Reviews in Mathematical Physics

We give a method to obtain estimates on distorted resolvents which we apply to the Stark effect with general potentials. This paper also contains a geometric perturbation theory which together with the estimates mentioned above, shows the existence of resonances and gives an upper bound on the width of resonances in various situations.

Starting from the one-dimensional Schrodinger equation with symmetric potential V_s(x), a general method is presented in order to obtain a family of partially isospectral hamiltonians. Arguments concerning supersymmetric transformations, factorization procedures and Riccati equations are invoked along the article. As a result of the appearance of singular superpotentials, the physical meaning of our method can be summarized as follows: only the odd wave-functions of the original potential V_s(x) are transported via supersymmetry into the Hilbert space associated with the partner V_p(x). In such a case the degeneracy of energy levels is partially broken. Supersymmetry is neither exact nor spontaneously broken but realizes itself acting on wave functions vanishing at x=0. While the domain of the original hamiltonian H extends along the whole real axis, the susy partner H_p reduces to the half-line (x≤0 or x≥0). To illustrate how the procedure works in practice we resort to a symmetric potential in the Posch-Teller class containing both discrete and continuous spectra.

We consider a ferromagnetic Ising spin system, consisting of m+1, d-dimensional, layers with “–” boundary condition on the bottom layer and “+” on the top layer. When β is larger than β_cr, the inverse critical temperature for the d-dimensional Ising model, the interface generated by the boundary conditions is expected to be halfway between bottom and top, for m odd, and just above or below the middle layer, for m even (each possibility with probability ). In this paper, we prove the above assertion under the condition that β≥const . m and partly for β>β_cr.

We shortly review the progress in the domain of stochastic dynamics for quantum spin systems on a lattice. We also present some new results obtained in the framework of noncommutative spaces. In particular, using noncommutative Radon-Nikodym theorem of A. Connes we construct Markov generators of stochastic dynamics of spin flip type for systems at high temperatures or on one-dimensional lattice and with interactions of finite range at arbitrary temperatures.

For singular potentials in quantum mechanics it can happen that the Schrödinger operator is not esssentially self-adjoint on a natural domain, i.e., each self-adjoint extension is a candidate for the right physical Hamiltonian. Traditional way to single out this Hamiltonian is the removing cut-offs for regularizing potential. Connecting regularization and extension theory we develop an abstract operator method to treat the problem of the right Hamiltonian. We show that, using the notion of the maximal (with respect to the perturbation) Friedrichs extension of unperturbed operator, one can classify the above problem as wellposed or ill-posed depending on intersection of the quadratic form domain of perturbation and deficiency subspace corresponding to restriction of unperturbed operator to stability domain. If this intersection is trivial, then the right Hamiltonian is unique: it coincides with the form sum of perturbation and the Friedrich extension of the unperturbed operator restricted to the stability domain. Otherwise it is not unique: the family of “right Hamiltonians” can be described in terms of symmetric extensions reducing the ill-posed problem to the well-posed problem.