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Scattering on Compact Manifolds With Infinitely Thin Horns
"... In the paper [25]... In this paper we consider the quantum mechanical scattering in a hedgehogshaped space which is constructed by gluing a finite number of halflines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum particle in such a sy ..."
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Cited by 21 (5 self)
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In the paper [25]... In this paper we consider the quantum mechanical scattering in a hedgehogshaped space which is constructed by gluing a finite number of halflines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum particle in such a system coincides with a Schrödinger operator on the punctured manifold (the points of gluing are removed) and with the free Schrödinger operator on each halfline. At the gluing points, some boundary conditions are imposed. In particular, the Schröodinger operator in a magnetic field is included in our scheme. The approach we use is based on the Krein resolvent formula from operator extension theory [50], therefore in Sec. 1 we give a very brief sketch of results needed from this theory. Sec. 2 is devoted to the construction of Schrödinger operators on the hedgehogshaped space; we use the theory of boundary value spaces [35] to describe all possible kinds of boundary conditions defining the Schrödinger operators. We distinguish among them operators of "Dirichlet" and of "Neumann" type. It is worth noting that the results of Sec. 2 are valid for all Riemannian manifolds of dimension less than four, not only for the compact ones. In principle, the definition of the Schrödinger operator on a hedgehogshaped space may be given in the framework of pseudodifferential operator theory on such a space [66], but our approach is more convenient for investigating the scattering parameters and connected with the approach to spectral problems for point perturbations on Riemannian manifolds [8], [9]...
Translation representations for automorphic solutions of the wave equation in nonEuclidean spaces
 thecaseoffinitevolume, Trans. Amer. Math. Soc
, 1985
"... Abstract. Let T be a discrete subgroup of automorphisms of H", with fundamental polyhedron of finite volume, finite number of sides, and N cusps. Denote by Ar the LaplaceBeltrami operator acting on functions automorphic with respect to T. We give a new short proof of the fact that Ar has absol ..."
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Cited by 18 (1 self)
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Abstract. Let T be a discrete subgroup of automorphisms of H", with fundamental polyhedron of finite volume, finite number of sides, and N cusps. Denote by Ar the LaplaceBeltrami operator acting on functions automorphic with respect to T. We give a new short proof of the fact that Ar has absolutely continuous spectrum of uniform multiplicity N on (00,«11 l)/2)2), plus a standard discrete spectrum. We show that this property of the spectrum is unchanged under arbitrary perturbation of the metric on a compact set. Our method avoids Eisenstein series entirely and proceeds instead by constructing explicitly a translation representation for the associated wave equation. Introduction. Using the methods developed in Parts I and II of [5], we obtain a short proof for the existence and completeness of incoming and outgoing translation representations for the wave equation acting on automorphic functions with fundamental polyhedron of finite volume and a finite number of sides. As a byproduct we show that the associated LaplaceBeltrami operator has a standard discrete spectrum
An adelic causality problem related to abelian Lfunctions
, 2000
"... We associate to the global field K a LaxPhillips scattering which has the property of causality if and only if the Riemann Hypothesis holds for all the abelian Lfunctions of K. As a Hilbert space closure problem this provides an adelic variation on a theme initiated by Nyman and Beurling. The ad ..."
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Cited by 9 (6 self)
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We associate to the global field K a LaxPhillips scattering which has the property of causality if and only if the Riemann Hypothesis holds for all the abelian Lfunctions of K. As a Hilbert space closure problem this provides an adelic variation on a theme initiated by Nyman and Beurling. The adelic aspects are related to previous work by Tate, Iwasawa and Connes.
Statistical Mechanics of 2+1 Gravity From Riemann Zeta Function and Alexander Polynomial: Exact Results
, 2008
"... In the recent publication (Journal of Geometry and Physics, 33 (2000) 23102) we have demonstrated that dynamics of 2+1 gravity can be described in terms of train tracks. Train tracks were introduced by Thurston in connection with description of dynamics of surface automorphisms. In this work we pro ..."
