Heat , Resolvent and Wave Kernels with Multiple Inverse Square Potential on the Euclidian Space n

In this paper, the heat, resolvent and wave kernels associated to the Schrödinger operator with multi-inverse square potential on the Euclidian space n  are given in explicit forms.


Introduction
This article is devoted to the explicit formulas for the Schwartz integral kernels of the heat, resolvent and wave operators e t ν ∆ , ( ) , , ,  .Note that the Schrödinger operator with bi-inverse square potential in the Euclidian plane is considered in Boyer [1] and Ould Moustapha [2].
For future use we recall the following formulas for the modified Bessel function of the first kind I ν and the Hankel function of the first kind (see Temme [3], p. 237).
Proposition 1.1.The Schwartz integral kernel of the heat operator with multiple-inverse square potential e t ν ∆ can be written for ( ) , , , where j I ν is the modified Bessel function of the first kind and of order j ν .Proof.The Formula (1.12) is a direct consequence of the Formula (1.11) and the properties of the operator (1.1).

Resolvent Kernel with Multiple-Inverse Square Potential on the Euclidian
Space n  Theorem 2.1.The Schwartz integral kernel for the resolvent operator ( ) is given by the formula.
where ( ) H ν is the Hankel function of the first kind and Proof.We use the well known formula connecting the resolvent and the heat kernels: We combine the Formulas (2.2), (1.12) and (1.2) then use the Formulas (1.3) and (1.4) to appley the Fubini theorem and in view of the Formula (1.6) we get the Formula (2.1) and the proof of the Theorem 2.1 is finished.
Proof.We use the Formulas (2.1) and (1.5) as well as the Fubini theorem to arrive at the announced Formula (2.3).

Wave Kernel with Multiple-Inverse Square Potential on the Euclidian Space
It is known that the energy and information can only be transmitted with finite speed, smaller or equal to the speed of light.The mathematical framework, which allows an analysis and proof of this phenomenon, is the theory the wave equation.The result, which may be obtained, runs under the name finite propagation speed (see Cheeger et al. [8]).The following theorem illustrates the principle of the finite propagation speed in the case of the Schrödinger operator with multiple-inverse square potential.
We recall the formula [9], p. 50 where Using the Formula (1.3) we see that the last integral ( ) 2 J τ converge absolutely and is analytic in τ for For the first integral ( ) and from the Formula (1.4) we see that , hence the integral ( ) where j I ν is the first kind modified Bessel functions of order ν .Proof.We start by recalling the formula (see Magnus et al. [5], p. 73).Finally making use of (1.12) in (3.19), we get the Formula (3.14).
To see the Formula (3.15) set ( ) to the Schrödinger operator with Multiple-inverse square potential on the Euclidian space n et al.[7], p. 68).
The proof of this result use an argument of analytic continuation from the identity

′
is the heat kernel with the multiple-inverse square potential given by (1.12).
function of first kind and of order ν defined by (see Magnus et al.[5], p. 83).
and argz ≤ π .Here we should note that the integral in (3.17) can be extended over a contour starting at ∞ , going clockwise around 0, and returning back to ∞ without cutting the real negative semi-axis.For 1 2 ν = − the Equation (3.17) can be combined with Equation (3.16) to derive the following formula.