Geodesic
A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space
Sbornik. Mathematics, Tome 205 (2014) no. 8, pp. 1080-1106 Cet article a éte moissonné depuis la source Math-Net.Ru

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The function $\Psi(x, y, s)=e^{iy}\Phi(-e^{iy},s,x)$, where $\Phi(z,s,v)$ is Lerch's transcendent, satisfies the following two-dimensional formally self-adjoint second-order hyperbolic differential equation: $$ L[\Psi]=\frac{\partial^2\Psi}{\partial x\,\partial y}+i(x-1)\frac{\partial\Psi}{\partial x}+\frac{i}{2}\Psi=\lambda\Psi, $$ where $s={1}/{2}+i\lambda$. The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space $L_2(\Pi)$, where $\Pi=(0,1)\times(0,2\pi)$. 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 $\Psi(x,y,s)$. 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.
Keywords: Lerch's transcendent, Hilbert space, symmetric operator, eigenfunction. 
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V. M. Kaplitskii. A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space. Sbornik. Mathematics, Tome 205 (2014) no. 8, pp. 1080-1106. http://geodesic.mathdoc.fr/item/SM_2014_205_8_a1/

[1] J. B. Conrey, “The Riemann hypothesis”, Notices Amer. Math. Soc., 50:3 (2003), 341–353 | MR | Zbl

[2] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, Based, in part, on notes left by H. Bateman, v. I, McGraw-Hill Book Co., Inc., New York–Toronto–London, 1953, xxvi+302 pp. | MR | MR | Zbl | Zbl

[3] B. S. Pavlov, L. D. Faddeev, “Teoriya rasseyaniya i avtomorfnye funktsii”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 6, Zap. nauchn. sem. LOMI, 27, Izd-vo «Nauka», Leningrad. otd., L., 1972, 161–193 | MR | Zbl

[4] I. V. Volovich, V. V. Kozlov, “Square integrable solutions to the Klein–Gordon equation on a manifold”, Dokl. Math., 73:3 (2006), 441–444 | DOI | MR | Zbl

[5] V. V. Kozlov, I. V. Volovich, “Finite action Klein–Gordon solutions on Loretzian manifolds”, Int. J. Geom. Methods Mod. Phys., 3:7 (2006), 1349–1357 | DOI | MR | Zbl

[6] O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math. Sci., 49, Springer-Verlag, New York, 1985, xxx+322 pp. | DOI | MR | MR | Zbl | Zbl

[7] N. I. Akhiezer, I. M. Glazman, Theory of linear operators in Hilbert space, v. I, II, Frederick Ungar Publishing Co., New York, 1961, 1963, xi+147 pp., v+218 pp. | MR | MR | MR | Zbl | Zbl

[8] S. L. Sobolev, “O dvizhenii simmetrichnogo volchka s polostyu, napolnenoi zhidkostyu”, PMTF, 3 (1960), 20–55 | Zbl

[9] V. I. Arnol'd, “Small denominators. I. Mappings of the circumference onto itself”, Amer. Math. Soc. Transl. Ser. 2, 46, Amer. Math. Soc., Providence, RI, 1965, 213–284 | MR | Zbl

[10] S. D. Troitskaya, “On non-almost-periodicity of solutions of the Sobolev problem in domains with edges”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 97–124 | DOI | MR | Zbl

[11] M. V. Fokin, “Hamiltonian systems in the theory of small oscillations of a rotating ideal fluid. I”, Siberian Adv. Math., 12:1 (2002), 1–50 | MR | MR | Zbl | Zbl

[12] M. V. Fokin, “Hamiltonian systems in the theory of small oscillations of a rotating ideal fluid. II”, Siberian Adv. Math., 12:2 (2002), 1–37 | MR | MR | Zbl

[13] V. P. Burskii, A. S. Zhedanov, “Dirichlet and Neuman problems for string equation, Poncelet problem and Pell–Abel equation”, SIGMA, 2 (2006), 041, 5 pp. | DOI | MR | Zbl

[14] V. M. Kaplitskii, “Asymptotic behaviour of the discrete spectrum of a quasi-periodic boundary value problem for a two-dimensional hyperbolic equation”, Sb. Math., 200:2 (2009), 215–228 | DOI | DOI | MR | Zbl

[15] V. M. Kaplitskiǐ, “Asymptotics of the distribution of eigenvalues of a selfadjoint second order hyperbolic differential operator on the two-dimensional torus”, Siberian Math. J., 51:5 (2010), 830–846 | DOI | MR | Zbl