1D Green's Function Model in Quantum Systems

• The 1D Green's function model is a one-dimensional kernel-based framework that employs the resolvent operator to solve inhomogeneous equations and extract spectral information.
• It uses paired left/right homogeneous solutions and a constant Wronskian to enforce boundary/jump conditions and reveal quantized energy levels.
• The model extends to handle localized defects and multi-site interactions, providing a versatile algebraic architecture for diverse physical and computational applications.

Searching arXiv for the specified paper and closely related work on 1D Green’s function models. A 1D Green’s function model is a one-dimensional kernel-based formulation in which the resolvent $G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle$ or its analog for a boundary-value operator is used as the central object for solving inhomogeneous equations, extracting spectra, and incorporating localized defects. In the quantum harmonic-oscillator setting, this model consists of the differential equation $(H-E)G=\delta(x-x')$, its boundary and jump conditions, and the associated left/right homogeneous solutions whose Wronskian fixes the kernel; in that form, the Green’s function encodes eigenvalues as poles, eigenfunctions as residues, and perturbations such as $\delta$-function interactions through explicitly modified resolvents (Chua et al., 2017).

1. Operator formulation and one-dimensional structure

For a one-dimensional quantum system with Hamiltonian

$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$

the time-independent Green’s function is defined by

$$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$

In the unperturbed simple harmonic oscillator (SHO),

$$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$

the corresponding Green’s function $G_0(x,y;E)$ satisfies

$$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$

with decay conditions

$\lim_{x\to\pm\infty}G_0(x,y;E)=0,$

continuity at the source,

$G_0(y^+,y;E)=G_0(y^-,y;E),$

and derivative jump

$(H-E)G=\delta(x-x')$0

Using $(H-E)G=\delta(x-x')$1 and $(H-E)G=\delta(x-x')$2, the governing equation becomes

$(H-E)G=\delta(x-x')$3

which is the basic one-dimensional Green’s function model for the SHO (Chua et al., 2017).

A defining feature of the 1D setting is that once two homogeneous solutions are chosen to satisfy left and right boundary conditions, the Green’s function assumes the universal form

$(H-E)G=\delta(x-x')$4

The constant $(H-E)G=\delta(x-x')$5 is fixed by the jump condition, and because the ODE contains no first-derivative term, the Wronskian $(H-E)G=\delta(x-x')$6 is constant. This converts the distributional matching problem into a closed algebraic normalization problem.

2. Harmonic-oscillator realization

Away from the source point, the SHO Green’s function satisfies the homogeneous equation

$(H-E)G=\delta(x-x')$7

With $(H-E)G=\delta(x-x')$8, linearly independent solutions can be expressed through generalized Hermite functions $(H-E)G=\delta(x-x')$9. The physically admissible choices are the decaying solutions

$\delta$0

which satisfy

$\delta$1

For this pair, the Wronskian is

$\delta$2

Imposing the derivative jump yields the standard one-dimensional representation

$\delta$3

Substitution of the explicit $\delta$4, $\delta$5, and $\delta$6 gives a compact expression in terms of $\delta$7 and $\delta$8. The resulting kernel is symmetric under exchange of arguments and under parity,

$\delta$9

so the SHO Green’s function becomes the canonical central object of the 1D model (Chua et al., 2017).

The construction is representative of a broader 1D pattern. One first solves the homogeneous ODE, then selects left/right solutions adapted to decay or boundary data, and finally enforces the source discontinuity through the Wronskian. The harmonic oscillator is distinctive not because the method changes, but because the special functions and their Wronskian are known explicitly.

3. Spectrum from poles and residues

In the SHO case, the poles of $H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$0 arise from the factor $H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$1. Since the Gamma function has simple poles at non-positive integers,

$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$2

the energies are

$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$3

These are exactly the SHO energy levels. The same quantization can be read from the Wronskian: at $H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$4, the Hermite functions reduce to polynomials obeying

$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$5

so the left and right solutions become linearly dependent and $H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$6. In the Green’s function language, the discrete bound-state spectrum is therefore the zero set of the Wronskian and the pole set of the resolvent (Chua et al., 2017).

Near a pole,

$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$7

and hence

$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$8

Using the limiting behavior of $H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$9 near $$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$0, one finds

$$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$1

with $$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$2 the normalized SHO eigenfunctions. The framework therefore identifies eigenvalues with poles and eigenfunctions with square-roots of residues. This is one of the central uses of a 1D Green’s function model for bound states.

A common misconception is that the Green’s function is merely an inverse operator for forced problems. In one dimension, and especially in the SHO example, it is equally a spectral object: the same kernel that solves the inhomogeneous equation also carries quantization, parity, normalization, and bound-state structure.

4. Localized perturbations: a single $$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$3-function potential

The model extends directly to a localized perturbation

$$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$4

with perturbed Green’s function $$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$5. Eliminating the unknown $$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$6 yields the explicit resolvent formula

$$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$7

The denominator contains only the local object $$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$8, which means that the full spectral effect of the point interaction is encoded through the unperturbed Green’s function evaluated at the defect site (Chua et al., 2017).

At the wave-function level, the $$G(x,x';E)=\langle x|(H-E)^{-1}|x'\rangle, \qquad (H-E)G(x,x';E)=\delta(x-x').$$9-interaction preserves continuity and imposes a derivative jump,

$$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$0

In the Green’s function formulation this matching data are built into the denominator above. The new bound states are determined by the zeros of

$$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$1

which becomes the transcendental equation

$$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$2

For $$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$3, the second term vanishes and the unperturbed SHO spectrum is recovered.

The qualitative consequences depend on position and sign. For $$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$4, the potential is not parity-symmetric, so the eigenfunctions are not purely even or odd and all levels shift. For $$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$5, parity is preserved: the odd SHO eigenfunctions vanish at the origin, so odd levels remain unchanged, whereas even levels move upward for $$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$6 and downward for $$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$7. The levels remain ordered, and no level crossing occurs (Chua et al., 2017).

5. Wronskians, quantization, and multi-site generalization

For a second-order ODE without first derivative,

$$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$8

the Wronskian satisfies $$V_0(x)=\frac12 m\omega^2x^2, \qquad H_0=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2,$$9. In 1D Green’s function theory this constancy has three simultaneous roles. It normalizes the kernel through the source jump, it diagnoses linear dependence of left/right solutions, and its zeros provide the quantization condition for bound states. In the SHO example,

$G_0(x,y;E)$0

so the same denominator that normalizes the resolvent also generates its poles (Chua et al., 2017).

The method generalizes to several point interactions. For two additional deltas,

$G_0(x,y;E)$1

the wave function is constructed in three regions using

$G_0(x,y;E)$2

with $G_0(x,y;E)$3. Matching at $G_0(x,y;E)$4 and $G_0(x,y;E)$5 produces two simultaneous equations for $G_0(x,y;E)$6 and $G_0(x,y;E)$7. In the symmetric configuration

$G_0(x,y;E)$8

parity is restored and the parameter $G_0(x,y;E)$9 collapses to $$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$0 for even states and $$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$1 for odd states. Repulsive deltas push levels upward, attractive deltas pull them downward, and no level crossings occur (Chua et al., 2017).

The same structure admits a compact matrix-resolvent form,

$$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$2

with $$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$3 acting in the finite-dimensional space of defect sites. This suggests that the phrase “1D Green’s function model” refers not to a single closed formula, but to a transferable algebraic architecture: homogeneous left/right solutions, a constant Wronskian, local matching data, and finite-dimensional reduction for point interactions.

6. Extensions, reinterpretations, and contemporary uses

The 1D Green’s function model has been extended far beyond the SHO. In the half-line acoustic impedance problem, the Green’s function solves

$$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$4

with boundary condition

$$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$5

and exact solution

$$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$6

In this case the direct and reflected eikonals are $$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$7 and $$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$8, and the “slowly varying amplitudes” reduce to constants, so the higher-dimensional decomposition into exponential factors times slowly varying terms becomes exact in 1D (Lin et al., 2024).

Other extensions preserve the same one-dimensional logic while changing the operator class. In nonlinear or complex Dirac-$$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +\frac12 m\omega^2x^2-E\right)G_0(x,y;E)=\delta(x-y),$$9 models, the free kernels

$\lim_{x\to\pm\infty}G_0(x,y;E)=0,$0

reduce scattering and bound-state problems with finitely many point interactions to finite-dimensional algebraic systems for the site values $\lim_{x\to\pm\infty}G_0(x,y;E)=0,$1, with transmission amplitude $\lim_{x\to\pm\infty}G_0(x,y;E)=0,$2 in the linear and nonlinear cases alike (Erman et al., 2019). In quasi-1D nanophotonics, the projected electromagnetic Green’s function $\lim_{x\to\pm\infty}G_0(x,y;E)=0,$3 becomes the matrix of atom-atom couplings, and transmission factorizes over collective eigenmodes of that Green’s matrix (Asenjo-Garcia et al., 2016). In slender blood-vessel theory, the exterior pressure is given by an explicit Green’s function involving the interior pressure, while the interior variable satisfies a novel 1D integrodifferential equation; the 3D–1D model converges to the 1D Green’s function model with rate proportional to $\lim_{x\to\pm\infty}G_0(x,y;E)=0,$4, and the full 3D–3D model converges to it at rate proportional to $\lim_{x\to\pm\infty}G_0(x,y;E)=0,$5 (Ohm et al., 17 Jul 2025, Ohm et al., 17 Jul 2025).

Computationally, contemporary work treats 1D Green’s functions both as objects to be learned and as reduced models in their own right. A data-driven approach learns $\lim_{x\to\pm\infty}G_0(x,y;E)=0,$6 from input-output pairs and approximates it in low rank as

$\lim_{x\to\pm\infty}G_0(x,y;E)=0,$7

using either POD modes or an operator-valued randomized SVD, with manifold interpolation across parameter values (Praveen et al., 2022). Symbolic regression with physical hard constraints has also recovered exact or near-exact 1D Green’s functions for Laplace and Helmholtz operators and produced candidate kernels for periodic Helmholtz and jump-condition problems, with solution error on the order of $\lim_{x\to\pm\infty}G_0(x,y;E)=0,$8 in the reported cases (Gu et al., 2024). A further reinterpretation views the Green’s function itself as the unique defect state of an auxiliary problem with a self-consistently determined $\lim_{x\to\pm\infty}G_0(x,y;E)=0,$9-potential, so that the kernel is recast as an eigenstate rather than merely an inverse operator (Rivero et al., 2021).

Taken together, these developments suggest that a 1D Green’s function model is best understood as an operator-theoretic template. Its classical SHO realization is especially transparent: the kernel is built from left/right decaying solutions, fixed by a Wronskian, and modified by local defect data; but the same template reappears in impedance boundaries, point-scatterer systems, inverse and data-driven modeling, photonic transport, and multiscale perfusion theory.