Multipliers with inverse square potential and applications I

TL;DR

This paper derives explicit formulas for Schwartz integral kernels of certain Schrödinger multipliers with inverse square potential, utilizing integral transforms and special functions like Bessel and hypergeometric functions.

Contribution

It provides new explicit formulas for these kernels, connecting classical special functions with Schrödinger operators with inverse square potential.

Findings

Explicit formulas for Schwartz integral kernels derived

Connections established between integral transforms and special functions

New formulas involving Bessel and hypergeometric functions obtained

Abstract

In this work we give explicit formulas for the Schwartz integral kernels of some multipliers of the Schr\"odinger operator with inverse square potential on $\R^\ast_+$. By using the integral transforms connecting these multipliers we obtain old and new formulas involving Bessel and hypergeometric functions

Equations

$$L_{\nu }=\frac{{\partial }^2}{\partial x^2}+\frac{1/4-{\nu }^2}{x^2},\nu \in IR.$$

$$H_{\nu }^0(t,x,x^{\prime })=\frac{(x{x}^{\prime }{)}^{1/2}}{2t}{e}^{\frac{-(x^2+x^{\prime 2})}{4t}}{I}_{\nu }(\frac{x{x}^{\prime }}{2t}).$$

$$R_{\nu}^{0}(\lambda,x,x^{\prime})=\frac{i\pi}{2}\sqrt{xx^{\prime}}\left\{\begin{array}[]{cc}J_{\nu}(\lambda x)H^{(1)}_{\nu}(\lambda x^{\prime})&\mbox{$x<x^{\prime}$}\\ J_{\nu}(\lambda x^{\prime})H^{(1)}_{\nu}(\lambda x)&\mbox{$x>x^{\prime}$}\end{array}\right.$$

$$\int_{0}^{\infty}e^{\lambda^{2}t}t^{-1}e^{\frac{-(x^{2}+x^{\prime 2})}{4t}}I_{\nu}(\frac{xx^{\prime}}{2t})dt=i\pi\left\{\begin{array}[]{cc}J_{\nu}(\lambda x)H^{(1)}_{\nu}(\lambda x^{\prime})&\mbox{$x<x^{\prime}$}\\ J_{\nu}(\lambda x^{\prime})H^{(1)}_{\nu}(\lambda x)&\mbox{$x>x^{\prime}$}\end{array}\right.$$

$$I_{\nu }(x)\approx \frac{x^{\nu }}{2^{\nu }\Gamma (1+\nu )},x\to 0;I_{\nu }(x)\approx \frac{e^x}{\sqrt{2\pi x}},x\to \infty .$$

$$\left(H_{\nu }f\right)(\omega )={\int }_0^{\infty }(x\omega {)}^{1/2}{J}_{\nu }(x\omega )f(x)dx$$

$$K_{\nu }(\varphi ,x,x^{\prime })=(x{x}^{\prime }{)}^{1/2}{\int }_0^{\infty }{J}_{\nu }(\omega x)J_{\nu }(\omega x^{\prime })\varphi (\omega )\omega d\omega .$$

$$\int_{0}^{\infty}\frac{J_{\nu}(x\omega)J_{\nu}(x^{\prime}\omega)}{-\omega^{2}+\lambda^{2}}\omega d\omega=\frac{i\pi}{2}\left\{\begin{array}[]{cc}J_{\nu}(\lambda x)H^{(1)}_{\nu}(\lambda x^{\prime})&x<x^{\prime}\\ J_{\nu}(\lambda x^{\prime})H^{(1)}_{\nu}(\lambda x)&x>x^{\prime}\end{array}\right.$$

$$J_{\nu }(x)\approx \frac{x^{\nu }}{2^{\nu }\Gamma (1+\nu )},x\to 0,J_{\nu }(x)\approx \sqrt{\frac{2}{\pi x}},x\to \infty .$$

$${\Psi }_2\left(a;c,c^{\prime },x,y\right)=\sum _{n;m\ge 0}\frac{(a{)}_{n+m}}{(c{)}_m(c^{\prime }{)}_{n}m!n!}{x}^m{y}^n,\text{hfil}|x|<\infty ,|y|<\infty .$$

