Narrow Results

The 3-dimensional Schrödinger operator H corresponding to a uniform magnetic field and a periodic electric potential V is considered. Under the condition of magnetic flux rationality, and with a mild regularity assumption on V, it is shown that the number of gaps in the spectrum of H is finite.

Let H be a self-adjoint operator on a Hilbert space , T be a symmetric operator on and K(t)(t ∈ ℝ) be a bounded self-adjoint operator on . We say that (T, H, K) obeys the generalized weak Weyl relation (GWWR) if e^-itHD(T) ⊂ D(T) for all t ∈ ℝ and Te^-itHψ = e^-itH(T+K(t))ψ, ∀ψ ∈ D(T) (D(T) denotes the domain of T). In the context of quantum mechanics where H is the Hamiltonian of a quantum system, we call T a generalized time operator of H. We first investigate, in an abstract framework, mathematical structures and properties of triples (T, H, K) obeying the GWWR. These include the absolute continuity of the spectrum of H restricted to a closed subspace of , an uncertainty relation between H and T (a "time-energy uncertainty relation"), the decay property of transition probabilities |〈ψ,e^-itHϕ〉|² as |t| → ∞ for all vectors ψ and ϕ in a subspace of , where 〈·,·〉 denotes the inner product of . We describe methods to construct various examples of triples (T, H, K) obeying the GWWR. In particular, we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces and fermion Fock spaces) which may have applications to quantum field models with interactions.

We consider the magnetic Schrödinger operator on R². The magnetic field is the sum of a homogeneous magnetic field and periodically varying pointlike magnetic fields on a lattice. We shall give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. This condition is also necessary for the lowest Landau level. In the threshold case, we see that the spectrum near the lowest Landau level is purely absolutely continuous. Moreover, we shall give an estimate for the density of states for Landau levels and their gaps. The proof is based on the method of Geyler and Šťovíček, the magnetic Bloch theory, and canonical commutation relations.

An abstract operator theory is developed on operators of the form A_H(t) := e^itHAe^-itH, t ∈ ℝ, with H a self-adjoint operator and A a linear operator on a Hilbert space (in the context of quantum mechanics, A_H(t) is called the Heisenberg operator of A with respect to H). The following aspects are discussed: (i) integral equations for A_H(t) for a general class of A; (ii) a sufficient condition for D(A), the domain of A, to be left invariant by e^-itH for all t ∈ ℝ; (iii) a mathematically rigorous formulation of the Heisenberg equation of motion in quantum mechanics and the uniqueness of its solutions; (iv) invariant domains in the case where H is an abstract version of Schrödinger and Dirac operators; (v) applications to Schrödinger operators with matrix-valued potentials and Dirac operators.

We investigate Schrödinger operators with δ- and δ′-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result, we prove an operator inequality for the Schrödinger operators with δ- and δ′-interactions which is based on an optimal coloring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ′-interactions.

This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator, we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrödinger operator on the boundary with an external Yang–Mills potential and a curvature-induced potential.

We investigate spreading rates of one-dimensional quantum states under the Schrödinger time-evolution. The focus of this paper is on the states that either have finite support or decay exponentially at $\pm \infty$. In particular, we extend results of Damanik and Tcheremchantsev on estimating transport exponents that were originally proved to hold for the initial states supported on a single site. These general upper and lower estimates are then applied to several classes of models, including Sturmian, quasi-periodic and substitution-generated potentials, and the random polymer model.

We study the behavior of solutions of the Helmholtz equation $(-{\mathrm{\Delta }}_{\mathrm{\text{disc}},h}-E)u_h=f_h$ on a periodic lattice as the mesh size $h$ tends to 0. Projecting to the eigenspace of a characteristic root ${\lambda }_h(\xi )$ and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution $u_h$ converges to that for the equation $(P(D_x)-E)v=g$ for a continuous model on $R^d$, where ${\lambda }_h(\xi )\to P(\xi )$. For the case of the hexagonal and related lattices, in a suitable energy region, it converges to that for the Dirac equation. For the case of the square lattice, triangular lattice, hexagonal lattice (in another energy region) and subdivision of a square lattice, one can add a scalar potential, and the solution of the lattice Schrödinger equation $(-{\mathrm{\Delta }}_{\mathrm{\text{disc}},h}+V_{\mathrm{\text{disc}},h}-E)u_h=f_h$ converges to that of the continuum Schrödinger equation $(P(D_x)+V(x)-E)u=f$.

