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We study realizations generated by the original Weyl–Titchmarsh functions m ∞ (z) and m α (z) . It is shown that the Herglotz–Nevanlinna functions (- m ∞ (z)) and (1 / m ∞ (z)) can be realized as the impedance functions of the corresponding Shrödinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz–Nevanlinna functions (- m α (z)) and (1 / m α (z)) . We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrödinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrödinger L-systems with equal boundary parameters. A condition for two Shrödinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper. [ABSTRACT FROM AUTHOR]

We consider a system of N interacting fermions in R 3 confined by an external potential and in the presence of a homogeneous magnetic field. The intensity of the interaction has the mean-field scaling 1/N. With a semi-classical parameter ħ ∼ N - 1 / 3 , we prove convergence in the large N limit to the appropriate magnetic Thomas–Fermi-type model with various strength scalings of the magnetic field. [ABSTRACT FROM AUTHOR]

It is well known that the resolvent of the free Schrödinger operator on weighted L 2 spaces has norm decaying like λ - 1 2 at energy λ . There are several works proving analogous high frequency estimates for magnetic Schrödinger operators, with large long or short range potentials, in dimensions n ≥ 3 . We prove that the same estimates remain valid in all dimensions n ≥ 2 . [ABSTRACT FROM AUTHOR]

We study the bottom of the spectrum of the Anderson Hamiltonian H L : = - ∂ x 2 + ξ on [0, L] driven by a white noise ξ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as L → ∞ , the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on R with intensity e x d x , and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentered around its maximum and properly rescaled, is given by the inverse of a hyperbolic cosine. We also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate, and we obtain very precise information on the zeros and local maxima of these eigenfunctions. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits. [ABSTRACT FROM AUTHOR]

We study linear perturbations of Donoghue classes of scalar Herglotz–Nevanlinna functions by a real parameter Q and their representations as impedance of conservative L-systems. Perturbation classes M Q , M κ Q , M κ - 1 , Q are introduced and for each class the realization theorem is stated and proved. We use a new approach that leads to explicit new formulas describing the von Neumann parameter of the main operator of a realizing L-system and the unimodular one corresponding to a self-adjoint extension of the symmetric part of the main operator. The dynamics of the presented formulas as functions of Q is obtained. As a result, we substantially enhance the existing realization theorem for scalar Herglotz–Nevanlinna functions. In addition, we solve the inverse problem (with uniqueness condition) of recovering the perturbed L-system knowing the perturbation parameter Q and the corresponding non-perturbed L-system. Resolvent formulas describing the resolvents of main operators of perturbed L-systems are presented. A concept of a unimodular transformation as well as conditions of transformability of one perturbed L-system into another one are discussed. Examples that illustrate the obtained results are presented. [ABSTRACT FROM AUTHOR]

We consider the wave equation with a boundary condition of memory type. Under natural conditions on the acoustic impedance of the boundary a corresponding semigroup of contractions is known to exist. With the help of quantified Tauberian theorems we establish energy decay rates via resolvent estimates on the generator of the semigroup. Using a variational approach, we reduce resolvent estimates to estimates for a sesquilinear form induced by an operator characteristic function arising form the matrix representation of the generator. Under not too strict additional assumptions on the acoustic impedance we establish an upper bound on the resolvent. For the wave equation on the interval or the disc we prove our estimates to be sharp. [ABSTRACT FROM AUTHOR]

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh-Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. [ABSTRACT FROM AUTHOR]

Let ( E, h) be a holomorphic, Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of E. If E is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian. [ABSTRACT FROM AUTHOR]

We study the impedance functions of conservative L-systems with the unbounded main operators. In addition to the generalized Donoghue class $${\mathfrak {M}}_\kappa $$ of Herglotz-Nevanlinna functions considered by the authors earlier, we introduce 'inverse' generalized Donoghue classes $${\mathfrak {M}}_\kappa ^{-1}$$ of functions satisfying a different normalization condition on the generating measure, with a criterion for the impedance function $$V_\Theta (z)$$ of an L-system $$\Theta $$ to belong to the class $${\mathfrak {M}}_\kappa ^{-1}$$ presented. In addition, we establish a connection between 'geometrical' properties of two L-systems whose impedance functions belong to the classes $${\mathfrak {M}}_\kappa $$ and $${\mathfrak {M}}_\kappa ^{-1}$$ , respectively. In the second part of the paper we introduce a coupling of two L-system and show that if the impedance functions of two L-systems belong to the generalized Donoghue classes $${\mathfrak {M}}_{\kappa _1}$$ ( $${\mathfrak {M}}_{\kappa _1}^{-1}$$ ) and $${\mathfrak {M}}_{\kappa _2}$$ ( $${\mathfrak {M}}_{\kappa _2}^{-1}$$ ), then the impedance function of the coupling falls into the class $${\mathfrak {M}}_{\kappa _1\kappa _2}$$ . Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class $${\mathfrak {M}}={\mathfrak {M}}_0$$ is coupled with any other L-system, the impedance function of the coupling belongs to $${\mathfrak {M}}$$ (the absorbtion property). Observing the result of coupling of n L-systems as n goes to infinity, we put forward the concept of a limit coupling which leads to the notion of the system attractor, two models of which (in the position and momentum representations) are presented. All major results are illustrated by various examples. [ABSTRACT FROM AUTHOR]