Schrödinger-like Perturbation Equation

• The Schrödinger-like perturbation equation is a framework for linearizing and analyzing corrections in systems governed by Schrödinger operators, preserving key spectral and dynamic properties.
• It is applied to study phenomena like soliton stability, mode-mixing dynamics, and nonlocal effects in quantum, astrophysical, and fractional models.
• Numerical and iterative methods, including parameter flow techniques, enable exploration of high perturbation regimes and aid convergence in complex quantum systems.

A Schrödinger-like perturbation equation describes the dynamics of perturbations or corrections to a system governed by a Schrödinger-type operator, typically in quantum mechanics, mathematical physics, or certain nonlinear dispersive settings. These equations arise both in the analysis of stability of nonlinear solitons (as in self-gravitating systems), linear or nonlinear response theory, fractional quantum models, and iterative or numerical approaches to perturbation around known solutions. The “Schrödinger-like” adjective indicates that the perturbed equation retains the essential operator structure, dynamics, or spectral properties of the foundational Schrödinger equation, albeit typically linearized or otherwise generalized.

1. Foundational Formulation: Linearization and Soliton Perturbations

The canonical example arises in the study of self-gravitating quantum matter such as solitons of the Schrödinger–Poisson system. The unperturbed system is

$$i\hbar\,\partial_t\psi(\mathbf{x},t) = -\frac{\hbar^2}{2m}\nabla^2\psi + m\,\Phi(\mathbf{x},t)\psi, \qquad \nabla^2\Phi = 4\pi G|\psi|^2$$

with $\psi$ the wave field and $\Phi$ the (Newtonian) potential. For a ground-state soliton $\psi_0(\mathbf{x},t) = \psi_0(r) e^{-iE_0 t/\hbar}$, perturbations are defined by $\psi = \psi_0 + \delta\psi$, $\Phi = \Phi_0 + \delta\Phi$. Linearizing to first order yields the fundamental Schrödinger-like perturbation equation: $$i\hbar\,\partial_t\,\delta\psi = H_0\,\delta\psi + \delta H(t)\,\psi_0$$ where $H_0 = -\frac{\hbar^2}{2m}\nabla^2 + m\Phi_0$ is the stationary “background” Hamiltonian and $\delta H(t) = m\,\delta\Phi(\mathbf{x},t)$. This form directly reflects the structure of the linearized dynamics for perturbations about a stationary solution and is widely employed in astrophysical, cosmological, and soliton stability settings (Zagorac et al., 2021).

2. Spectral Decomposition and Mode-Mixing Dynamics

Given the self-adjoint $H_0$ with eigenfunctions $\phi_{n\ell m}(\mathbf{x})$: $H_0\,\phi_{n\ell m} = E_n\,\phi_{n\ell m}$ the perturbation $\delta\psi$ is expanded in this eigenbasis: $$\delta\psi(\mathbf{x},t) = \sum_{n\ell m} c_{n\ell m}(t)\,\phi_{n\ell m}(\mathbf{x})$$ Projecting onto this basis and using the perturbation operator leads to a coupled mode-system (after transformation to the interaction picture): $$i\hbar\,\dot{a}_{n\ell m}(t) = \sum_{n'\ell' m'} V_{n\ell m,\,n'\ell' m'}(t) a_{n'\ell' m'}(t)$$ with $$V_{n\ell m,\,n'\ell' m'}(t) = \langle\phi_{n\ell m}|\delta H(t)|\phi_{n'\ell' m'}\rangle\,e^{i(E_n-E_{n'})t/\hbar}$$. This finite-dimensional Schrödinger equation (in mode coefficients) captures the resonant and non-resonant mixing of linear modes induced by the perturbation (Zagorac et al., 2021).

Physically, select multipole perturbations correspond to distinctive system responses: $\ell=0$ induces breathing (radial) oscillations, $\ell=1$ induces translational shifts (“random walk”), and $\ell=2$ yields quadrupolar deformations.

3. Generalizations: Fractional Operators, Nonlinear and Non-local Perturbations

Schrödinger-like perturbation equations also encompass non-classical generalizations. In the fractional-in-time nonlinear Schrödinger equation with Hartree-type perturbation (Prado et al., 2019),

$$i^\alpha D_t^\alpha u = (-\Delta)^{\beta/2}u + \lambda\, J^\alpha[K_\gamma(|u|^2)u], \qquad u(0) = u_0(x)$$

the perturbative effects appear through both the non-local Hartree convolution and the Caputo fractional time derivative $D_t^\alpha$ (with $0<\alpha<1$). The nonlinear nonlocal structure requires fixed-point and convolution estimates for local well-posedness analysis.

