^{1,a)}and Julius Ruseckas

^{1}

### Abstract

We solve the bound state problem for the Hamiltonian with the spin-orbit and the Raman coupling included. The Hamiltonian is perturbed by a one-dimensional short-range potential V which describes the impurity scattering. In addition to the bound states obtained by considering weak solutions through the Fourier transform or by solving the eigenvalue equation on a suitable domain directly, it is shown that ordinary point-interaction representations of V lead to spin-orbit induced extra states.

The authors gratefully acknowledge Dr. G. Juzeliūnas who bears much of the credit for the genesis of the present paper. It is a pleasure to thank Dr. I. Spielman for useful discussions. R.J. acknowledges Dr. I. Spielman for warm hospitality extended to him during his visit at the University of Maryland, where the part of this work has been done. The present work was supported by the Research Council of Lithuania (No. VP1-3.1-ŠMM-01-V-02-004).

I. INTRODUCTION

II. PRELIMINARIES

III. FERMI PSEUDOPOTENTIAL

IV. SPIN-ORBIT COUPLING INDUCED STATES

V. DISCRETE SPECTRUM

VI. SUMMARY AND DISCUSSION

### Key Topics

- Eigenvalues
- 18.0
- Spin orbit interactions
- 11.0
- Boundary value problems
- 10.0
- Dirac equation
- 8.0
- Fourier transforms
- 7.0

## Figures

Computed spectrum of operator A 0 (see Eq. (3.1) and Theorem 1) for the point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = c = 1 units). The eigenvalues divided by η > 0 are those of the one-dimensional Dirac-like operator for the particle of spin one-half and mass Ω/(2η) moving in the Fermi pseudopotential (3.7) . In the figure, red lines show the border of the essential spectrum of A 0, which is ±Ω/2. The blue ɛ+ (green ɛ−) line, showing the bound state as a function of the Raman coupling Ω > 0, corresponds to the eigenfunction with a zero-valued lower (upper) component at the origin x = 0.

Computed spectrum of operator A 0 (see Eq. (3.1) and Theorem 1) for the point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = c = 1 units). The eigenvalues divided by η > 0 are those of the one-dimensional Dirac-like operator for the particle of spin one-half and mass Ω/(2η) moving in the Fermi pseudopotential (3.7) . In the figure, red lines show the border of the essential spectrum of A 0, which is ±Ω/2. The blue ɛ+ (green ɛ−) line, showing the bound state as a function of the Raman coupling Ω > 0, corresponds to the eigenfunction with a zero-valued lower (upper) component at the origin x = 0.

Computed lower branch of dispersion in (4.4) for the spin-orbit-coupling strength η = 0.6 (in ℏ = c = 1 units), for a range of Raman couplings Ω ⩾ 0. As Ω increases (Ω > η2), the two dressed spin states ( Lin et al., 2011 ) are merged into a single minimum −Ω/2 at p = 0. This is a regime when the spin-orbit coupling induced states σso(A), Theorem 2, are observed below the continuous spectrum as well as above it. For Ω ⩽ η2, the spin states have two minima −[η2 + (Ω/η)2]/4 at , and the spin-orbit induced states are embedded into the essential spectrum of A.

Computed lower branch of dispersion in (4.4) for the spin-orbit-coupling strength η = 0.6 (in ℏ = c = 1 units), for a range of Raman couplings Ω ⩾ 0. As Ω increases (Ω > η2), the two dressed spin states ( Lin et al., 2011 ) are merged into a single minimum −Ω/2 at p = 0. This is a regime when the spin-orbit coupling induced states σso(A), Theorem 2, are observed below the continuous spectrum as well as above it. For Ω ⩽ η2, the spin states have two minima −[η2 + (Ω/η)2]/4 at , and the spin-orbit induced states are embedded into the essential spectrum of A.

Computed spin-orbit coupling induced states σso(A) ⊂ σdisc(A) (refer to Theorem 2) for the point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = c = 1 units). In the figure, red line shows the border of the essential spectrum of A (Lemma 2). The eigenvalues λ(ɛ) ∈ σso(A) (ɛ ∈ σdisc(A 0)), as functions of the Raman coupling Ω > 0, are drawn by the blue (σ>(A)) and green (σ<(A)) lines. Resonant states of A are drawn by yellow curves (R).

Computed spin-orbit coupling induced states σso(A) ⊂ σdisc(A) (refer to Theorem 2) for the point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = c = 1 units). In the figure, red line shows the border of the essential spectrum of A (Lemma 2). The eigenvalues λ(ɛ) ∈ σso(A) (ɛ ∈ σdisc(A 0)), as functions of the Raman coupling Ω > 0, are drawn by the blue (σ>(A)) and green (σ<(A)) lines. Resonant states of A are drawn by yellow curves (R).

The eigenvalues of A associated with everywhere continuous eigenfunctions. The point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = c = 1 units). In the figure, red line shows the border of the essential spectrum of A (Lemma 2). The blue λ+ (green λ−) line, showing the bound state as a function of the Raman coupling Ω ⩾ 0, corresponds to the eigenfunction with a zero-valued lower (upper) component at the origin x = 0 (Theorem 3). The eigenvalue λ+ approaches at Ω = η2 + γ2/4 and then disappears (for details, refer to Remark 4). Resonant states of A are drawn by the yellow curve (R).

The eigenvalues of A associated with everywhere continuous eigenfunctions. The point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = c = 1 units). In the figure, red line shows the border of the essential spectrum of A (Lemma 2). The blue λ+ (green λ−) line, showing the bound state as a function of the Raman coupling Ω ⩾ 0, corresponds to the eigenfunction with a zero-valued lower (upper) component at the origin x = 0 (Theorem 3). The eigenvalue λ+ approaches at Ω = η2 + γ2/4 and then disappears (for details, refer to Remark 4). Resonant states of A are drawn by the yellow curve (R).

## Tables

All possible solutions of (5.7) with respect to for a j , b j = ±1 for j = 1, …, 4 given in (5.8) and (5.9) .

All possible solutions of (5.7) with respect to for a j , b j = ±1 for j = 1, …, 4 given in (5.8) and (5.9) .

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