## Abstract

In the present manuscript, we prove that the singular numbers of the Cauchy transform Lσfz=−1π∫Dfξξ−zlog1∣z−ξ∣1−ξξ¯σ−2dμξ (2) defined on the space L2,σD of complex-valued measurable functions, which are 1−ξξ¯σ−2dμξ-square integrable on D where σ>1 is a fixed parameter, are asymptotically ≈Ckm−4ν+1,ask→∞ where C is a constant.

### Keywords

- the logarithmic potential transform
- the singular values
- Cauchy transform.

## 1. Introduction

Let

This operator is very important as the transformed Cauchy and it often appears in Analysis [1].

The dimensional analysis [1, 2] and scaling arguments form an integral part of theoretical physics to solve some important problems without doing much calculation.

The logarithmic potential in physics forms an interesting one as it provides some unusual predictions about the system. Moreover, this potential can be used suitably to illustrate some of the important features of field theory such as dimensional regularization and renormalization. In most of our textbooks, this potential is not discussed in detail; although the calculations are quite simple to demonstrate some of its unique features. We have obtained the bound state energy of this logarithmic potential through uncertainty principle, phase space quantization, and the Hellmann-Feynman theorem.

In Ref. [3] the authors have been dealing with the restriction of

Now, consider the following * weighted* logarithmic potential transform

defined on the space

of the second order differential operator

known as the

with their corresponding eigenspaces

are here called * generalized Bergman* spaces since …

After noticing that, we here deal with analogous questions as in Ref. [3] in the context of the weighted Cauchy transform (2) and for its restriction to the space

Firstly, we find that the singular values of

where

and

For

where

Secondly, we show that these singular values behave like

where

The paper is organized as follows: In Section 2, we review the definition of the weighted logarithmic potential transform, as well as some of its needed properties. Section 3 deals with some basic facts on the spectral theory of Mass Laplacians on the Poincaré disk. In Section 4, a precise description of the generalized Bergmann spaces is reviewed. Section 5 is devoted to the computation of the singular values of the weighted logarithmic potential transform. The asymptotic behavior of these singular values is established in Section 6.

## 2. The weighted logarithmic potential transform L ν

### 2.1 The case ν = 1

Let

The Logarithmic Potential operator

### 2.2 The case of ν ≥ 1

We fix a real parameter

defined on the space

## 3. The Landau Hamiltonian H ν on the Poincaré disk D

Let

By Ref. [5] the Schrödinger operator on

which is also called Maass Laplacian on the disk. A slight modification of

acting in the Hilbert space

For our purpose, we shall consider the unitary equivalent realization

which is defined by

where * scattering states*,

*) of the form*hyperbolic Landau levels

with infinite degeneracy, provided that * bound states* since the particle in such a state cannot leave the system without additional energy. A concrete description of these bound states spaces will be the goal of the next section.

## 4. The bound states spaces A ν , m 2 D

Here, we consider the eigenspace

See Refs. [6, 7], for the following proposition.

* an orthogonal basis* of

is given by the functions

* in terms of Jacobi polynomials constitute an orthonormal basis* of

** Proof.** Write the connection between the

then the functions

constitute an orthonormal basis of

Here,

is an orthonormal basis of

by making appeal to the identity

for _{,} and

## 5. Computation of the singular values λ k

Elements of this basis are given in terms of Jacobi polynomials as

The norm square of

Here,

So that we consider the elements

### 5.1 The action L ν

By ref. [3], it remains to prove that this lemma for

We have

The function

□

We observe that

□

As

or

For all

□

Just use

□

If

where

and

For

where

and

Calculus of

We use the formula

We have

By Ref. [8], we have

implies that

Calculus of

Use the previous formula in Ref. [8] and the integration by part gives

Calculus of

Use the following formula, which has place in [9]

We put

which implies

implies that

By the change

In [8], p. 44,

Since

and by the change

we set

Finally

Now if

We set

and

Calculus of

By the formula

we have

Calculus of

As the previous

also

Now if

By the formula

and put

substituting in the expression of

it is the same formula for

* By the previous formula in [*9

], we have

### 5.2 The spectrum of L ν

where

and

where

If

We set

Calculus of

Since

then

where

Thus

Use the fact that

implies

Calculus of

In the same,

Calculus of

If

Since

where

## 6. Asymptotic behavior of singular values λ k as k → ∞

where

If

where

and

The limit of

We use the formula

we have

In the same

Therefore

where

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