The Hamiltonians of Pseudorelativistic Atoms with Finite-Mass Nuclei: The Structure of the Discrete Spectrum

We study the structure of the discrete spectrum of pseudorelativistic Hamiltonians H for atoms and positive ions with finite-mass nuclei and with n electrons, where n 1 is arbitrary. The center-of-mass motion cannot be separated, and hence we study the spectrum of the restriction H _P of H to the subspace of states with given value P of the total momentum of the system. For the operators H _P we discover a) two-sided estimates for the counting function of the discrete spectrum _d (H _P ) of H _P in terms of the counting functions of some effective two-particle operators; b) the leading term of the spectral asymptotics of _d (H _P ) near the lower bound inf _ess(H _P ) of the essential spectrum of H _P . The structure of the discrete spectrum of such systems was known earlier only for n=1.     

Multiplication Operators on Invariant Subspaces of Function Spaces

Let M_φ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator M_z, we derive some spectral properties of the multiplication operator M_φ : F → F. We characterize norm, spectrum, essential norm and essential spectrum of such operators when F has the codimension n property with n ∈ {1, 2, …, + ∞}.     

A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L ₂(T ^ν ×T ^ν ), where T ^ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrödinger operator ? = H₀ + H₁ + H₂, where H₀ is the operator of multiplication by a function and H₁ and H₂ are partial integral operators. We prove several theorems concerning the essential spectrum of ?. We study the discrete and essential spectra of the Hamiltonians H^t and h arising in the Hubbard model on the three-dimensional lattice.     

The Birman-Schwinger principle on the essential spectrum

Let H₀ and H be self-adjoint operators in a Hilbert space. We consider the spectral projections of H₀ and H corresponding to a semi-infinite interval of the real line. We discuss the index of this pair of spectral projections and prove an identity which extends the Birman-Schwinger principle onto the essential spectrum. We also relate this index to the spectrum of the scattering matrix for the pair H₀, H.     

Discrete analysis on a radial lattice

A discrete model for analytic functions is constructed using lattice points of the complex plane arranged in radial form. The discrete analytic functions are defined as solutions of a finite-difference approximation to the polar Cauchy-Riemann equations. The resulting discrete powerz⁽ⁿ⁾ (an analogue of zⁿ) has a simple algebraic form (a direct analogue of ?ⁿ exp{inθ}) and has some surprising properties. For example every discrete polynomial ∑₀^ma_nz⁽ⁿ⁾ has a factorization in terms of the zeros of its classical counterpart ∑₀^ma_nzⁿ every discrete entire function has a power series representation ∑ a_nz⁽ⁿ⁾.     

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

Spectral theory for commutative algebras of differential operators on Lie groups

The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L₁,…,Ln on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a “weighted subcoercive operator” of ter Elst and Robinson (1998) [52]. The joint spectrum of L₁,…,Ln in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L₁,…,Ln. Connections with the theory of Gelfand pairs are established in the case L₁,…,Ln generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G).     

Lower bound for the spectrum and the presence of pure point spectrum of a singular discrete Hamiltonian system

In this paper, criteria for limit-point (n) case of a singular discrete Hamiltonian system are established. Furthermore, the lower bound of the essential spectrum is obtained and the present of pure point spectrum is discussed for such system by using the spectral theory of self-adjoint operators in a Hilbert space.     

An addition theorem for the manifolds with the Laplacian having discrete spectrum

The question of the preservation of discreteness of the spectrum of the Laplacian acting in a space of differential forms under the cutting and gluing of manifolds reduces to the same problem for compact solvability of the operator of exterior derivation. Along these lines, we give some conditions on a cut Y dividing a Riemannian manifold X into two parts X ₊ and X _? under which the spectrum of the Laplacian on X is discrete if and only if so are the spectra of the Laplacians on X ₊ and X _?.     

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.     

Zonal spherical functions on the complex reflection groups and (n+1,m+1)-hypergeometric functions

The pair of groups, complex reflection group G(r,1,n) and symmetric group S_n, is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called (n+1,m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the (n+1,m+1)-hypergeometric functions.     

Spectral geometry for Riemannian foliations

Let F be a Riemannian foliation on a Riemannian manifold (M, g), with bundle-like metric g. Aside from the Laplacian △_g associated to the metric g, there is another differential operator, the Jacobi operator J▽, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum is discrete as a consequence of the compactness of M. Hence one has two spectra, spec (M, g) = spectrum of △_g (acting on functions), and spec (F, J▽) = spectrum of J▽. We discuss the following problem: Which geometric properties of a Riemannian foliation F on a Riemannian manifold (M, g) are determined by the two types of spectral invariants?     

On spectra of neither Pisot nor Salem algebraic integers

We investigate the class of  ± 1 polynomials evaluated at a real number q> 1 defined as: $$A(q)=\{\epsilon_0+\epsilon_1q+\cdots+\epsilon_k q^k : \epsilon_i\in\{-1,1\}\}$$ and usually called spectrum. Λ(q) is defined by analogy where the coefficients are extended to ± 1,0. In this paper an algorithm for finding the spectrum of a real algebraic integer q without a conjugate of modulus 1 is given. The algorithm can check and terminate if A(q) or Λ(q) is not discrete. Using this criterion a part of a conjecture of Borwein and Hare is proved: if 1 < q < 2 and A(q) is discrete, then all real conjugates of q are of modulus less than q. For an infinite sequence r _2m Zaimi has proved that A(r _2m ) is discrete, and here it is proved that Λ(r _2m ) is not discrete. For given ${q\in (1,2)}$ , an algorithm counting a sequence of  ± 1 polynomials P _n (x) such that, solutions α _n  > q of equations ${P_n(x)=\frac{1}{x-1}}$ satisfy α _n → q, is presented.     

