On the Angular Density of Three Dimensional Scattering Resonances

which is of the form I + A(λ), where A(λ) is of trace class depending meromorphically on λ ∈ C. The poles are called the scattering resonances which share the same multiplicity at each pole as resolvent resonances. It is a subject of great interest in mathematical physics to describe the scattering resonances approximately inside a disc of radius r or in certain region in complex plane C. Therefore, we count the poles of the meromorphically defined scattering determinant det S(λ). In any case, we consider the determinant


Introduction
In this paper, we study the distribution of the scattering resonances of a certain class of elliptic operators arousing from Schrödinger operator.We have where  ∈ C ∞ 0 (R 3 ; R).Let us denote the physical plane by It is well-known from spectral analysis that the resolvent operator ( −  2 ) −1 :  2 (R 3 ) →  2 (R 3 ) is bounded in P except for some finite set { 1 , . . .,   } such that { 2 1 , . . .,  2  } are the pure point spectrum of .The resolvent ( −  2 ) −1 can be meromorphically extended from P to C as an operator: with poles of finite rank.All such meromorphic poles in C are called resolvent resonances in mathematical physics literature.There are scattering theories in more generalized formalism.We refer to [1][2][3].Let all of the meromorphic poles of () be denoted as R := { 1 , . . .,   ,  1 ,  2 , . ..} repeated according to the multiplicity such that the only accumulation point is at infinity.The possible infinite set { 1 ,  2 , . ..} is in the lower half complex plane.
The resolvent operator () defines a scattering matrix

ISRN Mathematical Physics
The growth estimate on () has only upper bound as proved in [2] which is an optimal upper bound.The actual lower bound is unknown to the author.(P.5) is the most nontrivial hypothesis.Let us define where 0 ≤  0 ≤ 2 is chosen minimally such that s() has no zero at  = 0. Surely, s() is a regular function of order three in C + := {I ≥ 0}.Because of (P.1), the zeros of s() are substitutes in the study of poles of () in C − := {I < 0}.
Theorem 1.Let (, , , s) be the number of the zeros of s() inside the sector  ≤ arg  ≤  and || ≤ .Let ℎ() be the generalized indicator function of s() with respect to proximate order ().We assume the properties (P.1) to (P.5).Then, s() is of completely regular growth in [0, ] and the following asymptotics hold: The definition of a proximate order () of a regular function is to be given in Definition 3 and we define the generalized indicator function ℎ() in Definition 4. The connection of Cartwright's theory to the location of resonances is firstly mentioned in [11] and followed by [12,13].In [11, page 278], Zworski studied the resonances using the theory of zeros of certain Fourier transform developed by Cartwright and Titchmarsh.In [13, page 269], Froese computed the indicator function in one dimensional potential scattering and used the fundamental theorem on the distribution of the zeros of a function of completely regular growth [9, page 152] to prove his results.In [12], this fundamental theorem is applied to study the location of resonances in sectors.In this paper, we study the indicator function and then the density function for the scattering resonances.
The existence of infinitely many resonances is known in several settings.See [1,3,4,12].In particular, we previously have lower bound [4]

Cartwright's Theory
We collect many classic theorems from [5][6][7][8][9][10]14] We say () is a function of finite order if there exists a positive constant  such that the inequality is valid for all sufficiently large values of .The greatest lower bound of such numbers  is called the order of the function ().By the type   of an entire function () of order , we mean the greatest lower bound of positive number  for which asymptotically we have That is, If 0 <   < ∞, then we say () is of normal type or mean type.
Definition 3. Let  ∈ R and () : R + → R + .We say () is a Lindelöf proximate order to  if  () is real, continuous, and piecewise differentiable for  >  > 0; where   () is a right-or left-hand derivative wherever different.
and for all values , except perhaps one.
If 0 <   < ∞, then we say arg  =  is a direction of Borel of maximum kind.
We review the following Cartwright theorem [5, page 504, Theorem A; page 507, Theorem V] or, more generally, as in [9, page 155].However, one should notice the typography in the corollary on page 155.The density of the zero set inside an open angle with a sinusoidal indicator function is zero.We will examine the condition for a direction of Borel has an exceptional value or not.Theorem 10.Suppose that () is an integral function of proximate order (), where  > 1, and that Then, for any  > 0, In particular, () has no direction of Borel of proximate order () inside || < /2.
There is another theorem for the nonexistence of direction of Borel.See [14, page 201].
Theorem 12.If () is of finite proximate order () for  ≤ arg  ≤ , || ≥ , where  −  > /, then there is at least one direction of Borel of maximum kind for which  < arg  < .This is stated the same as in [6, page 425].We state the following fundamental theorem in [9, page 152].
Theorem 13.If a holomorphic function () of order () has completely regular growth within an angle ( 1 ,  2 ), then for all ISRN Mathematical Physics values ,  such that  1 <  <  <  2 , except possibly for a denumerable set, the following limit exists: where The exceptional denumerable set can only consist of points for which ℎ  F ( + 0) ̸ = ℎ  F ( − 0).
We may find ℎ() by the following lemma in
which is of the form  + (), where () is of trace class depending meromorphically on  ∈ C. The poles are called the scattering resonances which share the same multiplicity at each pole as resolvent resonances.It is a subject of great interest in mathematical physics to describe the scattering resonances approximately inside a disc of radius  or in certain region in complex plane C. Therefore, we count the poles of the meromorphically defined scattering determinant det ().In any case, we consider the determinant () := det  ()(6)satisfying the following properties [1-4]: (P.1) (−) = () = 1/(); (P.2) the point set R and |()| is symmetric about the imaginary axis; (P.3) there is no pole on the real axis except possibly a double pole at  = 0; (P.4) there are only exceptionally finitely many poles { 1 , . . .,   } in C + := {I > 0}; infinitely many poles in C − := {I < 0}; (P.5) the functional determinant () is of order 3, the number of space dimension.