Search results for "Mathematical physics"
showing 10 items of 2687 documents
Nowhere differentiable intrinsic Lipschitz graphs
2021
We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.
The annular decay property and capacity estimates for thin annuli
2016
We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted $\mathbf{R}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar\'e inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincar\'e inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted $\mathbf{R}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also character…
Hardy-Orlicz Spaces of conformal densities
2014
We define and prove characterizations of Hardy-Orlicz spaces of conformal densities.
Intertwining operators for non-self-adjoint hamiltonians and bicoherent states
2016
This paper is devoted to the construction of what we will call {\em exactly solvable models}, i.e. of quantum mechanical systems described by an Hamiltonian $H$ whose eigenvalues and eigenvectors can be explicitly constructed out of some {\em minimal ingredients}. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in infinite-dimensional Hilbert spaces, are discussed.
Non-self-adjoint hamiltonians defined by Riesz bases
2014
We discuss some features of non-self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that the eigenvectors form a Riesz basis of Hilbert space. Among other things, {we give conditions under which these Hamiltonians} can be factorized in terms of generalized lowering and raising operators.
Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights.
2018
The fractional Laplacian $(- \Delta)^{\alpha /2}$, $\alpha \in (0,2)$ has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of $\alpha $-stable stochastic processes in $R^n$. On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data respecting fractional Laplacian should actually be. This ambiguity holds true not only for each specific choice of the process behavior at the boundary (like e.g. absorbtion, reflection, conditioning or boundary taboos), but extends as well to its particular technical implementation (Dirchlet, Neumann, etc. problems). The inferred jump-type …
$O^\star$-algebras and quantum dynamics: some existence results
2008
We discuss the possibility of defining an algebraic dynamics within the settings of O -algebras. Compared to our previous results on this subject, the main improvement here is that we are not assuming the existence of some Hamiltonian for the full physical system. We will show that, under suitable conditions, the dynamics can still be defined via some limiting procedure starting from a given regularized sequence. © 2008 American Institute of Physics.
The Poisson embedding approach to the Calderón problem
2020
We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calder\'on type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result by Lassas and Uhlmann (2001) solving the Calder\'on problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.
RADEMACHER'S THEOREM IN BANACH SPACES WITHOUT RNP
2017
Abstract We improve a Duda’s theorem concerning metric and w *-Gâteaux differentiability of Lipschitz mappings, by replacing the σ-ideal 𝓐 of Aronszajn null sets [ARONSZAJN, N.: Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147–190], with the smaller σ-ideal 𝓐 of Preiss-Zajíček null sets [PREISS, D.—ZAJÍČEK, L.: Directional derivatives of Lipschitz functions, Israel J. Math. 125 (2001), 1–27]. We also prove the inclusion C̃ o ⊂ 𝓐, where C̃ o is the σ-ideal of Preiss null sets [PREISS, D.: Gâteaux differentiability of cone-monotone and pointwise Lipschitz functions, Israel J. Math. 203 (2014), 501–534].
Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions
2015
We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the first kind. The coefficients of different expansions obey four-, five-, or six-term recurrence relations that are reduced to ones involving less number of terms only in a few exceptional cases. The conditions for deriving finite-sum solutions via termination of the series are discussed.