1. Introduction

Among the classical equations of mathematical physics the heat equation , the Schrodinger equation , the Laplace equation , the wave equation and the Helmoltz equation , where is the Laplace operator in . These equations are studied with and without initial conditions by several authors since long times ^[1]. For exemple the heat, the Schrodinger and the Laplace equations are often coupled with the initial condition u(0,x) = u₀(x), the wave equation with initial conditions u(0,x) = u₀(x), u_t(0,x) = u₁(x) and the Helmoltz equation with a boundarie condition at infinity called Sommerfeld radiation condtion (^{[4, 6]} and ^[7]). The aim of this paper is first to give explicit formulas of the Schwartz kernels of the following multipliers called here respectively the weighted heat, Schrodinger, Laplace, wave, resolvent and generalized resolvent kernels on

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

(1.7)

with is the Laplacian on . Note that we can define if the function is such that is a tempered distribution (^[4], p. 149) where F is the Fourier transform (see below for a precise statement).

Recall that the classical heat, Schrodinger, Laplace, wave and resolvent Schwartz kernels are given respectively by (^[4], p. 146 et 170-171) and (^[6], p. 59).

(1.8)

(1.9)

(1.10)

(1.11)

(1.12)

(1.13)

(1.14)

(1.15)

where is the Hankel function of the first kind. The end of this section is devoted to the preliminaries on the Fourier transform on and formulas evaluating the Hankel transform of some functions.

The Fourier transform of a function is defined by the integral

(1.16)

with is the inner product on . The Fourier inverse transform is given by

(1.17)

Recall also that the Fourier transform of a radial function f on is radial and it can be written in terms of the Hankel transform (^[7], p. 226) as

(1.18)

where is the Bessel function of the first kind and order v. For more informations on the Fourier transform the reader can consults the book ^[7].

Proposition 1.1. The Schwartz kernel of the operator is given at last formally by

(1.19)

The proof of this proposition uses essentially the formula (1.18) and in consequence is left to the reader. Recall the following formulas evaluating some Hankel transform (^[3] p. 24, 29, and 30).

(1.20)

with ₁F₁(a,c,z) is the first kind conuente hypergeometric function.

(1.21)

where ₂F₁ (a,b,c;z) is the Gauss hypergeometric with

(1.22)

for a > 0, where

(1.23)

(1.24)

(1.25)

are the hypergeometric functions.

2. Weighted Heat and Schrodinger Evolution Operators on

In this section we give an exact formula for the Schwartz integral kernel of the weighted heat and Schrodinger evolution operators and on .

Theoreme 2.1. For Rep > -n, the Schwartz integral kernel of the weighted heat evolution operator on is given in terms of the first order Kummer conuent hypergeometric function ₁F₁(a, c; z) by

(2.1)

Proof. By making use of the formula (1.19) with we can write

(2.2)

using the formula (1.20) with and we get the formula (2.1).

Corollary 2.2. For Rep >-n, the Schwartz integral kernel of the weighted Schrodinger evolution operator on is given in terms of the first order Kummer conuent hypergeometric function ₁F₁(a, c; z) by

(2.3)

3. Weighted Poisson Operator on

This section is devoted to the computation of the weighted Poisson operator .

Theoreme 3.1. For Rep >-n, The Shwartz integral kernel of the weighted Poisson operator on is given in terms of the Gauss hypergeometric function ₂F₁ by

(3.1)

with ₂F₁ is the Gauss hypergeometric function.

Proof. From the formula (1.19) with we get

(3.2)

and by the formula (1.21) with and a = y one can easily deduce the formula (3.1) .

Proposition 3.2. are respectively the weighted heat and the weighted. Poisson kernels given above then we have

(3.3)

and

(3.4)

Proof. We recall the formula (^[6], p. 50)

(3.5)

By setting in (3.5) we can write for the Poisson semi-group

(3.6)

and this gives the formula (3.3). To prove the formula (3.4) we use the formula (3.3), (2.1) and (3.1).

4. Weighted Wave Evolution Operator on

In this section we shall compute explicitely the Schwartez integral kernel of the weighted wave evolution operators and

Theorem 4.1. For t > 0 we have

(4.1)

Here we should note that the integral in (4.1) can be extended over a contour starting at 1, going clockwise around 0, and returning back to 1 without cutting the real negative semi-axis.

Proof. We start by recalling the formulas (^[5], p. 73)

(4.2)

where J_v(.) is the Bessel function of first kind and of order v given by (^[5], p. 83)

(4.3)

provided that and . Moreover, we have the following formula:

(4.4)

Putting α= 1 and replacing the variable z by the symbol (4.4) we obtain the formula (4.1).