$$F_{C:D}^{A:B}\left({}_{c_1,...,c_C:d_1,...d_D;d_1^{\prime },...d_D^{\prime }}^{a_1,...a_A:b_1,...b_B;b_1^{\prime },...b_B^{\prime }};x,x^{\prime }\right)=$$

$$\sum _{n,m\ge 0}\frac{{\prod }_{j=1}^A(a_j{)}_{m+n}{\prod }_{j=1}^B(b_j{)}_m{\prod }_{j=1}^B(b_j^{\prime }{)}_n}{{\prod }_{j=1}^C(c_j{)}_{m+n}{\prod }_{j=1}^D(d_j{)}_m{\prod }_{j=1}^D(d_j^{\prime }{)}_{n}m!n!}{x}^m{x}^{\prime n}.$$

$${\int }_0^{\infty }{e}^{-\gamma t}{t}^{\alpha -1}{\Psi }_2\left(a,b_1,b_2,\frac{X}{t},\frac{Y}{t}\right)dt=$$

$$\frac{\Gamma (\alpha )}{{\gamma }^{\alpha }}{F}_{1:1}^{1:0}\left({}_{1-\alpha ,b_1,b_2}^{a:-,-};-\gamma X,-\gamma Y\right).$$

$${\int }_0^{\infty }{e}^{-\gamma t}{t}^{\alpha -1}{\Psi }_2\left(a,b_1,b_2,\frac{X}{t},\frac{Y}{t}\right)dt={\gamma }^{-\alpha }\sum _{n,m\ge 0}\frac{(a{)}_{n+m}\Gamma (\alpha -n-m)}{(b_1{)}_m(b_2{)}_{n}m!n!}(\gamma x{)}^m(\gamma y{)}^n.$$

$${\int }_0^{\infty }{e}^{-\gamma t}{t}^{\alpha -1}{\Psi }_2\left(a,b_1,b_2,\frac{X}{t},\frac{Y}{t}\right)dt=$$

$$\frac{\Gamma (\alpha )}{{\gamma }^{\alpha }}\sum _{n,m\ge 0}\frac{(a{)}_{n+m}}{(1-\alpha {)}_{n+m}(b_1{)}_m(b_2{)}_{n}m!n!}(-\gamma x{)}^m(-\gamma y{)}^n.$$

$$H_{\nu }^p(t,x,x^{\prime })=\frac{\Gamma (p/2+1+\nu )(x/2{)}^{\nu +1/2}(x^{\prime }/2{)}^{\nu +1/2}}{[\Gamma (\nu +1){]}^2{t}^{p/2+\nu +1}}\times$$

$${\Psi }_2\left(p/2+1+\nu ,\nu +1,\nu +1;x^2/4t,x^{\prime 2}/4t\right)$$

$$H_{\nu }^p(t,x,x^{\prime })=(x{x}^{\prime }{)}^{1/2}{\int }_0^{\infty }{J}_{\nu }(\omega x)J_{\nu }(\omega x^{\prime })e^{-t{\omega }^2}{\omega }^{p+1}d\omega .$$

$${\int }_0^{\infty }{e}^{-pt}{t}^{\nu -1}{J}_{2{\mu }_1}(2(a_{1}t{)}^{1/2})J_{2{\mu }_2}(2(a_{2}t{)}^{1/2})dt=$$

$$\frac{\Gamma (\nu +M)}{\Gamma (2{\mu }_1+1)\Gamma (2{\mu }_2+1)}{p}^{-\nu -M}{a}_1^{{\mu }_1}{a}_2^{{\mu }_2}{\Psi }_2\left(\nu +M,2{\mu }_1+1,2{\mu }_2+1,a_1/p,a_2/p\right).$$

$$K_{\nu }^p(t,x,x^{\prime })=\frac{\Gamma (p/2+1+\nu )(x/2{)}^{\nu +1/2}(x^{\prime }/2{)}^{\nu +1/2}}{[\Gamma (\nu +1){]}^2(it{)}^{p/2+\nu +1}}times$$

