Group theoretical aspects of $L^2(\mathbb{R}^+)$, $L^2(\mathbb{R}^2)$ and associated Laguerre polynomials

TL;DR

This paper explores the algebraic structures underlying certain function spaces and associated Laguerre polynomials, revealing Lie algebra connections and representation theories relevant to mathematical physics.

Contribution

It introduces a ladder algebraic structure for $L^2( eal^+)$, constructs quadratic generators for various Lie algebras, and links these to representations in $L^2( eal^2)$.

Findings

Ladder algebraic structure for $L^2( eal^+)$ with Lie algebra closure

Construction of quadratic generators for $so(3)$, $so(2,1)$, $so(3,2)$

Unitary irreducible representations in $L^2( eal^2)$ similar to spherical harmonics

Abstract

A ladder algebraic structure for $L^2(\mathbb{R}^+)$ which closes the Lie algebra $h(1)\oplus h(1)$, where $h(1)$ is the Heisenberg-Weyl algebra, is presented in terms of a basis of associated Laguerre polynomials. Using the Schwinger method the quadratic generators that span the alternative Lie algebras $so(3)$, $so(2,1)$ and $so(3,2)$ are also constructed. These families of (pseudo) orthogonal algebras also allow to obtain unitary irreducible representations in $L^2(\mathbb{R}^2)$ similar to those of the spherical harmonics.

Equations

$$\left[x\frac{d^2}{d{x}^2}+(1+\alpha -x)\frac{d}{dx}+n\right]L_n^{(\alpha )}(x)=0.$$

$$\left[-\frac{d}{dx}+1\right]L_n^{(\alpha )}(x)=L_n^{(\alpha +1)}(x),\left[x\frac{d}{dx}+\alpha \right]L_n^{(\alpha )}(x)=(n+\alpha )L_n^{(\alpha -1)}(x).$$

$${\int }_0^{\infty }dx{x}^{\alpha }{e}^{-x}{L}_n^{(\alpha )}(x)L_{n^{\prime }}^{(\alpha )}(x)=\frac{\Gamma (n+\alpha +1)}{n!}{\delta }_{n{n}^{\prime }}.$$

$$L_n^{(-\alpha )}(x):=\frac{\Gamma (n-\alpha +1)}{\Gamma (n+1)}(-x{)}^{\alpha }{L}_{n-\alpha }^{(\alpha )}(x).$$

$$M_n^{(\alpha )}(x):=\sqrt{\frac{\Gamma (n+1)}{\Gamma (n+\alpha +1)}}{x}^{\alpha /2}{e}^{-x/2}{L}_n^{(\alpha )}(x).$$

$${\int }_0^{\infty }{M}_n^{(\alpha )}(x)M_m^{(\alpha )}(x)dx={\delta }_{nm},\sum _{n=0}^{\infty }{M}_n^{(\alpha )}(x)M_n^{(\alpha )}(x^{\prime })=\delta (x-x^{\prime }).$$

$$\begin{array}[]{lll}\displaystyle\left[-\sqrt{x}\frac{d}{dx}+\frac{1}{2\sqrt{x}}(\alpha+x)\right]\;M^{(\alpha)}_{n}(x)&=&\;\sqrt{n+\alpha+1}\;M^{(\alpha+1)}_{n}(x)\,,\\[8.5359pt] \displaystyle\left[\sqrt{x}\frac{d}{dx}+\frac{1}{2\sqrt{x}}(\alpha+x)\right]\;M^{(\alpha)}_{n}(x)&=&\;\sqrt{n+\alpha}\;M^{(\alpha-1)}_{n}(x)\,,\end{array}$$

$$\begin{array}[]{lll}\displaystyle\left[-\sqrt{x}\frac{d}{dx}+\frac{\sqrt{x}}{2}+\frac{p-n}{2\sqrt{x}}\right]\;{\cal M}_{n,p}(x)&=&\;\sqrt{p+1}\;{\cal M}_{n,p+1}(x)\,,\\[14.22636pt] \displaystyle\left[\sqrt{x}\frac{d}{dx}+\frac{\sqrt{x}}{2}+\frac{p-n}{2\sqrt{x}}\right]\;{\cal M}_{n,p}(x)&=&\;\sqrt{p}\;{\cal M}_{n,p-1}(x)\,.\end{array}$$

$$\begin{array}[]{lllll}\displaystyle X\;{\cal M}_{n,p}(x)&=&\;x\;{\cal M}_{n,p}(x)\,,\qquad D_{x}\;{\cal M}_{n,p}(x)&=&\;\frac{d{\cal M}_{n,p}(x)}{dx}\,,\\[8.5359pt] N\;{\cal M}_{n,p}(x)&=&\;n\;{\cal M}_{n,p}(x)\,,\qquad P\;{\cal M}_{n,p}(x)&=&\;p\;{\cal M}_{n,p}(x)\,.\end{array}$$

$$\mathbb{E}{\mathcal{M}}_{n,p}(x)=0,$$

$$\mathbb{E}:=X{D}_x^2+D_x+\frac{N+P+1}{2}-\frac{1}{4X}(P-N{)}^2-\frac{X}{4}.$$

$$b^{\pm }:=\mp \sqrt{X}{D}_x+\frac{\sqrt{X}}{2}+\frac{1}{2\sqrt{X}}(P-N),$$

$$a^{+}{\mathcal{M}}_{n,p}(x)=\sqrt{n+1}{\mathcal{M}}_{n+1,p}(x),a^{-}{\mathcal{M}}_{n,p}(x)=\sqrt{n}{\mathcal{M}}_{n-1,p}(x).$$

$$J_{\pm }=\mp D_x(N-P\pm 1)+\frac{1}{2X}(N-P\pm 1)(N-P)-\frac{1}{2}(N+P+1).$$

$$J_{+}{\mathcal{M}}_{n,p}(x)=\sqrt{(n+1)p}{\mathcal{M}}_{n+1,p-1}(x),J_{-}{\mathcal{M}}_{n,p}(x)=\sqrt{n(p+1)}{\mathcal{M}}_{n-1,p+1}(x).$$

$$K_{+}=X{D}_x+\frac{1}{2}(N+P+2-X),K_{-}=-X{D}_x+\frac{1}{2}(N+P-X).$$

$$[K_3,K_{\pm }]=\pm K_{\pm },[K_{+},K_{-}]=-2K_3,$$

$$K_{+}{\mathcal{M}}_{n,p}(x)=\sqrt{(n+1)(p+1)}{\mathcal{M}}_{n+1,p+1}(x),K_{-}{\mathcal{M}}_{n,p}(x)=\sqrt{np}{\mathcal{M}}_{n-1,p-1}(x).$$

$$R_{\pm }:=\pm [J_{\pm },K_{\pm }],S_{\pm }:=\pm [J_{\mp },K_{\pm }].$$

$$[R_{+},R_{-}]=-4R_3,[R_3,R_{\pm }]=\pm 2R_{\pm },$$

$${\mathcal{L}}_j^m(x):={\mathcal{M}}_{j+m,j-m}(x)=\sqrt{\frac{(j+m)!}{(j-m)!}}{x}^{-m}{e}^{-x/2}{L}_{j+m}^{(-2m)}(x).$$

$$J_3{\mathcal{L}}_j^m(x)=m{\mathcal{L}}_j^m(x),J_{\pm }{\mathcal{L}}_j^m(x)=\sqrt{(j\mp m)(j\pm m+1)}{\mathcal{L}}_j^{m\pm 1}(x).$$

$${\mathcal{Z}}_j^m(r,\varphi ):=e^{im\varphi }{\mathcal{L}}_j^m(r^2),\varphi \in \mathbb{R},-\pi \le \varphi <\pi .$$

$$\left[\frac{d^2}{d{r}^2}+\frac{1}{r}\frac{d}{dr}-\frac{4m^2}{r^2}-r^2+4(j+\frac{1}{2})\right]{\mathcal{Z}}_j^m(r,\varphi )=0.$$

$$\begin{array}[]{l}\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\,d\phi\int_{0}^{\infty}\;2r\;dr\;\;{\cal Z}^{m}_{j}(r,\phi)^{*}\,\,{\cal Z}^{m^{\prime}}_{j^{\prime}}(r,\phi)\;=\;\delta_{j,j^{\prime}}\;\;\delta_{m,m^{\prime}}\;,\\[14.22636pt] \displaystyle\sum_{j,m}\;\;{\cal Z}^{m}_{j}(r,\phi)^{*}\,\,{\cal Z}^{m}_{j}(r^{\prime},\phi^{\prime})\;=\;\frac{\pi}{r}\,\delta(r-r^{\prime})\;\delta(\phi-\phi^{\prime})\,.\end{array}$$

$${\mathbb{J}}_{\pm }{\mathcal{Z}}_j^m(r,\varphi )=\sqrt{(j\mp m)(j\pm m+1)}{\mathcal{Z}}_j^{m\pm 1}(r,\varphi ),{\mathbb{J}}_3{\mathcal{Z}}_j^m(r,\varphi )=m{\mathcal{Z}}_j^m(r,\varphi ),$$

Full text

$so(2,1)$ and $so(3,2)$ are also constructed. These families of (pseudo) orthogonal algebras also allow to obtain unitary irreducible representations in $L^{2}(\mathbb{R}^{2})$ similar to those of the spherical harmonics.

1 Introduction

The associated Laguerre polynomials (ALP) [1], $L_{n}^{(\alpha)}(x)$ $\left(x\in[0,\infty\right)$, $n=0,1,2,\cdots$ and $\alpha$ real fixed parameter, continuous and $>-1$), are defined by the 2nd order differential equation (DE)

[TABLE]

The ALPs reduce to the Laguerre polynomials for  $\alpha=0$. From the many recurrence relations that they verify [1, 2, 3], we start from the following ones

[TABLE]

For $\alpha>-1$ and fixed, the ALP $L_{n}^{(\alpha)}(r)$ are orthogonal in the label $n$ with respect the weight measure $d\mu(x)=x^{\alpha}\,e^{-x}\,dx$

[TABLE]

For $\alpha$ integer such that   $0\leq\alpha\leq n$ , we have the generalization [1]

[TABLE]

Hereafter we assume here $n\in\mathbb{N},\,\,\alpha\in\mathbb{Z}\,,\,\,n-\alpha\in\mathbb{N}$ and we consider  $\alpha$  as a label, like $n$, and not a parameter fixed at the beginning.

Following the approach of previous works [4, 5, 6, 7] we introduce now a set of alternative functions including also the weight measure, in such a way to obtain the orthonormal bases we are used to in Quantum Mechanics

[TABLE]

For each fixed value of $\alpha\geq-n$ and $n\in\mathbb{N}$, the set of $M_{n}^{(\alpha)}(x)$, is a basis of $L^{2}(\mathbb{R}^{+})$

[TABLE]

2 The symmetry algebra

$h(1)_{n}\oplus h(1)_{p}$

The eqs. (2) rewritten in terms of $M_{n}^{(\alpha)}$ take the form

[TABLE]

where $p:=n+\alpha$ plays, for $n$ fixed, the role of eigenvalue of the number operator in a Heisenberg-Weyl algebra, $h(1)$, realized on the space of functions $M^{\alpha}_{n}(x)$. It is indeed a positive integer like $n$, so that we can define the new functions ${{\cal M}_{n,p}(x):=M_{n}^{(p-n)}(x)}$, that by inspection are symmetric in the interchange $n\Leftrightarrow p$, i.e. ${\cal M}_{n,p}(x)\,=\,(-1)^{p-n}\,{\cal M}_{p,n}(x)\,.$ The previous recurrence relations (3) can thus be rewritten

[TABLE]

To construct the operatorial structure corresponding to the recurrence relations we define now four operators $X$, $D_{x}$, $N$ and $P$

[TABLE]

Then, the 2nd order DE (1) becomes

[TABLE]

where

[TABLE]

Moreover from (4) we get the differential operators (DOs)

[TABLE]

that act on the functions ${\cal M}_{n,p}(x)$ in such a way that $\Delta n=0$ and $\Delta p=\pm 1$. Since $[{\mathbb{b}}^{-},{\mathbb{b}}^{+}]=\mathbb{I}\,$ they close an $h(1)$ algebra, ($h(1)_{p}$) with quadratic Casimir ${\cal C}_{p}=\{{\mathbb{b}}^{-},{\mathbb{b}}^{+}\}-2(P+1/2)$ verifying ${\cal C}_{p}\,{\cal M}_{n,p}(x)=-2\,\mathbb{E}\,{\cal M}_{n,p}(x)=0\,.$

Now taking into account the symmetry under the interchange $n\Leftrightarrow p$ of ${\cal M}_{n,p}(x)$ we can define the operators ${\mathbb{a}}^{\pm}(N,P):=-{\mathbb{b}}^{\pm}(P,N)$ that change the labels of ${\cal M}_{n,p}(x)$ as $\Delta p=0$ and $\Delta n=\pm 1$. Their explicit action on ${\cal M}_{n,p}(x)$ is indeed

[TABLE]

The two operators ${\mathbb{a}}^{\pm}$ determine thus another HW algebra, $h(1)_{n}$. Since these bosonic operators ${\mathbb{a}}^{\pm}$ and ${\mathbb{b}}^{\pm}$ commute among them we have obtained in this way the global algebra $h(1)_{n}\oplus h(1)_{p}$.

Moreover inside the Universal Enveloping Algebra $UEA\left[h(1)_{n}\oplus h(1)_{p}\right]$ other algebras preserving the parity of $n+p$ can be found by the Schwinger procedure [8] as we will do in the next section.

3 $so(3)$, $so(2,1)$ and $so(3,2)$ symmetries

$so(3)$ symmetry

We start from $J_{\pm}:={\mathbb{a}}_{\pm}\,{\mathbb{b}}_{\mp}$ obtaining 2nd order DOs that, taking into account eq. (5), can be rewritten in the space $\{{\cal M}_{n,p}(x)\}$ as 1st order DOs

[TABLE]

Defining $J_{3}:=({\mathbb{a}}_{-}\,{\mathbb{a}}_{+}-{\mathbb{b}}_{-}\,{\mathbb{b}}_{+})/2\;\equiv\;{(N-P)}/{2}$ we see that $\{J_{\pm},J_{3}\}$ close a $su(2)$ algebra in the space $\{{\cal M}_{n,p}(x)\}$ since $[J_{+},J_{-}]=2J_{3}-\frac{8}{X}\,J_{3}\,\mathbb{E}.$ The action of $J_{\pm}$ is

[TABLE]

Also the Casimir of $su(2)$, ${\cal{C}}_{su(2)}=J_{3}^{2}+\frac{1}{2}\{J_{+},J_{-}\}$ is closely related to eq. (5) as ${\cal{C}}_{su(2)}=J(J+1)+\frac{1}{X}\,(4J_{3}^{2}+1)\,\mathbb{E}$, where $J$ is the diagonal operator $J:=(N+P)/2$.

$so(2,1)$ symmetry

In a similar way we can define the operators $K_{\pm}:={\mathbb{a}}_{\pm}\,{\mathbb{b}}_{\pm}$, such that, like in the case of the operators $J_{\pm}$, we find in the space $\{{\cal M}_{n,p}(x)\}$

[TABLE]

Both operators together with $K_{3}:=({\mathbb{a}}_{-}\,{\mathbb{a}}_{+}+{\mathbb{b}}_{+}\,{\mathbb{b}}_{-})/2\;\equiv\;{(N+P+1)}/{2}$ determine a $su(1,1)$ algebra

[TABLE]

since the action on the functions ${\cal M}_{n,p}(x)$ is

[TABLE]

The Casimir of $su(1,1)$, ${\cal{C}}_{su(1,1)}=K_{3}^{2}-\frac{1}{2}\{K_{+},K_{-}\}$, is also connected with eq. (5) as ${\cal{C}}_{su(1,1)}=(M^{2}-\frac{1}{4})+{X}\,\mathbb{E}$, where $M=J_{3}:={(N-P)}/{2}$.

More $so(2,1)$ symmetries

The commutators of $J_{\pm}$ and $K_{\pm}$ give the new operators

[TABLE]

Provided that we define $R_{3}:=J+M+1/2$ and $S_{3}:=J-M+1/2$, they close two $so(2,1)$ algebras with commutators

[TABLE]

and Casimir ${\cal C}_{R}=R_{3}^{2}-\frac{1}{2}\{R_{+},R_{-}\}\,=\,-\frac{3}{4}+\frac{1}{X}\,(1+(X+2M)^{2})\,\mathbb{E}$ and similarly for $\{S_{\pm},S_{3}\}$. Note that under the interchange $m\leftrightarrow-m$ we have $\{R_{\pm},R_{3}\}\leftrightarrow\{S_{\pm},S_{3}\}$.

$so(3,2)$ symmetry

All the operators $\{K_{\pm},L_{\pm},R_{\pm},S_{\pm},J,M\}$ can be written on the space $\{{\cal M}_{n,p}(x)\}$ as 1st order DOs. All together they determine on $\{{\cal M}_{n,p}(x)\}$ the representation of the Lie algebra $so(3,2)$ with $C^{so(3,2)}_{2}=-5/4$ .

4 Representations of $so(3)$, $so(2,1)$ and $so(3,2)$ on the plane

We introduce now the operators directly related to $so(3)$, $J:=(N+P)/2$ and $J_{3}\equiv M:=(N-P)/2$, and define

[TABLE]

The operators $J_{3}$ and $J_{\pm}$ (7), rewritten in terms of $J$ and $M$, act on $\{{\cal L}^{m}_{j}(x)\}$ as

[TABLE]

So, $\{{\cal L}^{m}_{j}(x)\}$ with $j\in\mathbb{N}$ and $|m|\leq j$ supports the representation ${\cal D}_{j}$ of $so(3)$.

Similar results can be obtained for the other algebras $so(2,1)$ and $so(3,2)$. For instance, for the $so(2,1)$ spanned by $\{K_{\pm},K_{3}\}$, $\{{\cal L}^{m}_{j}(x)\}$ supports the irreducible representation of the discrete series with Casimir ${\cal{C}}_{su(1,1)}:=m^{2}-\frac{1}{4}$ with $m$ fixed and $j\geq|m|$.

On the other hand, in general these representations are not faithful because ${\cal L}_{j}^{m}(x)={\cal L}_{j}^{-m}(x)$. The same difficulty is also present in the spherical harmonic where the associated Legendre polynomial ${P}_{l}^{m}$ is related to ${P}_{l}^{-m}$. There the degeneration was removed by introducing an angle variable. Here we follow the same procedure by considering the new functions

[TABLE]

Under the change of variable $x\to r^{2}$ the DE (5) becomes

[TABLE]

Normalization and orthogonality of the ${\cal Z}_{j}^{m}(r,\phi)$ are similar to the ones of $Y_{j}^{m}(\theta,\phi)$

[TABLE]

This means that the set $\{{\cal Z}_{j}^{m}(r,\phi)\}$ is a basis in the space of square integrable functions defined on the plane, $L^{2}(\mathbb{R}^{2})$, like $\{Y_{j}^{m}(\Omega)\}$ is a basis of $L^{2}(\mathbb{S}^{2})$.

Moreover, with a convenient introduction of phases we can define the operators $\mathbb{J}_{\pm}:=e^{\pm i\phi}\,J_{\pm}$ and $\mathbb{J}_{3}:=J_{3},$ in the finite dimensional space $\{{\cal Z}_{j}^{m}(r,\phi)\}$ wih fixed $j$

[TABLE]

and analogously for the remaining operators. So $\{{\cal Z}_{j}^{m}(r,\phi)\}$ support irreducible representations of $so(3)$, $so(2,1)$ and $so(3,2)$ on the plane as $\{Y_{j}^{m}(\theta,\phi)\}$ are on the sphere. For more details see [7, 9, 10].

From the physical point of view, in spite of the analogy with the angular momentum, $\mathbb{J}_{\pm}$ and $\mathbb{J}_{3}$ can be related to a one-dimensional Morse system, where $m$ and $j$ are connected with the potential [9].

Conclusions

A relationship between Lie algebras and square integrable functions has been found. Indeed we need to restrict ourselves to ${L}^{2}(\mathbb{R}^{+})$ and ${L}^{2}(\mathbb{R}^{2})$, where $\mathbb{E}$ is identically zero, to obtain differential representations of Lie algebras in the spaces of functions defined in $\mathbb{R}^{+}$ and $\mathbb{R}^{2}$ .

Acknowledgements

This work was partially supported by the Ministerio de Economía y Competitividad of Spain (Project MTM2014-57129-C2-1-P with EU-FEDER support).