# A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space

## Abstract

The function Ψ(x,y,s)=e{sup iy}Φ(−e{sup iy},s,x), where Φ(z,s,v) is Lerch's transcendent, satisfies the following two-dimensional formally self-adjoint second-order hyperbolic differential equation, where s=1/2+iλ. The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space L{sub 2}(Π), where Π=(0,1)×(0,2π). We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of Ψ(x,y,s). We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

- Authors:

- Southern Federal University, Rostov-on-Don (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22364910

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Sbornik. Mathematics; Journal Volume: 205; Journal Issue: 8; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ANALYTIC FUNCTIONS; DIFFERENTIAL EQUATIONS; EIGENFUNCTIONS; EIGENVALUES; HILBERT SPACE; MATHEMATICAL OPERATORS; MATHEMATICAL SOLUTIONS; SYMMETRY; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Kaplitskii, V M.
```*A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space*. United States: N. p., 2014.
Web. doi:10.1070/SM2014V205N08ABEH004411.

```
Kaplitskii, V M.
```*A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space*. United States. doi:10.1070/SM2014V205N08ABEH004411.

```
Kaplitskii, V M. Fri .
"A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space". United States.
doi:10.1070/SM2014V205N08ABEH004411.
```

```
@article{osti_22364910,
```

title = {A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space},

author = {Kaplitskii, V M},

abstractNote = {The function Ψ(x,y,s)=e{sup iy}Φ(−e{sup iy},s,x), where Φ(z,s,v) is Lerch's transcendent, satisfies the following two-dimensional formally self-adjoint second-order hyperbolic differential equation, where s=1/2+iλ. The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space L{sub 2}(Π), where Π=(0,1)×(0,2π). We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of Ψ(x,y,s). We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.},

doi = {10.1070/SM2014V205N08ABEH004411},

journal = {Sbornik. Mathematics},

number = 8,

volume = 205,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}