We show that any symmetric operator H has a dense maximal b-stability domain (i.e. , b∈R¹) if and only if H is unbounded from above. This abstract result allows an application to singular perturbed Schrödinger operators which are not semi-bounded from below, i.e., to the so-called "fall to the center problem". It turns out that in this case the regularization problem is always ill-posed which implies that there is no unique "right Hamiltonian" for corresponding perturbed system. We give an example of singular perturbed Schrödinger operator for which stability domains are described explicitly.

This work deals with the inverse scattering problems for the two-dimensional Schrödinger equation

This paper is devoted to a generalized Hartree–Fock model in the Euclidean space. For large classes of free energy functionals, minimizers are obtained as long as the total charge of the system does not exceed a threshold which depends on a parameter that we call the temperature in analogy with models based on a thermodynamical approach. The usual Hartree–Fock model is recovered in the zero temperature limit. An orbital stability result for the Cauchy problem is deduced from the variational approach.

We study Bohr’s formula for fractal Schrödinger operators on the half line. These fractal Schrödinger operators are defined by fractal measures with overlaps and a locally bounded potential that tends to infinity. We first derive an analog of Bohr’s formula for these Schrödinger operators under some suitable conditions. Then we demonstrate how this result can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, m-fold convolution of Cantor-type measures, and a family of graph-directed self-similar measures that are essentially of finite type.

In this paper, it is proved the existence of a transmutation operator between two schrödinger equations with perturbed exactly solvable potential. An explicit formula for the solution of nucleus function by using Varsha and Jafari's method is also provided.

We consider the quantum mechanics of a charged particle in the presence of Dirac’s magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the Feynman–Kac formula expressing solutions of the imaginary time Schrödinger equation as stochastic integrals.

Let be a metric space with doubling measure, L a nonnegative self-adjoint operator in satisfying the Davies–Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type p_ω∈(0, 1] and ρ(t) = t^-1/ω^-1(t^-1) for all t∈(0, ∞). The authors introduce the Orlicz–Hardy space via the Lusin area function associated to the heat semigroup, and the BMO-type space . The authors then establish the duality between and ; as a corollary, the authors obtain the ρ-Carleson measure characterization of the space . Characterizations of , including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let and L = -Δ+V be a Schrödinger operator, where is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L^-1/2 is bounded from H_{ω, L}(ℝⁿ) to L(ω). Moreover, if there exist q₁, q₂∈(0, ∞) such that q₁<1<q₂ and [ω(t^q₂)]^q₁ is a convex function on (0, ∞), then several characterizations of the Orlicz–Hardy space H_{ω, L}(ℝⁿ), in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = t^p for all t ∈ (0, ∞) and p ∈ (0, 1).

The main purpose of this paper is to establish the existence of solutions for the semilinear Schrödinger equation

We consider a Schrödinger differential expression L₀ = Δ_M + V₀ on a Riemannian manifold (M,g) with metric g, where Δ_M is the scalar Laplacian on M and V₀ is a real-valued locally square integrable function on M. We consider a perturbation L₀ + V, where V is a non-negative locally square-integrable function on M, and give sufficient conditions for L₀ + V to be essentially self-adjoint on . This is an extension of a result of T. Kappeler.

We generalise the compactness of isospectral metrics in the C^∞-topology modulo gauge equivalence in dimension three to the case for Schrödinger operators with bounded potentials, as well as the finiteness of diffeomorphism types of isospectral manifolds.

In this paper, we investigate the spectrum and spectrality of the one-dimensional Schrödinger operator with a periodic PT-symmetric complex-valued potential.

A central hyperplane arrangement in ℂ² with multiplicity is called a "locus configuration" if it satisfies a series of "locus equations" on each hyperplane. Following [4], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting particle system on ℂ*: the charged trigonometric Calogero-Moser system. When the particles lie on S¹ ⊂ ℂ*, there is a unique equilibrium for this system. For certain classes of particle weight, this is enough to show that all the locus equations are satisfied, producing explicit examples of real locus configurations. This in turn produces new examples of Schrödinger operators with Baker–Akhiezer functions.