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Cited by 8 (1 self)
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In the recent publication (Journal of Geometry and Physics, 33 (2000) 23102) we have demonstrated that dynamics of 2+1 gravity can be described in terms of train tracks. Train tracks were introduced by Thurston in connection with description of dynamics of surface automorphisms. In this work we provide an example of utilization of general formalism developed earlier. The complete exact solution of the model problem describing equilibrium dynamics of train tracks on the punctured torus is obtained. Being guided by similarities between the dynamics of 2d liquid crystals and 2+1 gravity the partition function for gravity is mapped into that for the Farey spin chain. The Farey spin chain partition function, fortunately, is known exactly and has been thoroughly investigated recently. Accordingly, the transition between the pseudoAnosov and the periodic dynamic regime (in Thurston’s terminology) in the case of gravity is being reinterpreted in terms of phase transitions in the Farey spin chain whose partition function is just the ratio of two Riemann zeta functions. The mapping into the spin chain is facilitated by recognition of a special role of the Alexander polynomial for knots/links in study of dynamics of self homeomorphisms of surfaces. At the end of paper, using some facts from the theory of arithmetic hyperbolic 3manifolds ( initiated by Bianchi in 1892), we develop systematic extension of the obtained results to noncompact Riemann surfaces of higher genus. Some of the obtained results are also useful for 3+1 gravity. In particular, using the theorem of Margulis, we provide new reasons for the black hole existence in the Universe: black holes make our Universe arithmetic. That is the discrete Lie groups of motion are arithmetic.
Number Theory, Dynamical Systems and Statistical Mechanics
, 1998
"... We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/ ..."
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Cited by 8 (2 self)
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We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/2Z) k, with Z/2Z = ({0, 1}, +). h0: = 1, hk+1(σ, 0): = hk(σ) and hk+1(σ, 1): = hk(σ) + hk(1 − σ), (1) where σ = (σ1,..., σk) ∈ Gk and 1 − σ: = (1 − σ1,..., 1 − σk) is the inverted configuration. The sequences hk(σ) of integers, written in lexicographic order, coincide with the denominators of the modified Farey sequence. We now formally interpret σ ∈ Gk as a configuration of a spin chain with k spins and energy function Hk: = ln(hk). Thus we may interpret
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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Generalized Weierstrass representation for surfaces in terms of DiracHestenes spinor field
 J. Geom. Phys
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Quantization of the Riemann zetafunction and cosmology
"... Quantization of the Riemann zetafunction is proposed. We treat the Riemann zetafunction as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of padic strings and by recent works on stringy cosmologi ..."
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Quantization of the Riemann zetafunction is proposed. We treat the Riemann zetafunction as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of padic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zetafunction field is equivalent to the sum of the KleinGordon Lagrangians with masses defined by the zeros of the Riemann zetafunction. Quantization of the mathematics of FermatWiles and the Langlands program is indicated. The Beilinson conjectures on the values of Lfunctions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zetafunction field theory are discussed. 1 1
The Schrödinger Operator with Morse Potential on the Right Half Line
, 2009
"... This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4 e2u + ke u on a right halfline [u0, ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittak ..."
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This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4 e2u + ke u on a right halfline [u0, ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittaker function Wκ,µ(x), for fixed real parameters κ,x with x> 0, viewed as an entire function in the complex variable µ. In this case all zeros lie on the imaginary axis, with the exception, if k < 0 of a finite number of real zeros which lie in the interval κ  < k. We obtain an asymptotic formula for the number of zeros N(T) = {ρ  Wκ,ρ(x) = 0, Im(ρ)  < T} of the form N(T) = 2 2 πT log T + π (2log 2−1−log x)T +O(1). Parallels are observed with zeros of the Riemann zeta function.
Discrete wave scattering on stargraph
 J. Phys. A: Math. Gen
"... In this paper we consider the spectral problem for the adjacency matrix of a graph composed of a compact part with a few semiinfinite periodic leads attached. Based on the spectral properties of the adjacency matrix we develop LaxPhillips scattering theory for the corresponding discrete wave equa ..."
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In this paper we consider the spectral problem for the adjacency matrix of a graph composed of a compact part with a few semiinfinite periodic leads attached. Based on the spectral properties of the adjacency matrix we develop LaxPhillips scattering theory for the corresponding discrete wave equation. 1