Theorme 4.2. For p > 1-n the Schwartz integral kernel of the weighted wave operator is given by the following formulas

(4.5)

and

(4.6)

where is the weighted heat kernel given in (2.1)

and ₂F₁ is the classical Gauss hypergeometric function.

Proof. The formula (4.5) is a consequence of (4.1), to prove the formula (4.6) set

(4.7)

where

(4.8)

then we have

(4.9)

(4.10)

(4.11)

and

(4.12)

where the paths γ1, γ2 and γ3 are given by

(above the cut)

(below the cut)

(rund the small circle)

as , we have and .

Adding the integrals estabilishes the following formula

(4.13)

with

(4.14)

Recalling the formula ^[5], p. 24 Reα> 0, Reα >, Rek, Rez > 0

(4.15)

with α = 1/2, a = (n + p)/2, c = n/2, z = (n + p-1)/2 and to get

(4.16)

Combining (4.7), (4.13) and (4.16) we get the formula (4.5) .

Corollary 4.3. The Schwartz integral kernel for the weighted wave evolution operator on the Euclidian space can be written on the following form

(4.17)

with

Proof. In view of the Formula we can use the formula (^[5], p. 41)

(4.18)

to obtain the formula (4.17) from the formula (4.6).

5. Weighted Resolvent Operator on

In this section we give explicit formula for the Schwartz integral kernel of the weighted resolvent operator .

Theorem 5.1. For Imλ > 0 The Schwartz integral kernel for the weighted resolvent operator is given by

(5.1)

where ₁F₂ is the hypergeometric series given in (1.25).

Proof. Using the formula (1.19) with we get

(5.2)

and by the formula (1.22) with and we get the result of the theorem.

Proposition 5.2. Let be the Schwartz kernel of the weighted resolvent operator then we have the following integral representations

(5.3)

(5.4)

(5.5)

(5.6)

where are respectively the Schwartz integral kernel of the weighted heat, Schrodinger, and wave evolution operators.

Proof. We use respectively the following formulas

(5.7)

(5.8)

(5.9)

(5.10)

Corollary 5.3. For Reλ² < 0 we have the following formula

(5.11)

Proof. This is a consequence of the proposition (5.2) formula (5.3), (5.1) and (2.1).

On can write the weighted resolvent in terms of the weighted wave kernel.

Corollary 5.4. We have

(5.12)

(5.13)

where is as in the theorem 4.2.

Proof. The proof of this corollary can be seen from the proposition 5.2 (5.5), (5.6), (4.6), (4.17) and (5.1).

6. Weighted Generalized Resolvent Operators on

In this section we generalize some results of the section 5 by give an explicit expression of the weighted generalized resolvent kernels .

Theorem 6.1. For Reλ² < 0 The Schwartz kernel of the weighted generalized resolvent operator is given by

(6.1)

where ₁F₂ is the hypergeometric function given in (1.25).

Proof. Using the formula (1.19) with we get

(6.2)

and to see the formula (6.1) we use (1.22) with and k² =-λ².

Proposition 6.2. We have the following formula connecting the weighted generalized resolvent kernel to the weighted heat kernel

(6.3)

Proof. We use the formula

(6.4)

Corollary 6.3. We have

(6.5)

Proof. We use the formulas (6.3), (2.1) and (6.1).

7. Commentaries and Applications

The subject of study of this paper is situated at the meeting point of the partial differential equations and the special functions of the mathematical physics.

Firstly explicit solutions of the following partial differential equations are given in terms of the hypergeometric functions.

and

Secondly some old and new formulas involving the hypergeometric functions are given by comparing these solutions. The formulas (5.1) and (6.1) gives the Laplace transform of Kummer hypergeometric function with argument 1/x and extend the well known formula giving the Laplace transform of the exponential with the argument 1/x.

(7.1)

The formulas (5.12) gives the Fourier transform of the Gauss hypergeometric function with argument 1/x² and extend the known formula

(7.2)

and to the best of our knowledge there is no such relation in the mathematical litterature. Note that the formulas (7.1) and (7.2) are a consequences respectively of the formula

(7.3)

(see Erdely et al ^[2], p. 83).

(7.4)

Imz > 0 and Imα²z > 0 (see Magnus et al ^[5], p. 84).

We can also derive the formulas (7.1) and (7.2) from (5.3) (1.9), (1.15) and (5.5), (1.12), (1.15).

We finish this section by the following corollary.

Corollary 7.1. We have the following formula connecting the Hankel and the hypergeomtric functions ₁F₂.

(7.5)

Proof. By comparing the formulas (1.15) and (5.1) with p = 0 we obtain (7.5).

Remark 7.2. By using the formula

(7.6)

we can deduce from the formula (7.5) the following classical formula

(7.7)

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