1. Schroedinger operators : spectra, scattering and semiclassical behaviour.- Negative discrete spectrum of the Schroedinger operator with large coupling constant : qualitative discussion.- Discrete spectrum in the gaps of the continuous one in the large-coupling-constant limit.- On the asymptotics of discrete spectrum for the Schroedinger operator in electric and homogeneous magnetic fields.- Asymptotics of the discrete spectrum of Hamiltonians of quantum systems with a homogeneous magnetic field.- Asymptotics of eigenvalues for many-particle Hamiltonians at symmetry subspaces.- Spectral asymptotics with highly accurate remainder estimates.- Bound states and resonances in quantum wires.- Spectral properties of the operators H? = ??XX + p(x)? + v(?x), p is periodic.- Discrete spectrum for a periodic Schroedinger operator perturbed by a decreasing potential.- On a complete description of the principal discrete series of spectral invariants of the Hill operator.- WKB-approximations from the perturbation theory viewpoint.- Moving potentials and the completeness of wave operators. Part II : Propagating observables on scattering states.- Representation of the three-body S-matrix in terms of effective amplitudes.- A three-body one-dimensional system with increasing interaction : wave function asymptotics.- Surprises of quantum tunneling (something about the velocity of sub-barrier motion).- 2. Point and contact interactions, self-adjoint extensions.- Lifshitz tails and non-Lifshitz tails for one-dimensional random point interactions.- Point interactions with an internal structure as limits of nonlocal point interactions.- Green's function for the Aharonov-Bohm effect with a non-Abelian gauge group.- A model of zero-width slits and the real diffraction problem.- Construction of an inelastic scatterer in nanoelectronics by the extension-theory methods.- On boundary theory for Schroedinger operators and stochastic processes.- Dirac Hamiltonian with Coulomb potential and contact intraction on a sphere.- Self-adjoint extensions of Schroedinger operators with singular potentials.- Currents and the extension theory.- 3. Quantum chaos.- Chaotic quantum billiards.- Relevance of the localization to quasienergy statistics in quantum chaotic systems.- Singular continuous quasi-energy spectrum in the kicked rotator with separable perturbation : onset of quantum chaos ?.- Relation between correlation functions and spectrum statistics in the region of quantum chaos.- 4. Other topics.- Localization effects in non-homogeneous dielectrics.- A self-adjoint dilation of the linear Boltzmann operator.- Kato problem for functional-differential equations and difference Schroedinger operators.- An existence theorem for some nonlinear nonlocal Schroedinger operators and the solitonlike dynamics for the corresponding dynamical systems.- The Sturm-Liouville problem with a potential linear in spectral parameter.- Stochastic model of tree growth.- Critical phenomenon in the stationary model of random growth.- On a quantum-classical connection, hidden symmetries and a model of Josephson junction.- Open multiquantum systems. Method of a generating functional.- Generalized functions and their applications.- List of unpublished contributions.- List of participants.
Dynamical C*-systems without norm-continuity properties (e.g., the dynamics of a boson-type system) are studied. The respective dynamics cannot be perturbed by (bounded, self-adjoint) elements of the underlying abstract C*-algebra. Hence, it is necessary to define notions like ‘‘invariant state,’’ ‘‘ground state,’’ ‘‘KMS-state,’’ etc. with respect to the perturbed dynamics in suitable Hilbert space representations. Here, the original dynamical system is supplemented by an auxiliary pointwise norm-continuous dynamical system in such a way that invariant states of the original system correspond to certain invariant states of the auxiliary system. This bijective correspondence is sequentially continuous and preserves the KMS- (ground state) characterizing conditions. As a typical application it is verified that Spohn’s ground states of the spin-boson model (arising as a temperature to zero limit of thermic equilibrium states) are ground states in the algebraic sense, i.e., are eigenstates of the respective (Gel’fand–Naimark–Segal) Hamiltonian with an eigenvalue at the lower end of the Hamiltonian’s spectrum.
It remains a difficult task to establish contact between the rigorous quantum theory built on sound foundations by Birkhoff and von Neumann and others and the models usually used to describe resonances. A theorem is proved which gives a set of necessary and sufficient conditions for the existence of solutions of formal perturbation and kindred equations in atomic physics and establishes certain uniqueness properties of solutions which exist. In another theorem it is proved that whenever a state vector belonging to a point eigenvalue of the Hamiltonian is perturbed, one has the phenomenon of spectral concentration and it is possible that in cases where the resulting spectral concentration does not constitute a point eigenvalue of the perturbed Hamiltonian, one has the phenomenon which probably corresponds to a resonance. It is shown that in the autoionising 2s2p 1.3 P states of helium, a solution of the formal perturbation equations exists in the underlying Hilbert space only in the zero order, though a formal solution which is outside the underlying Hilbert space exists and belongs to the space of distributions on a subset of the Hilbert space. It is also shown that in cases where a complex eigenvalue is obtained by interfering with a self-adjoint operator, the phenomenon is not related to deficiency indices. The mathematical background of the work of Sharma and Bowtell (1975) has been examined and some progress has been made towards finding the rigorous mathematical framework in which the theory should find a proper formulation.
A new method for treating the most important eigenvalue problems in quantum mechanics is developed. The solutions can be found immediately once the equations are factorized by means of linear operators. These operators acting on a normalized eigenfunction change it into a new normalized eigenfunction and all solutions can be found once the basic eigenfunction is known. This basic eigenfunction is a solution of a simple differential equation of the first order. The underlying theory is explained more fully on a special case (Section 1) and then the rules of procedure are formulated explicitly (Section 2). The rest of the paper contains applications including the Kepler problem treated according to Dirac's theory.
This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials
V∈Hs, where
s>max{d/2−2,−1} and
d denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order
2s+2 for eigenvalues and order
s+1 for eigenfunctions in the
H1 norm. More importantly, the
H1 error with respect to the projected eigenfunction converges with a higher order
s+1+b, where
b=min{s+2−d/2−ε,s+1,2}>0 for arbitrarily small
ε>0, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders
2s+2+2b for eigenvalues and
s+1+b for eigenfunctions in the
H1 norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in
Hs+4 together with
d+1 asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.