sommerfeld conditions
Holographic QCD	Dilaton profile $Q(x,E)=0$0	RKR/WKB inversion of spectra

2. Relative inverse estimation in radioactive decay

In alpha decay of superheavy nuclei, a distinctly inverse formulation avoids some of the uncertainties of absolute WKB prediction by estimating an unknown half-life from a neighboring nucleus with known half-life. The method is built on the Royer and Viola-Seaborg semi-empirical formulas,

$Q(x,E)=0$1

and

$Q(x,E)=0$2

From these, relative formulas are obtained for a target nucleus $Q(x,E)=0$3 and a reference nucleus $Q(x,E)=0$4. For the Royer form,

$Q(x,E)=0$5

For the VSS form,

$Q(x,E)=0$6

The practical procedure is to use the half-life of a nearby isotope or a nucleus in the same alpha-decay chain, together with known or measured $Q(x,E)=0$7-values, so that uncertainties associated with absolute preformation factors largely cancel in the logarithmic difference (Dong et al., 2011).

The same work benchmarked WKB penetrability against numerical solutions of the Schrödinger equation obtained by discretizing the barrier into $Q(x,E)=0$8 thin square barriers and recursively propagating the amplitudes. The relative deviation

$Q(x,E)=0$9

showed that WKB underestimates penetrability by about $$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$0–$$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$1 for alpha decay, by about $$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$2–$$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$3 for proton emission, and by $$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$4 to $$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$5 for cluster emission. The deviation is nearly constant across wide nuclear ranges, which is precisely why the relative formulation remains effective: systematic error is present, but it is sufficiently stable to cancel or be absorbed when neighboring systems are compared.

3. Energy-dependent potentials and the structure of non-uniqueness

For energy-dependent potentials, inverse semiclassical reconstruction requires a modification of the standard energy-independent picture. The turning points must be found self-consistently from $$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$6, and the reconstructed object is again a width function $$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$7, obtained from bound-state spectra or transmission data. The inverse construction proceeds by computing observables for the original energy-dependent potential, applying the inversion formulas to derive $$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$8, fixing one turning-point function, and reconstructing an energy-independent potential $$\int_{x_0}^{x_1}\sqrt{Q(x,E_n)}\,dx=\pi\left(n+\frac12\right),$$9 that is WKB-equivalent to the original system (Albuquerque et al., 2024).

A central result is that width-equivalent or WKB-equivalent potentials are not isospectral anymore when the original potential depends explicitly on energy. Instead, one obtains quasi-isospectral potentials: the reconstructed energy-independent potentials closely match the spectrum or transmission of the original energy-dependent potential over a range, but not identically. The paper demonstrates this explicitly for the energy-dependent harmonic-oscillator-type potential

$$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$0

and the energy-dependent Pöschl-Teller potential

$$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$1

This point corrects a common misconception inherited from the energy-independent theory. In the classical energy-independent setting, width-equivalence and semiclassical isospectrality go together. In the energy-dependent setting, that implication fails. The reconstructed effective potentials nevertheless preserve key asymptotics and local behavior near extrema, and the best agreement is found near the well minimum for spectral inversion and near the barrier peak for transmission inversion. The method also inherits the usual WKB limitation that inverse reconstruction fails if $$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$2 is not monotonic.

4. Phaseless inverse scattering and global convexification

In phaseless inverse scattering for the three-dimensional Helmholtz equation,

$$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$3

the measured data are the modulus of the total field,

$$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$4

with single-point-source illumination and no direct phase information. The WKB ansatz at large $$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$5,

$$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$6

supplies the leading-order phase model, and the eikonal equation

$$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$7

reduces to $$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$8 in the measurement region where $$T(E)=\exp\left(2i\int_{x_0}^{x_1}\sqrt{Q(x,E)}\,dx\right).$$9. This yields the initial phase guess

$L(E)=x_1(E)-x_0(E)$0

The recovered boundary field is then obtained by nonlinear optimization,

$L(E)=x_1(E)-x_0(E)$1

which enforces consistency with both the magnitude data and the Helmholtz equation. After phase retrieval, a logarithmic transformation

$L(E)=x_1(E)-x_0(E)$2

is expanded in an orthonormal polynomial-exponential basis,

$L(E)=x_1(E)-x_0(E)$3

thereby reducing the problem to a finite nonlinear elliptic system for the coefficients $L(E)=x_1(E)-x_0(E)$4. The resulting Cauchy problem is stabilized by Carleman convexification, which introduces the Carleman weight $L(E)=x_1(E)-x_0(E)$5 and produces a strictly convex weighted cost functional $L(E)=x_1(E)-x_0(E)$6 for sufficiently large $L(E)=x_1(E)-x_0(E)$7. The corresponding error estimate is

$L(E)=x_1(E)-x_0(E)$8

The numerical results are reported for synthetic data with $L(E)=x_1(E)-x_0(E)$9 uniform random noise. The reconstructions recover both the geometric location and the dielectric contrast of hidden scatterers, including single, multiple, and elongated inclusions, with relative error of $L_1(E)=\frac{\partial}{\partial E}I(E),$0–$L_1(E)=\frac{\partial}{\partial E}I(E),$1 in maximum contrast for high-contrast targets. The distinctive feature of this approach is that the WKB ansatz is not the terminal approximation; it is the phase-retrieval front end of a globally convergent inverse scheme based on Fourier filtering and Carleman convexification (Le et al., 26 Jun 2025).