$${\Psi }_2\left(p/2+1+\nu ,\nu +1,\nu +1;x^2/4it,x^{\prime 2}/4it\right).$$

$$H_{\nu }^0(t,x,x^{\prime })=\frac{(x/2{)}^{\nu +1/2}(x^{\prime }/2{)}^{\nu +1/2}}{\Gamma (\nu +1)t^{\nu +1}}\times$$

$${\Psi }_2\left(\nu +1,\nu +1,\nu +1;x^2/4t,x^{\prime 2}/4t\right).$$

$${\Psi }_2\left(\nu +1,\nu +1,\nu +1;x,y\right)=\Gamma (\nu +1)(xy{)}^{-\nu /2}{e}^{-x-y}{I}_{\nu }(2\sqrt{xy}).$$

$$R_{\nu }^p(\lambda ,x,x^{\prime })={\int }_0^{\infty }{e}^{{\lambda }^{2}t}{H}_{\nu }^p(t,x,x^{\prime })dt,\mathcal{R}e{\lambda }^2<0.$$

$$(-{\lambda }^2{)}^{p/2+\nu }(x{x}^{\prime }/4{)}^{\nu +1/2}{F}_{0:1}^{0:0}\left({}_{-:\nu +1;\nu +1}^{-:-;-};\frac{{\lambda }^2{x}^2}{4},\frac{{\lambda }^2{x}^{\prime 2}}{4}\right).$$

$$(-{\lambda }^2{)}^{p/2+\nu }(x{x}^{\prime }/4{)}^{\nu }{F}_{0:1}^{0:0}\left({}_{-:\nu +1;\nu +1}^{-:-;-};\frac{{\lambda }^2{x}^2}{4},\frac{{\lambda }^2{x}^{\prime 2}}{4}\right)$$

$$F_{0:1}^{0:0}\left({}_{-:\nu +1;\nu +1}^{-:-;-};\frac{{\lambda }^2{x}^2}{4},\frac{{\lambda }^2{x}^{\prime 2}}{4}\right).$$

$$R_{\nu }^{\mu ,p}(\lambda ,x,x^{\prime })=\frac{1}{\Gamma (\mu +1)}{\int }_0^{\infty }{e}^{{\lambda }^{2}t}{t}^{\mu }{H}_{\nu }^p(t,x,x^{\prime })dt;\mathcal{R}e{\lambda }^2<0.$$

$$R_{\nu }^{\mu ,p}(\lambda ,x,x^{\prime })=\frac{\Gamma (\mu -p/2-\nu )[\Gamma (p/2+\nu +1)]}{\Gamma (\mu +1)[\Gamma (\nu +1){]}^2}\times$$

$$(-{\lambda }^2{)}^{p/2+\nu -\mu }(x{x}^{\prime }/4{)}^{\nu +1/2}{F}_{1:1}^{1:0}\left({}_{p/2+\nu +1-\mu :\nu +1;\nu +1}^{p/2+\nu +1:-;-};\frac{{\lambda }^2{x}^2}{4},\frac{{\lambda }^2{x}^{\prime 2}}{4}\right)$$

Full text

**Key words **: Bessel operator, Multipliers, Weighted heat kernel, Weighted resolvent kernels, Bessel functions, Kampe de Feriét generalized hypergeometric functions.

1 Introduction

The Bessel operator is an interesting operator which arises in several contexts, one of them being the Schrödinger equation in non relativistic quantum mechanics [13]. For the recent papers on the Bessel operator see([3, 9, 9, 15]). The aim of this paper is twofold: first we give explicit formulas for the Schwartz integral kernels of the following multipliers $H_{\nu}^{p}(t):=e^{tL_{\nu}}(\sqrt{-L_{\nu}})^{p},R_{\nu}^{p}(\lambda):=\left(L_{\nu}+\lambda^{2}\right)^{-1}(\sqrt{-L_{\nu}})^{p},R_{\nu}^{\mu,p}(\lambda):=\left(L_{\nu}+\lambda^{2}\right)^{-1-\mu}(\sqrt{-L_{\nu}})^{p}$ called here respectively the weighted heat, weighted resolvent and weighted generalised resolvent operator associated to the Bessel operator on $I\!\!R^{\ast}_{+}$:

[TABLE]

and secondly, by using the integral transforms connecting these multipliers we obtain old and new formulas involving Bessel and hypergeometric functions. For finding formulas involving Bessel and hypergeometric functions using multipliers on the Euclidian space $I\!\!R^{n}$ see[11].

