**1. Introduction**

Let be the complex unit disk endowed with its Lebesgue measure *μ* and let ∂ID be its boundary. Denote by *L*<sup>2</sup> ð Þ , *dμ* the space of complex-valued measurable functions, which are *dμ* square integrable on . The logarithmic potential transform: *L*2 ð Þ! *<sup>L</sup>*<sup>2</sup> ð Þ is defined by

$$\mathcal{L}[f](z) = -\frac{1}{\pi} \int\_{\mathcal{D}} \frac{f(\xi)}{\xi - z} \log \left( \frac{1}{|z - \xi|} \right) d\mu(\xi). \tag{1}$$

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 L to the space *L*2 *<sup>a</sup>*ð Þ of analytic *μ*-square integrable on . They precisely have considered the projection operator *P*0: *L*<sup>2</sup> ð Þ! *<sup>L</sup>*<sup>2</sup> *<sup>a</sup>*ð Þ and they have proved that the singular values *λ<sup>k</sup>* of L*P*0, (which turn out to be eigenvalues of the operator ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ LP<sup>0</sup> <sup>∗</sup> ð Þ <sup>L</sup>*P*<sup>0</sup> p behave like *k*�<sup>1</sup> as *k* goes to ∞. They also concluded that L*P*<sup>0</sup> belongs to the Schatten class *S*1,∞.

Now, consider the following *weighted* logarithmic potential transform

$$\mathcal{L}\_{\sigma}[f](x) = -\frac{1}{\pi} \int\_{\mathcal{D}} \frac{f(\xi)}{\xi - x} \log \left( \frac{1}{|x - \xi|} \right) \left( 1 - \xi \overline{\xi} \right)^{\sigma - 2} d\mu(\xi), \tag{2}$$

defined on the space *<sup>L</sup>*2,*<sup>σ</sup>*ð Þ of complex-valued measurable functions, which are <sup>1</sup> � *ξξ* � �*<sup>σ</sup>*�<sup>2</sup> *dμ ξ*ð Þ-square integrable on where *σ* >1 is a fixed parameter. We observe that the subspace *L*2,*<sup>σ</sup> <sup>a</sup>* ð Þ of analytic functions on and belonging to *<sup>L</sup>*2,*<sup>σ</sup>*ð Þ coincides with the eigenspace

$$\mathcal{A}\_0^{\sigma}(\mathbb{D}) \coloneqq \left\{ \boldsymbol{\psi} \in L^{2,\sigma}(\mathbb{D}), \ \Delta\_{\sigma} \boldsymbol{\psi} = \mathbf{0} \right\}, \tag{3}$$

of the second order differential operator

$$
\Delta\_{\sigma} \coloneqq -4(\mathbf{1} - z\overline{z}) \left( (\mathbf{1} - z\overline{z}) \frac{\partial^2}{\partial z \, d\overline{z}} - \sigma \overline{z} \frac{\partial}{\partial \overline{z}} \right), \tag{4}
$$

known as the *σ*-weight Maass Laplacian and its discrete eigenvalues are given by

$$e\_m \coloneqq 4m(\sigma - 1 - m), \ m = 0, 1, 2, \dots, \lfloor (\sigma - 1)/2 \rfloor,\tag{5}$$

with their corresponding eigenspaces

$$\mathcal{A}\_m^{\sigma}(\mathbb{D}) := \left\{ \boldsymbol{\mu} \in L^{2,\sigma}(\mathbb{D}) \text{ and } \Delta\_{\sigma} \boldsymbol{\mu} = \varepsilon\_m^{\sigma} \boldsymbol{\mu} \right\},\tag{6}$$

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 A*σ <sup>m</sup>*ð Þ . That is, we are concerned with the operator <sup>C</sup>*σP<sup>σ</sup> <sup>m</sup>* where *P<sup>σ</sup> <sup>m</sup>* is the projection *<sup>L</sup>*2,*<sup>σ</sup>*ð Þ! <sup>A</sup>*<sup>σ</sup> <sup>m</sup>*ð Þ . The results achieved are as follows:

Firstly, we find that the singular values of L*σP<sup>σ</sup> <sup>m</sup>*. For *k* 6¼ *m*, it can be expressed as

$$
\lambda\_k = \sqrt{J\_1 + J\_2 + J\_3} \,\mathrm{s}
$$

where

$$\begin{aligned} J\_1 &= \left(\frac{(1+k-m)\_m}{m!(k-m+1)}\right)^2 \sum\_{n=0}^{\infty} A\_n \frac{\Gamma(2n+2k-2m+6-1)\Gamma(4\nu-2m-1)}{\Gamma(2n+2k-4m+4\nu+6)}, \\ J\_2 &= \left(\frac{\alpha\_k^{\nu,m}}{2\nu-m-1}\right)^2 \sum\_{n=0}^{\infty} A\_n \frac{\Gamma(4\nu-2m-1)\Gamma(2n+2)}{\Gamma(2n+4\nu-2m+1)}, \end{aligned}$$

*The Singular Values of the Logarithmic Potential Transform on Bound States Spaces DOI: http://dx.doi.org/10.5772/intechopen.107090*

and

$$J\_3 = \frac{(1+k-m)\_m a\_k^{\nu, m}}{m!(k-m+1)(2\nu-m-1)} \left( \sum\_{n=0}^{\infty} A\_n \frac{\Gamma(k-m+2)\Gamma(4\nu-2m-1)}{\Gamma(4\nu-k-3m)} \right).$$

For *k* ¼ *m* can be expressed as

$$
\lambda\_k^2 = \frac{a\_k^{\nu, m} (2(2\nu - m) - 1)}{8(\pi(2\nu - m + 1))} \sum\_{n=0}^{\infty} \frac{B\_n}{n + 2\nu - m},\tag{7}
$$

where

$$B\_n = \sum\_{n=0}^{\infty} \frac{\Gamma(-m+1)\Gamma(2\nu - m)\Gamma(2(\nu - m) + 1)}{n!\Gamma(2(\nu - m)\Gamma(2\nu - m + 2))},$$

$$a\_k^{\nu, m} = \frac{\Gamma(2)\Gamma(2(m - \nu) + 1)}{\Gamma(m+1)\Gamma(2 + m - 2\nu)}.$$

Secondly, we show that these singular values behave like

$$
\lambda\_k \sim C \sqrt{k^{m-4\nu+1}}, \text{ as } k \to \infty,
$$

where *C* is a constant.

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.