This broadened framework also accommodates time-dependent and spatially non-local perturbations, as in the reducibility of quasiperiodically forced Schrödinger equations with unbounded symbols (Bambusi, 2016).

4. Numerical and Iterative Approaches: ODE Systems and All-Order Resummation

The “Large Perturbation Method” (Mikaberidze, 2016) recasts the stationary Schrödinger equation with perturbation $H(\lambda) = H_0 + \lambda V$ as a flow in $\lambda$:

• For energies:

$$\frac{dE_n}{d\lambda} = \langle n(\lambda)|V|n(\lambda)\rangle$$

• For wavefunction amplitudes $Q_n^j(\lambda)$ in the $H_0$ basis, as coupled first-order ODEs.

Numerical integration of these Schrödinger-like ODEs in “perturbation parameter space” enables nonperturbative treatment of large $V$, higher-order effects, and regime transitions that are inaccessible to standard Rayleigh–Schrödinger expansions (Mikaberidze, 2016). Similar iterative strategies, including synthetic Hamiltonian constructions and convergence-optimized methods, further extend applicability to problems with degeneracies and divergent series (Kerley, 2013).

5. Discrete and Nonlinear Schrödinger-Like Systems

Discrete Schrödinger equations with finite-rank perturbations yield Schrödinger-like operator equations at the lattice level. For a Jacobi matrix with a rank-one perturbation,

$$\psi_{n+1} + \psi_{n-1} + \beta\,\delta_{n,k}\psi_n = E\psi_n$$

the perturbative calculation of scattering matrices and eigenvalues employs the distinct algebraic structure of finite-dimensional, Schrödinger-like operators (Borzov et al., 2016).

Additionally, in integrable nonlinear models subjected to localized perturbations (e.g., defocusing NLS with $\varepsilon a(x)|q|^\ell q$), the perturbed evolution of spectral data (e.g., reflection coefficient) can be expressed as an integral equation with linear and nonlinear Schrödinger-like terms (Chen et al., 15 Aug 2025).

6. Physical Interpretation and Applications

Schrödinger-like perturbation equations serve to:

• Analyze the stability and evolution of nonlinear coherent structures (solitons, breathers).
• Quantify mode-mixing, damping, and response in quantum, astrophysical, and plasma systems.
• Deliver tractable frameworks for both analytic (e.g. eigenfunction expansion, Wronskian techniques for phase shifts (Hoffmann, 2020)) and numerical investigation of complex perturbative regimes.
• Enable rigorous proofs of long-time asymptotic stability and equidistribution in semiclassical and chaotic backgrounds (Eswarathasan et al., 2014).
• Generalize to degenerate, nonlocal, and fractional-dynamics contexts, supporting broad applicability in mathematical and physical models.

7. Key Equations and Implementation Summary

The archetypal Schrödinger-like perturbation hierarchy is summarized in the following table:

Structural Level	Prototype Equation	Context
Linearized about soliton	$$i\hbar \partial_t \delta\psi = H_0 \delta\psi + \delta H \psi_0$$	Soliton stability (Zagorac et al., 2021)
Spectral expansion	$$i\hbar \dot a_{n\ell m} = \sum_{n'\ell'm'} V_{n\ell m,\,n'\ell'm'}(t) a_{n'\ell'm'}$$	Mode-mixing, parametric driving
Fractional/Nonlinear	$$i^\alpha D_t^\alpha u = (-\Delta)^{\beta/2}u + \lambda J^\alpha [K_\gamma(|u|^2)u]$$	Fractional time/NLSE (Prado et al., 2019)
ODE-based (parameter flow)	$$\frac{dE_n}{d\lambda} = \langle n(\lambda)|V|n(\lambda)\rangle$$, $$\, \frac{dQ_n}{d\lambda} = \sum_{m\neq n} Q_m [...]$$	All-orders, large V (Mikaberidze, 2016)
Discrete structure	$$\psi_{n+1} + \psi_{n-1} + \beta \delta_{n,k} \psi_n = E \psi_n$$	Lattice, finite-rank (Borzov et al., 2016)
Integral equation form	$|\psi\rangle = |\psi_0\rangle - G_0 V |\psi\rangle$	Inhomogeneous, anchored (Kauffmann, 2012)
Nonperturbative iteration	$$\langle\pi_i|\psi^{(n+1)}\rangle = \frac{\langle\pi_i|\psi_0\rangle}{1 + \langle\pi_i|G_0 V|\psi^{(n)}\rangle / \langle\pi_i|\psi^{(n)}\rangle}$$	Continued-fraction (Kauffmann, 2012)

These equations define the backbone for a broad class of problems where Schrödinger-like perturbation theory is central. Their explicit form and analytical structure enable rigorous study and application to quantum systems, nonlinear waves, fractional models, and astrophysics.