First of all we recall the following formulas for the classical heat kernel for the Bessel operator ([1], p. 68):

[TABLE]

and the resolvent kernel [9]

[TABLE]

Using the fact that the resolvent kernel is the Laplace transform of the heat kernel we have for ${\cal R}e\lambda^{2}<0$ and $\nu>-1$:

[TABLE]

where $I_{\nu}$ is the first kind modified Bessel function, $J_{\nu}$ and $H^{(1)}_{\nu}$ are respectively the first and the third kind Bessel functions ( see[5, 10] ).

We mention that the absolute convergence of the above integral is assured by the formulas ([10], p. 136)

[TABLE]

The end of this section is devoted to the preliminaries on the Hankel transform on $I\!\!R^{+}$.

For $\nu>-1$, the Hankel transform of order $\nu$ for a function $f\in C^{\infty}_{0}(I\!\!R^{+})$,is defined by the integral

[TABLE]

where $J_{\nu}$ is the first order Bessel function of order $\nu$.

Proposition 1.1**.**

[12*]** For $\nu>-1$, we have

$i)H_{\nu}^{2}=1$

$ii)H_{\nu}$ is self adjoint

$iii)H_{\nu}$ is an $L^{2}$ isometry

$iv)H_{\nu}L_{\nu}=-\omega^{2}H_{\nu}$.

For more informations on the Hankel transform the reader can consults the nice book by Davies [2].

Note that we can define $\phi(\sqrt{-L_{\nu}})$ for $\phi$ a well behaved Borel function by using the Hankel transform:

Proposition 1.2**.**

For $\nu>-1$, the Schwartz integral kernel of the operator $\phi(\sqrt{-L_{\nu}})$, is given at last formally by

[TABLE]

The proof of this proposition uses essentially Proposition 1.1 and in consequence is left to the reader.

Note that using (1.7) with $\phi(x)=\frac{1}{-\omega^{2}+\lambda^{2}}$ we obtain for ${\cal R}e\lambda^{2}<0$ and $\nu>-1$ the following formula

[TABLE]

The absolute convergence of the above integral is assured by the formulas ([10], p.134)

[TABLE]

The following lemma gives the Laplace transform of the two variables Humbert confluent hypergeometric function ([5], p.225):

[TABLE]

in term of the Kampé de Feriét generalized hypergeometric function $F^{A:B}_{C:D}$ given by EXTON ([7], p.29).

[TABLE]

with $A+B<C+D+1$, $|x|<\infty$ and $|x^{\prime}|<\infty$.

Lemma 1.1**.**

For $\gamma>0$, $\alpha>0$, $X,Y\in I\!\!R$ we have

[TABLE]

Proof.

Replacing the confluent hypergeometric $\Psi_{2}$ by its series (1.10) in the integral (1.1) and integrating term by term we obtain

[TABLE]

using the formula ([14], p. 22) $\frac{\Gamma(\alpha-n)}{\Gamma(\alpha)}=\frac{(-1)^{n}}{(1-\alpha)_{n}}$ we can write

[TABLE]

which gives the result in (1.1) and the proof of Lemma 1.1 is finished. ∎

The organization of the remaining of the paper is as follows, the Schwartz integral kernel of the weighted heat evolution operator $e^{tL_{\nu}}(\sqrt{-L_{\nu}})^{p}$ and the weighted Schrödinger evolution operator $e^{itL_{\nu}}(\sqrt{-L_{\nu}})^{p}$ will be given in section 2. In section 3 we will obtain a closed form of the Schwartz integral kernel of the weighted resolvent operator. The section 4 is devoted to the Schwartz integral kernel of the weighted generalized resolvent operator on $I\!\!R^{+}$.

