The Laplace operator on a hyperbolic manifold I. Spectral and scattering theory

Using techniques of stationary scattering theory for the Schrödinger equation, we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the Laplace-Beltrami operator on manifolds Mⁿ arising as the quotient of hyperbolic n-dimensional space by a geometrically finite, discrete group of hyperbolic isometries. We consider manifolds Mⁿ of infinite volume. In subsequent papers, we will use the techniques developed here to analytically continue Eisenstein series for a large class of discrete groups, including some groups with parabolic elements.     

Translation representations for automorphic solutions of the wave equation in non-euclidean spaces. II

In this part we prove that the translation representation defined in Part I for the non-Euclidean wave equation is complete. We show how one can derive from this translation representation a spectral representation for the Laplace-Beltrami operator over geometrically finite fundamental polyhedra with infinite volume, without the use of Eisenstein series.     

On Bounded Local Resolvents

It is known that each normal operator on a Hilbert space with nonempty interior of the spectrum admits vectors with bounded local resolvent. We generalize this result for Banach space operators with the decomposition property (δ) (in particular for decomposable operators). Moreover, the same result holds for operators with interior points in the localizable spectrum.     

On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight

We give a Katok-Sarnak type correspondence for Niebur type Poincaré series and Eisenstein series on the three-dimensional hyperbolic space.     

Eisenstein series and theta functions to the septic base

We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his “Lost Notebook.” We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function and the eta quotient η⁴(7τ)/η⁴(τ). This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series.     

Functional Model of a Closed Non-Selfadjoint Operator

We construct the symmetric functional model of an arbitrary closed operator with non-empty resolvent set acting on a separable Hilbert space. The construction is based on the explicit form of the Sz.-Nagy-Foiaş model of a closed dissipative operator, the Potapov-Ginzburg transform of characteristic functions, and certain resolvent identities. All considerations are carried out under minimal assumptions, and obtained results are directly applicable to problems typically arising in mathematical physics. Explicit formulae for all the objects participating in the model construction are provided.      

Eisenstein series attached to lattices¶and modular forms on orthogonal groups

We study certain vector valued Eisenstein series on the metaplectic cover of SL₂(ℝ), which transform with the Weil representation associated with the discriminant group of an even lattice L. We find a closed formula for the Fourier coefficients in terms of Dirichlet L-series and representation numbers of L modulo “bad” primes. Such Eisenstein series naturally occur in the context of Borcherds' theory of automorphic products. We indicate some applications to modular forms on the orthogonal group of L with zeros on Heegner divisors. Received: 27 September 2001     

Spectral Scattering Theory for Automorphic Forms

We construct a scattering process for L²-automorphic forms on the quotient of the upper half plane by a cofinite discrete subgroup Γ of . The construction is algebraic besides being analytic in the sense that we use some relations satisfied by real-analytic Eisenstein series with a complex parameter. Thanks to this feature, the construction of our operators and spaces is explicit. We show some properties of the Lax-Phillips generator on a scattering subspace carved out from this process. We prove that the spectrum of this operator consists only of eigenvalues, which correspond to the nontrivial zeros, counted with multiplicity, of the Dirichlet series appearing in the functional equation of the Eisenstein series. In particular, in the case of the (full) modular group , the Dirichlet series reduces to the Riemann zeta function ζ, thereby we obtain a spectral interpretation of the nontrivial zeros of ζ.      

Deformation of -cusp forms

We develop an algorithm for numerical computation of the Eisenstein series on a Riemann surface of constant negative curvature. We focus in particular on the computation of the poles of the Eisenstein series. Using our numerical methods, we study the spectrum of the Laplace-Beltrami operator as the surface is being deformed. Numerical evidence of the destruction of -cusp forms is presented.

     

The Rogers-Ramanujan continued fraction and a new Eisenstein series identity

With two elementary trigonometric sums and the Jacobi theta function θ₁, we provide a new proof of two Ramanujan's identities for the Rogers-Ramanujan continued fraction in his lost notebook. We further derive a new Eisenstein series identity associated with the Rogers-Ramanujan continued fraction.     

Some Local Properties of Spectrum of Linear Dynamical Systems in Hilbert Space

We prove some local properties of the spectrum of a linear dynamical system in Hilbert space. The semigroup generator, the control operator and the observation operator may be unbounded. We consider (i) the PBH test, (ii) the correspondence between the poles of the resolvent of the semigroup generator and the poles of the transfer function, and (iii) pole-zero cancellation between two transfer functions of the cascade connection of two dynamical systems. For our investigation we take well-posed linear systems and a subclass of them called weakly regular systems as the most general setting.     

An infinitesimal trace formula for the Laplace operator on compact Riemann surfaces

We derive an infinitesimal (or variational) version of the Selberg trace formula for compact Riemann surfaces, which gives information on the behaviour of the eigenvalues of the Laplace-Beltrami operator as the surface varies over the appropriate moduli space.     

Dissipative q-Dirac operator with general boundary conditions

In this paper, we consider the symmetric q-Dirac operator. We describe dissipative, accumulative, self-adjoint and the other extensions of such operators with general boundary conditions. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.     

Invariant subspaces and localizable spectrum

LetT L(X) be a continuous linear operator on a complex Banach spaceX. We show thatT possesses non-trivial closed invariant subspaces if its localizable spectrum _loc(T) is thick in the sense of the Scott Brown theory. Since for quotients of decomposable operators the spectrum and the localizable spectrum coincide, it follows that each quasiaffine transformation of a Banach-space operator with Bishop's property () and thick spectrum has a non-trivial invariant subspace. In particular it follows that invariant-subspace results previously known for restrictions and quotients of decomposable operators are preserved under quasisimilarity.     

Multiple Eisenstein series and multiple cotangent functions

We construct the multiple Eisenstein series and we show a relation to the multiple cotangent function. We calculate a limit value of the multiple Eisenstein series.     

Ramanujan and coefficients of meromorphic modular forms

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincaré series introduced by Petersson and a new family of Poincaré series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles.     

Browder’s Theorems through Localized SVEP

A bounded linear operator T ∈ L(X) on aBanach space X is said to satisfy “Browder’s theorem” if the Browder spectrum coincides with the Weyl spectrum. T ∈ L(X) is said to satisfy “a-Browder’s theorem” if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum. In this note we give several characterizations of operators satisfying these theorems. Most of these characterizations are obtained by using a localized version of the single-valued extension property of T. In the last part we shall give some characterizations of operators for which “Weyl’s theorem” holds.     

The 26th power of Dedekind's η-function

We prove a simple and explicit formula, which expresses the 26th power of Dedekind's η-function as a double series. The proof relies on properties of Ramanujan's Eisenstein series P, Q and R, and parameters from the theory of elliptic functions.The formula reveals a number of properties of the product , for example its lacunarity, the action of the Hecke operator, and sufficient conditions for a coefficient to be zero.     

Poles of Eisenstein series on quaternion groups

We show that the normalized Siegel Eisenstein series of quaternion groups have at most simple poles at certain integers and half integers. These Eisenstein series play an important role of Rankin-Selberg integral representations of Langlands L-functions for quaternion groups.     

On Smooth Local Resolvents

We exhibit an example of a bounded linear operator on a Banach space which admits an everywhere defined local resolvent with continuous derivatives of all orders.