Stability of <Emphasis Type="Italic">n</Emphasis>-particle pseudorelativistic systems

For a system Z_n of n identical pseudorelativistic particles, we show that under some restrictions on the pair interaction potentials, there is an infinite sequence of numbers n_s, s = 1, 2,..., such that the system Z_n is stable for _n = n_s, and the inequality sup _sn_s+1n _s ⁻¹ < + ∞ holds. Furthermore, we show that if the system Z_n is stable, then the discrete spectrum of the energy operator for the relative motion of the system Z_n is nonempty for some values of the total momentum of the particles in the system. The stability of n-particle systems was previously studied only for nonrelativistic particles. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 528–537, September, 2007.     

Statistical inference of spectral estimation for continuous-time MA processes with finite second moments

In this paper we investigate a continuous-time MA (moving average) process (X _t ) _t≥0 sampled at an equally spaced time grid {Δ,2Δ, …, nΔ}, where the grid distance Δ > 0 is fixed and n denotes the number of observations, in the frequency domain. We derive for the process (X _kΔ) _k∈? with finite second moments the asymptotic behavior of the periodogram and of the lag-window spectral density estimator. The periodogram is not a consistent estimator for the spectral density of (X _kΔ) _k∈?. Different periodogram frequencies are asymptotically independent and exponentially distributed like for ARMA processes in discrete time. This result is basic for frequency bootstraps. In contrast, the lag-window spectral density estimator is a consistent estimator for the spectral density of (X _kΔ) _k∈? and moreover, it is asymptotically normally distributed.     

Upper bounds on the depth of symmetric Boolean functions

A new method for implementing the counting function with Boolean circuits is proposed. It is based on modular arithmetic and allows us to derive new upper bounds for the depth of the majority function of n variables: 3.34log₂ n over the basis B ₂ of all binary Boolean functions and 4.87log₂ n over the standard basis B ₀ = {∧, ∨, ?}. As a consequence, the depth of the multiplication of n-digit binary numbers does not exceed 4.34log₂ n and 5.87log₂ n over the bases B ₂ and B ₀, respectively. The depth of implementation of an arbitrary symmetric Boolean function of n variables is shown to obey the bounds 3.34log₂ n and 4.88log₂ n over the same bases.     

Schrödinger operators with δ′-interactions and the Krein-Stieltjes string

We investigate one dimensional symmetric Schrödinger operator H _{X, β} with δ′-interactions of strength β = “β _n ” _{n = 1} ^∞ ? ? on a discrete set X = “x _n ” _{n = 1} ^∞ ? [0, b), b ≤ +∞ (x _n ↑ b). We consider H _{X, β} as an extension of the minimal operator H _min:= ?d ²/dx ²?W ₀ ^2.2 (?\X) and study its spectral properties in the frame-work of the extension theory by using the technique of boundary triplets and the corresponding Weyl functions. The construction of a boundary triplet for H _min ^* is given in the case d _*:= inf_{n ∈ ?}\x _n ? x _{n ? 1}\ = 0. We show that spectral properties like self-adjointness, lower semiboundedness, nonnegativity, and discreteness of the spectrum of the operator H _{X, β} correlate with the corresponding properties of a certain Jacobi matrix. In the case β _n > 0, n ∈ ?, these matrices form a subclass of Jacobi matrices generated by the Krein-Stieltjes strings. The connection discovered enables us to obtain simple conditions for the operator H _{X, β} to be self-adjoint, lower semibounded and discrete. These conditions depend significantly not only on β but also on X. Moreover, as distinct from the case d _* > 0, the spectral properties of Hamiltonians with δ- and δ′-interactions in the case d _* = 0 substantially differ.     

Numerical analysis of the spectrum of the Orr-Sommerfeld problem

A high-accuracy method for computing the eigenvalues λ _n and the eigenfunctions of the Orr-Sommerfeld operator is developed. The solution is represented as a combination of power series expansions, and the latter are then matched. The convergence rate of the expansions is analyzed by applying the theory of recurrence equations. For the Couette and Poiseuille flows in a channel, the behavior of the spectrum as the Reynolds number R increases is studied in detail. For the Couette flow, it is shown that the eigenvalues λ _n regarded as functions of R have a countable set of branch points R _k > 0 at which the eigenvalues have a multiplicity of 2. The first ten of these points are presented within ten decimals.     

On Taylor and other joint spectra for commuting n-tuples of operators

Let R and S be commuting n-tuples of operators. We will give some spectral relations between RS and SR that extend the case of single operators. We connect the Taylor spectrum, the Fredholm spectrum and some other joint spectra of RS and SR. Applications to Aluthge transforms of commuting n-tuples are also provided.     

Non-self-adjoint Jacobi matrices with a rank-one imaginary part

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank-one imaginary part. It is shown that given a set of n not necessarily distinct nonreal numbers in the open upper (lower) half-plane uniquely determines an n×n Jacobi matrix with a rank-one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity. Algorithms of reconstruction for such finite Jacobi matrices are presented. A new model complementing the well-known Livsic triangular model for bounded linear operators with a rank-one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix. We show that any bounded, prime, non-self-adjoint linear operator with a rank-one imaginary part acting on some finite-dimensional (respectively separable infinite-dimensional Hilbert space) is unitarily equivalent to a finite (respectively semi-infinite) non-self-adjoint Jacobi matrix. This obtained theorem strengthens a classical result of Stone established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy-Simon uniqueness theorems for finite Jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and nonreal. A unique Jacobi matrix, unitarily equivalent to the operator of integration in the Hilbert space L₂[0,l] is found as well as spectral properties of its perturbations and connections with the well-known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank-one imaginary part.