2 Weighted Heat evolution operator for Bessel operator on $I\!\!R^{+}$

In this section we give the Schwartz integral kernels of the weighted heat and Schrödinger evolution operators $e^{tL_{\nu}}(\sqrt{-L_{\nu}})^{p}$ and $e^{itL_{\nu}}(\sqrt{-L_{\nu}})^{p}$ in explicit forms.

Theorem 2.1**.**

*For ${\cal R}ep>-2(\nu+1)$ and $\nu>-1$, the Schwartz integral kernel $H_{\nu}^{p}(t,x,x^{\prime})$ of the weighted heat evolution operator $e^{tL_{\nu}}(\sqrt{-L_{\nu}})^{p}$ is given as:

[TABLE]

The function $\Psi_{2}\left(a,c,c^{\prime};x;y\right)$ denotes the Humbert’s confluent hypergeometric function of two variables given in (1.10).

Proof.

Using the formula (1.7) with $\phi(\omega)=e^{-t\omega^{2}}\omega^{p}$ we have

[TABLE]

Next we employ the formula, $M=\mu_{1}+\mu_{2}$, ${\cal R}e(\nu+M)>0$,( [6] p.187 )

[TABLE]

and we arrive at the formula (2.1). ∎

Corollary 2.1**.**

*The Schwartz integral kernel $K_{\nu}^{p}(t,x,x^{\prime})$ of the weighted Schrödinger evolution operator with inverse square potential $e^{itL_{\nu}}(\sqrt{-L_{\nu}})^{p}$ is given in terms of the two variables Humbert’s confluent hypergeometric function for ${\cal R}ep>-2(\nu+1)$ and $\nu>-1$ as

[TABLE]

By taking $p=0$ in Theorem 2.1 we have

[TABLE]

and by comparing this with (1.3) we have

[TABLE]

3 Weighted resolvent operator for the Bessel operator on $I\!\!R^{+}$

In this section we give explicit formula for the Schwartz integral kernel of the weighted resolvent operator $R_{\nu}^{p}(\lambda)=\left(L_{\nu}+\lambda^{2}\right)^{-1}(-L_{\nu})^{p/2}$ using the formula

[TABLE]

where $H_{\nu}^{p}(t,x,x^{\prime}$) is the Schwartz integral kernels of the weighted heat operator.

The Formula (3.1) is a consequence of the formula $(a^{2}+y^{2})^{-1}=\int_{0}^{\infty}e^{-(a^{2}+y^{2})t}\,dt$ valid for ${\cal R}ea^{2}>0$.

Theorem 3.1**.**

*For ${\cal R}e\lambda^{2}<0$, $-1<p/2+\nu<0$ and $\nu>-1$,

the Schwartz integral kernel for the weighted resolvent operator$\left(L_{\nu}+\lambda^{2}\right)^{-1}(-L_{\nu})^{p/2}$ is given by

$R_{\nu}^{p}(\lambda,x,x^{\prime})=\frac{[\Gamma(p/2+\nu+1)\Gamma(-p/2-\nu)]}{[\Gamma(\nu+1)]^{2}}\times$*

[TABLE]

where $F^{0:0}_{0:1}$ is the Kampe de Firiet hypergeometric function given in (Finding formulas connecting Bessel and hypergeometric functions using multipliers of Bessel operator I).

The proof of this theorem can be seen as is a direct application of Proposition 3.1, Theorem 2.1 and of Lemma 1.1.

Corollary 3.1**.**

*For ${\cal R}e\lambda^{2}<0$, $-1<p/2+\nu<0$ and $\nu>-1$

$\int_{0}^{\infty}\frac{J_{\nu}(x\omega)J_{\nu}(x^{\prime}\omega)}{-\omega^{2}+\lambda^{2}}\omega^{p+1}d\omega=\frac{\Gamma(p/2+\nu+1)\Gamma(-p/2-\nu)}{2[\Gamma(\nu+1)]^{2}}\times$*

[TABLE]

Proof.

