The uniqueness of the solution of the Schrodinger equation with discontinuous coefficients

Consider the Schroeodinger equation: - Du(x) - l(x)u + s(x)u = 0, where D is the Laplacian, l(x) > 0 and s(x) is dominated by l(x). We shall extend the celebrated Kato's result on the asymptotic behavior of the solution to the case where l(x) has unbounded discontinuity. The result will be used to establish the limiting absorption principle for a class of reduced wave operators with discontinuous coefficients.Comment: 29 (twenty-nine) pages; no figures; to appear in Reviews of Mathematical Physic

Perturbation theory for eigenvalues and resonances of Schrodinger hamiltonians

Suppose that e2[epsilon]|x|V [set membership, variant] ReLP(R3) for some p > 2 and for g [set membership, variant] R, H(g) = - [Delta] + g V, H(g) = -[Delta] + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues [lambda](g) as they are absorbed by the continuous spectrum, that is [lambda](g) [NE pointing arrow]6 0 as g [searr]5 g0 > 0. We find a series expansion in powers of (g - g0)1/2, [lambda](g) = [summation operator]n = 2[infinity] an(g - g0)n/2 whose values for g g0 correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23314/1/0000253.pd

Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators

Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for C^\om (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a H\"older continuity class C^{0,\al}. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.Comment: 8 page

Quantum Averaging I: Poincar\'e--von Zeipel is Rayleigh--Schr\"odinger

An exact analogue of the method of averaging in classical mechanics is constructed for self--adjoint operators. It is shown to be completely equivalent to the usual Rayleigh--Schr\"odinger perturbation theory but gives the sums over intermediate states in closed form expressions. The anharmonic oscillator and the Henon--Heiles system are treated as examples to illustrate the quantum averaging method.Comment: 12 pages, LaTeX, to appear in Journ. Phys.

Weighted Sobolev spaces of radially symmetric functions

We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.Comment: 38 page

Adiabatic Approximation for weakly open systems

We generalize the adiabatic approximation to the case of open quantum systems, in the joint limit of slow change and weak open system disturbances. We show that the approximation is ``physically reasonable'' as under wide conditions it leads to a completely positive evolution, if the original master equation can be written on a time-dependent Lindblad form. We demonstrate the approximation for a non-Abelian holonomic implementation of the Hadamard gate, disturbed by a decoherence process. We compare the resulting approximate evolution with numerical simulations of the exact equation.Comment: New material added, references added and updated, journal reference adde

Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials

We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to second-order interpolation inequalities with weights.Comment: 19 page

Optimal portfolios for financial markets with Wishart volatility

We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain explicitly the optimal portfolio strategy and the value function in some parameter settings. In particular when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is indeed to identify when the solution of the HJB equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the in uence of the investors\u27 risk aversion on the hedging demand

Regular singular Sturm-Liouville operators and their zeta-determinants

We consider Sturm-Liouville operators on the line segment [0, 1] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski- determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten-Loya-Park for general separated boundary conditions but only special regular singular potentials.Comment: 38 pages, 2 figures; Completely revised according to the referees comprehensive suggestions; v3: minor corrections, accepted for publication in Journal of Functional Analysi

Controllability of the discrete-spectrum Schrodinger equation driven by an external field

We prove approximate controllability of the bilinear Schr\"odinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials