A note on the Exact Green function for a quantum system decorated by two   or more impurities

M. L. Glasser

arXiv:1702.05111·quant-ph·February 20, 2017

A note on the Exact Green function for a quantum system decorated by two or more impurities

M. L. Glasser

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TL;DR

This paper derives the exact Green function for a quantum system with multiple delta function impurities, revealing how multiple impurities combine into a single effective potential when coinciding, and extending the analysis to N impurities.

Contribution

It provides a method to construct the exact Green function for systems with multiple impurities and elucidates their behavior when impurities coincide, extending previous single-impurity results.

Findings

01

Impurities behave as a single potential when coinciding.

02

The Green function can be explicitly constructed for multiple impurities.

03

Results are extended to systems with N impurities.

Abstract

The exact Green function is constructed for a quantum system, with known Green function, which is decorated by two delta function impurities.It is shown that when two such impurities coincide they behave as a single singular potential with combined amplitude. The results are extended to N impurities.

Equations30

G (x, x^{'}; E) = G_{0} (x, x^{'}) + λ \frac{G _{0} ( x , a ; E ) G _{0} ( a , x ^{'} )}{1 - λ G _{0} ( a , a ; E )}

G (x, x^{'}; E) = G_{0} (x, x^{'}) + λ \frac{G _{0} ( x , a ; E ) G _{0} ( a , x ^{'} )}{1 - λ G _{0} ( a , a ; E )}

G (x, x^{'}) = G_{0} (x, x^{'}) + \int G_{0} (x, y) V (y) G (y, x^{'}) d y,

G (x, x^{'}) = G_{0} (x, x^{'}) + \int G_{0} (x, y) V (y) G (y, x^{'}) d y,

G (x, x^{'}) = G_{0} (x, x^{'}) + λ G_{0} (x, a) G (a, x^{'}) + μ G_{0} (x, b) G (b, x^{'})

G (x, x^{'}) = G_{0} (x, x^{'}) + λ G_{0} (x, a) G (a, x^{'}) + μ G_{0} (x, b) G (b, x^{'})

G (a, x^{'}) = G_{0} (a, x^{'}) + λ G_{0} (a, a) G (a, x^{'}) + μ G_{0} (a, b) G (b, x^{'}),

G (a, x^{'}) = G_{0} (a, x^{'}) + λ G_{0} (a, a) G (a, x^{'}) + μ G_{0} (a, b) G (b, x^{'}),

G (b, x^{'}) = G_{0} (b, x^{'}) + λ G_{0} (a, b) G (a, x^{'}) + μ G_{0} (b, b) G (b, x^{'}) .

G (b, x^{'}) = G_{0} (b, x^{'}) + λ G_{0} (a, b) G (a, x^{'}) + μ G_{0} (b, b) G (b, x^{'}) .

G (a, x^{'}) = \frac{G _{0} ( a , x ^{'} ) + μ [ G _{0} ( b , x ^{'} ) G _{0} ( a , b ) - G _{0} ( a , x ^{'} ) G _{0} ( b , b )]}{D}

G (a, x^{'}) = \frac{G _{0} ( a , x ^{'} ) + μ [ G _{0} ( b , x ^{'} ) G _{0} ( a , b ) - G _{0} ( a , x ^{'} ) G _{0} ( b , b )]}{D}

G (b, x^{'}) = \frac{G _{0} ( b , x ^{'} ) + λ [ G _{0} ( a , x ^{'} ) G _{0} ( a , b ) - G _{0} ( b , x ^{'} ) G _{0} ( a , a )]}{D}

G (b, x^{'}) = \frac{G _{0} ( b , x ^{'} ) + λ [ G _{0} ( a , x ^{'} ) G _{0} ( a , b ) - G _{0} ( b , x ^{'} ) G _{0} ( a , a )]}{D}

D = [1 - λ G_{0} (a, a)] [1 - μ G_{0} (b, b)] - λ μ [G_{0} (a, b)]^{2} .

D = [1 - λ G_{0} (a, a)] [1 - μ G_{0} (b, b)] - λ μ [G_{0} (a, b)]^{2} .

G (x, x^{'}) = G_{0} (x, x^{'})

G (x, x^{'}) = G_{0} (x, x^{'})

+ \frac{1}{D} {λ G_{0} (x, a) G_{0} (a, x^{'}) + μ G_{0} (x, b) G_{0} (b, x^{'})

+ \frac{1}{D} {λ G_{0} (x, a) G_{0} (a, x^{'}) + μ G_{0} (x, b) G_{0} (b, x^{'})

+ λ μ [G_{0} (x, a) (G_{0} (a, b) G_{0} (a, x^{'}) - G_{0} (b, b) G_{0} (b, x^{'}))

+ λ μ [G_{0} (x, a) (G_{0} (a, b) G_{0} (a, x^{'}) - G_{0} (b, b) G_{0} (b, x^{'}))

+ G_{0} (x, b) (G_{0} (a, b) G_{0} (a, x^{'}) - G_{0} (a, a) G_{0} (a, x^{'}))]} .

+ G_{0} (x, b) (G_{0} (a, b) G_{0} (a, x^{'}) - G_{0} (a, a) G_{0} (a, x^{'}))]} .

V (x) = j = 1 \sum N λ_{j} δ (x - a_{j})

V (x) = j = 1 \sum N λ_{j} δ (x - a_{j})

D = j = 1 \prod 3 [1 - λ_{j} G_{0} (a_{j}, a_{j})] - i < j \sum λ_{i} λ_{j} G_{0} (a_{i}, a_{j}) G_{0} (a_{j}, a_{i})

D = j = 1 \prod 3 [1 - λ_{j} G_{0} (a_{j}, a_{j})] - i < j \sum λ_{i} λ_{j} G_{0} (a_{i}, a_{j}) G_{0} (a_{j}, a_{i})

+ λ_{1} λ_{2} λ_{3} G_{0} (a_{1}, a_{2}) G (a_{2}, a_{3}) G_{0} (a_{3}, a_{1}) .

+ λ_{1} λ_{2} λ_{3} G_{0} (a_{1}, a_{2}) G (a_{2}, a_{3}) G_{0} (a_{3}, a_{1}) .

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Taxonomy

TopicsAdvanced Thermodynamics and Statistical Mechanics · Surface and Thin Film Phenomena · Quantum and electron transport phenomena

Full text

A note on the Exact Green function for a quantum system

decorated by two or more impurities.

M.L. Glasser

Dpto de Física Teórica, Facultad de Ciencias

Universidad de Valladolid

Physics Department, Clarkson University,

Potsdam, NY 13699-5820

(USA)

Abstract

The exact Green function is constructed for a quantum system, with known Green function, which is decorated by two delta function impurities.It is shown that when two such impurities coincide they behave as a single singular potential with combined amplitude. The results are extended to N impurities.

Keywords: Quantum Green function, Delta function Potential

1 Introduction

The one dimensional harmonic oscillator or square well, for example, for which the energy -dependent Green function $G_{0}(x,x^{\prime};E)$ is known, have been taken for many years as solvable models for semi-conductor quantum wells[1]. Frequently delta function potentials are placed at various points to simulate defects or impurities. In the case of a single such potential, $V(x)=\lambda\delta(x-a)$ , the Green function for the composite system is known to be[2]

[TABLE]

In this note a corresponding formula is derived for the case $V(x)=\lambda\delta(x-a)+\mu\delta(x-b).$

2 Calculation

We first note that the same argument can be used for the time dependent-, as well as the energy-dependent Green functions, so we shall omit the third argument and write simply $G(x,x^{\prime})$ .

Beginning with the Dyson equation, noting that $G_{0}(x,y)=G_{0}(y,x)$

[TABLE]

where the integration extends over the system domain, one has the set of equations

[TABLE]

The linear equations (4) annd (5) are easily solved for $G(a,x^{\prime})$ and $G(b,x^{\prime}):$

[TABLE]

with

[TABLE]

By inserting (6) and (7) into (3) we obtain the desired expression

[TABLE]

3 Discussion

By setting $\lambda$ to [math] (9) reduces to (1), proving this expression as well. The most salient feature of (9) is the denominator $D$ whose zeros form the exact spectrum of the composite system. For example, when $a$ and $b$ coincide, $D$ reduces to $1-(\lambda+\mu)G(a,a)$ and (9) reduces to (1) with $\lambda$ replaced by the amplitude $\lambda+\mu$ . I.e. the two impurities combine to form one with combined amplitude. This generalizes the result of Rinaldi and Fasssari[3], for two identical defects symmetrically placed with respect to the center of a harmonic oscillator.

Two further points can be made. Nothing in the derivation of (9) restricts it to the line. If we accept the standard definition $\delta(\vec{x})=\Pi_{j=1}^{d}\delta(x_{j})$ , then (9), and its consequences, are valid for $d$ -dimensional quantum systems. This has been proven function-theoretically for the three dimensional harmonic oscillator with two symmetrically placed identical impurities by Albeverio, Fassari and Rinaldi4].

A second observation is that $D$ is simply the Cramer determinant for the pair of simultaneous linear equations (4) and (5). In the case of impurity potential

[TABLE]

there will be $N$ such equations and the determinant is easily evaluated. Thus, for $N=3$

[TABLE]

This reduces to the $N=1$ and $N=2$ cases appropriately and shows that any two coinciding impurities coalesce as indicated above.

Finally, by letting N become infinite, the analogue of (11) might offer a new approach to Kronig-Penney-type systems for periodic or random unit cells.

Acknowledgement: The author thanks Prof. S. Fassari and Prof. L.M. Nieto for helpful comments and acknowledges the financial support of MINECO (Project MTM2014-57129-C2-1-P) and Junta de Castilla y Leon (VA057U16)).

References

[1] S. Albeverio, F. Gesztesy, R.Hoegh-Krohn and H. Holden, Solvabe Models in Quantum Mechanics, [AMS-Chelsea, Providence (2004)]

[2] M.L. Glasser and L.M. Nieto, The energy level structures of a variety of one-dimensional confining potentials and the effects of a local singular perturbation, Can. J. Physics 93, 1-9 (2015).

[3] 3] S. Fassari and F. Rinaldi, On the spectrum of the Schroedinger Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identical attractive point interactions, Reports on Mathematical Physics 69, 353-370 (2012).

[4] S.Albeverio, S. Fassari and F. Rinaldi,Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attaractive impurities symmetrically situated around the origin, Nanosystems, Physics, Chemistry and Mathematics 7,268-289 (2016).