Using Proposition 1.2 with $\phi(\omega)=(-\omega^{2}+\lambda^{2})^{-1}\omega^{p}$ and Theorem 3.1,we obtain the result, where the absolute convergence of the above integral is assured by the formulas (1.9). ∎

Note that by taking $p=0$ in (3.2) and comparing with (1.3) the following formula is valid for $x<x^{\prime}$

$J_{\nu}(\lambda x)H^{(1)}_{\nu}(\lambda x^{\prime})=\frac{\Gamma(-\nu)(-\lambda^{2}xx^{\prime}/4)^{-\nu}}{i\pi\Gamma(\nu+1)}\times$

[TABLE]

4 Weighted generalized resolvent operator for the Bessel operator on $I\!\!R^{+}$

In this section we generalize some results of the section $3$ by giving an explicit expression of the weighted generalized resolvent kernels $R_{\nu}^{\mu,p}(\lambda)=\left(L_{\nu}+\lambda^{2}\right)^{-1-\mu}(-L_{\nu})^{p/2}$.

Proposition 4.1**.**

We have the following formula connecting the weighted generalized resolvent kernel to the weighted heat kernel

[TABLE]

Proof.

We use the formula $(a^{2}+y^{2})^{-1-\mu}=\frac{1}{\Gamma(\mu+1)}\int_{0}^{\infty}e^{-(a^{2}+y^{2})t}t^{\mu}\,dt$ for ${\cal R}ea^{2}>0$ ∎

Theorem 4.1**.**

*For ${\cal R}e\lambda^{2}<0$, $-1<p/2+\nu<\mu$, $\nu>-1$ and $\mu>-1$, the Schwartz integral kernel of the weighted generalized resolvent kernel with inverse square potential is given by

[TABLE]

where $F^{A:B}_{C:D}$ is the Kampe de Feriét generalized hypergeometric function given by(Finding formulas connecting Bessel and hypergeometric functions using multipliers of Bessel operator I).

Proof.

This theorem is a direct consequence of Proposition 4.1, Theorem 2.1 and Lemma 1.1. ∎

Corollary 4.1**.**

*For ${\cal R}e\lambda^{2}<0$, $-1<p/2+\nu<\mu$, $\nu>-1$ and $\mu>-1$, we have the following formula

[TABLE]

Proof.

Using Proposition 1.2 with $\phi(\omega)=(-\omega^{2}+\lambda^{2})^{-1-\mu}\omega^{p}$ and Theorem 4.1. Note that the absolute convergence of the integral is assured by the formulas (1.9) ∎

By taking $p=0$ in the formula (4.1), we see that the Schwartz integral kernel of generalized resolvent with inverse square potential is given by

[TABLE]

where ${\cal R}e\lambda^{2}<0$, $\nu>-1$ and $\mu>-1$ and $\mu>\nu$.

By taking $p+1=\alpha-1$ and $\mu+1=\rho$ we have

$\int_{0}^{\infty}x^{\alpha-1}\frac{J_{\nu}(cx)J_{\nu}(cx)}{(x^{2}+z^{2})^{\rho}}dx=(-1)^{\rho}\frac{\Gamma(\rho-\alpha/2-\nu)\Gamma(\alpha/2+\nu)}{\Gamma(\rho)[\Gamma(\nu+1)]^{2}}$

[TABLE]

Using the formulas [8] p. 672-673 we obtain

[TABLE]

$=\frac{1}{2}(\frac{c}{2})^{2\rho-\alpha}\Gamma\left[{}^{\nu+\alpha/2-\rho,1+2\rho-\alpha,\rho,\nu+1,\nu+1}_{-\alpha/2+\rho+1,\nu-\alpha/2+\rho+1,-\alpha/2+\rho+1,\rho-\alpha/2-\nu,\alpha/2+\nu}\right]\times$

${}_{2}F_{3}((1-\alpha)/2+\rho,\rho;\rho+1-\nu-\alpha/2,\rho+1-\alpha/2;c^{2}z^{2})+\\ \frac{z^{\alpha-2\rho}}{2}(\frac{cz}{2})^{2\nu}{}_{2}F_{3}(1/2+\nu,\alpha/2+\nu;1-\rho+\alpha/2+\nu,\nu+1,2\nu+1,c^{2}z^{2}).$

where the hypergeometric function

[TABLE]