One-dimensional hydrogenic ions with screened nuclear Coulomb field

The gist We study the spectrum of the Dirac hamiltonian in one space dimension for a single electron in the electrostatic potential of a point nucleus, in the Born-Oppenheimer approximation where the nucleus is assumed fixed at the origin.

Derivation of Self-Adjointness and Reduced Hamiltonian

The analysis begins by solving the one-dimensional Dirac equation with minimal coupling to an electrostatic potential, leading to an eigenvalue problem for the reduced Hamiltonian HDΨ = Eψ(s) (Page 1). The wave function components are shown to have equal magnitudes, R1(s) = R2(s) = R(s), which simplifies the system into a set of ordinary differential equations (Page 3). This simplification leads to the reduced Hamiltonian h:= m − eϕ − d/ds d/ds −m − eϕ (Page 3). The paper establishes that for a screened nuclear Coulomb field, specifically ϕ(s) = Ze squared exp(-s) (Page 2), the reduced Hamiltonian h is essentially self-adjoint, and its essential spectrum is shown to be (−∞, -m] ∪ [m, ∞) (Page 4).

Dynamical System and Spectrum Analysis

The problem of finding energy eigenvalues is transformed into studying a dynamical system on a finite cylinder via a Pr¨ufer transform (Page 2). The key components of this system are the equations for R′/R = m sin 2θ and θ′ = m cos(2θ) − eϕ(s) − E (Page 6). The analysis proceeds by compactifying the system, introducing a variable z = arctan(s), which yields a coupled system F(z, Θ) and GE(z, Θ) (Page 7). Linearization near equilibrium points reveals that for E < 1, the equilibria correspond to saddle-nodes (Page 8). The existence of bound states is linked to the existence of heteroclinic saddle-saddle connectors Wj with a well-defined winding number around the cylinder (Page 8).

Existence of Bound States and Winding Numbers

The paper proves that for any value of the parameter γ:= Ze2 > 0, there is a finite number N = N(γ) of solutions, corresponding to a ground state and N −1 excited states with energy eigenvalues E0 < E1 < · · · < EN−1 lying in the interval (−m, m) (Page 2). The existence of heteroclinic saddles connectors with any given winding number N is established using continuity arguments and barrier constructions (Page 9). Specifically, Theorem 4 states that for any γ > 0, there exist integers n ≥ 0 and j ≥ n + 1 such that γ ∈ [γj−1, γj) ∩ [Γn, Γn+1), leading to exactly N:= j−n saddles connectors with all winding numbers between n and j − 1 (Page 17).

Numerical Investigations and Conclusion

Numerical investigations confirm the theoretical results by using a binary search program to find approximate energy eigenvalues EN for specified winding numbers N (Page 18). The relationship between γ and the corresponding energy eigenvalues is illustrated in Figure 5, showing that only certain winding numbers exist for any given γ (Page 18). The paper concludes by stating that the discrete spectrum of the hamiltonian consists of finitely many simple eigenvalues Ei with −1 < E0 < E1 < · · · < EN−1 < 1 (Page 20).

References

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[17] Teschl [11], for the definition of adjoint and the criteria for self-adjointness (Page 4)

[18] Thaller [12], The Dirac Equation (Page 5)

[19] Weidmann’s [15] Theorems 16.5, 16.6, and 10.8, respectively (Page 4)

[20] Thaller [12], The Dirac Equation (Page 5)

[23] Theorem 4, Let γ > 0 be given, and let the sequence Γj be defined by (80), (101). Thus there exist integers n ≥ 0 and j ≥ n + 1 such that γ ∈ [γj−1, γj) ∩ [Γn, Γn+1). Let k(n) ≥ 1 be the smallest integer with the property that γn+k(n) > Γn (Page 20)

[24] Figure 4: Winding numbers of saddles connectors and number of bound states versus γ (Page 17)

[25] Figure 5: This figure shows the energy eigenvalue as a function of γ, for different winding numbers (Page 18)

[26] Figure 6: Plots for γ = 0.5, n = 0 (Page 19)

[27] Figure 7: Plots for γ = 7.5, n = 1 (Page 19)

[28] Figure 8: Plots for γ = 7.5, n = 2 (Page 19)

[30] Equation (30), Z ∞−∞ (u squared + v 2) ds = 1, where the reduced hamiltonian h is as in (29) (Page 4)

[31] Constraint, Z ∞−∞ (u squared + v 2) ds = 1, where the reduced hamiltonian h is as in (29) (Page 4)

[32] Hamiltonian can be written in the form h = J d/ds + P, where J:= 0-1/1 0 P(s):= m - eϕ(s) 0/0-m − eϕ(s) (Page 4)

[33] Definition of P(s), P(s):= m - eϕ(s) 0/0-m − eϕ(s) (Page 4)

[34] Essential spectrum, σess(HDC) = (−∞, m] ∪ [m, ∞) (Page 5)

[35] Discrete spectrum, σdisc(HDC) = E n ∞ n=1 subset of (−m, m) (Page 5)

1. Adapt molecular simulation models to compute one-dimensional hydrogenic ions with screened nuclear Coulomb field spectra, as this framework provides a rigorous mathematical description of bound states for a single electron in a one-dimensional potential, which can serve as a foundational testbed for more complex quantum systems.

2. Develop an AI system capable of mapping the number of bound states to the nuclear charge parameter gamma using the derived relationship: "the discrete spectrum of the hamiltonian consists of finitely many simple eigenvalues Ei with −1 < E0 < E1 < · · · < EN−1 < 1." This allows for predictive modeling of how atomic structure changes with nuclear size.

3. Implement a tool that uses the correspondence between eigenfunctions and dynamical systems: there are only finitely many eigenvalues in that gap, corresponding to ground and excited states for the system and there is a one-to-one correspondence between the eigenfunctions of this hamiltonian and the heteroclinic saddle-saddle connectors W0, W1,..., WN−1 of a certain dynamical system on a finite cylinder. This enables the characterization of physical states via topological features (winding numbers) in phase space.

4. Create an algorithm to determine the number of bound states N:= j − n saddles connectors with all winding numbers between n and j − 1 for any given parameter gamma, leveraging the structure described in Theorem 4 to precisely count the discrete spectrum based on the intersection of intervals defined by "γj and Γj".

5. Design a numerical solver that utilizes the boundary value problems (76) and (77) to find specific energy eigenvalues E for both odd and even winding number solutions, as these solutions are explicitly linked to physical states via the structure of the heteroclinic orbits.

We study the spectrum of the Dirac hamiltonian in one space dimension for a single electron in the electrostatic potential of a point nucleus, in the Born-Oppenheimer approximation where the nucleus is assumed fixed at the origin. The potential is screened at large distances so that it goes to zero exponentially at spatial infinity. We show that the hamiltonian is essentially self-adjoint, the essential spectrum has the usual gap (-mc squared,mc 2) in it, and that there are only finitely many eigenvalues in that gap, corresponding to ground and excited states for the system. We find a one-to-one correspondence between the eigenfunctions of this hamiltonian and the heteroclinic saddle-saddle connectors of a certain dynamical system on a finite cylinder. We use this correspondence to study how the number of bound states changes with the nuclear charge.