5. Exact WKB, non-self-adjoint Dirac operators, and inverse scattering for focusing NLS

For the focusing cubic NLS equation, the inverse scattering transform is governed by the spectral analysis of a non-self-adjoint Dirac, or Zakharov-Shabat, operator. In the zero-phase case, the operator

$L_1(E)=\frac{\partial}{\partial E}I(E),$2

is analyzed for real, positive, analytic, decaying $L_1(E)=\frac{\partial}{\partial E}I(E),$3. In the rapidly oscillating case, the relevant operator is

$L_1(E)=\frac{\partial}{\partial E}I(E),$4

with both $L_1(E)=\frac{\partial}{\partial E}I(E),$5 and $L_1(E)=\frac{\partial}{\partial E}I(E),$6 tending to constants as $L_1(E)=\frac{\partial}{\partial E}I(E),$7. Exact WKB constructs convergent amplitude series and rigorous connection formulas in the complex plane, while Olver’s theory is used for the reflection coefficient and for eigenvalues near $L_1(E)=\frac{\partial}{\partial E}I(E),$8 in the spectral plane (Fujiié et al., 2019, Fujiié et al., 2022, Fujiié et al., 10 Mar 2026).

In the zero-phase setting, the phase integral

$L_1(E)=\frac{\partial}{\partial E}I(E),$9

enters the exact WKB solutions, and the reflection coefficient is exponentially small for $$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$0 bounded away from the origin. The discrete spectrum on the imaginary axis obeys the modified Bohr-Sommerfeld condition

$$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$1

with $$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$2. In the rapidly oscillating case one introduces

$$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$3

and the action integral

$$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$4

governs quantization on unions of analytic spectral arcs through

$$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$5

The inverse-problem significance is direct. The scattering data—reflection coefficient, eigenvalues, and norming constants—are the inputs to the inverse scattering transform. WKB and exact WKB therefore operate as asymptotic reconstruction tools for the spectral data themselves. A further consequence is methodological: the spectral support in the semiclassical limit is organized by turning points, Stokes geometry, and action integrals rather than by elementary perturbation theory. This is why the same conceptual scheme applies across analytic decaying potentials, rapidly oscillating potentials, and spectral arcs in the complex $$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$6-plane.

6. Refinements, extensions, and objective limits

A major refinement is the structured backward-error interpretation of WKB for singularly perturbed second-order ODEs

$$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$7

At leading order, the WKB approximation is the exact solution of a nearby problem

$$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$8

with

$$L_2(E)=\frac{1}{\pi}\int_E^{E_{\max}}\frac{\partial T(E')/\partial E'}{T(E')\sqrt{E'-E}}\,dE'.$$9

This recasts WKB from a purely forward asymptotic device into an explicit inverse statement: given an approximate WKB solution, one can identify the perturbed potential for which that solution is exact. The same paper gives an iterative correction

$$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$0

and a hybrid method in which $$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$1 is approximated by Chebyshev polynomials and integrated term-wise (Corless et al., 2024).

For the finite difference Schrödinger equation,

$$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$2

the WKB program extends to an all-order construction of a quantum momentum and to connection formulae across turning points. A distinctive result is the existence of additional periodic factors with period $$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$3, so that the general solution is not simply two-dimensional in the ODE sense. Quantization conditions are then formulated in terms of quantum periods and periodic factors, which makes the inverse spectral problem richer and more ambiguous than in the differential case (Baldino, 2024).

In gravitational tunneling, the WKB-like method for Unruh radiation shows that the tunneling rate should be written as the closed path integral of the canonical momentum,

$$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$4

and that there is an additional temporal contribution arising from an imaginary change of the time coordinate upon horizon crossing. This temporal term has no analog in ordinary quantum mechanical WKB. The result is an objective correction to the idea that the inverse interpretation of tunneling data can be based on a spatial barrier integral alone (Gill et al., 2010).

Two application domains illustrate how strongly the validity of the inverse WKB picture depends on structure. In Inhomogeneous Travelling-Wave Accelerating Sections, a discrete WKB approximation for matrix difference equations links field amplitudes to slowly varying cell parameters and supports inverse design and tuning, but it fails for abrupt, non-smooth parameter changes (Ayzatsky, 2021). In holographic bottom-up QCD, a WKB inverse problem based on the Rydberg-Klein-Rees formula starts from nonlinear radial Regge trajectories,

$$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$5

reconstructs the asymptotic confining potential

$$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$6

and yields a non-quadratic dilaton

$$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$7

The method is tested on heavy quarkonia, with $$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$8 for $$P_{\mathrm{WKB}}=\exp\left[-\frac{2}{\hbar}\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\sqrt{2\mu(V(r)-Q)}\,dr\right].$$9 and $L(E)$0 for $L(E)$1, and reports RMS errors of $L(E)$2 and $L(E)$3 for $L(E)$4–$L(E)$5 fitted states; it is then extended to tetraquarks by superimposing an additional potential term derived from the Bethe-Salpeter equation for diquarks (Contreras et al., 5 Sep 2025).

Across these literatures, the limitations are consistent. WKB-based inversion is non-unique because it often determines only turning-point separations or effective equivalence classes; it deteriorates near coalescing turning points, near or above barrier tops, and when the structural assumption of slow variation fails; and, in radioactive decay, shell or deformation effects can upset the stability of preformation factors. The strength of the approach is therefore not exact uniqueness, but controlled semiclassical reconstruction with explicit action integrals, identifiable error structure, and a clear account of where the approximation is expected to hold.