Search results for "Square-integrable function"
showing 10 items of 11 documents
Generators of Random Processes in Ultrametric Spaces and Their Spectra
2009
The L 2(\( \mathbb{S} \)) space of square integrable functions on an ultrametric space \( \mathbb{S} \) has rather specific structure. As a consequence in a natural way there appear in L 2(\( \mathbb{S} \)) the operators of which unitary counterparts in L 2(ℝn) would be difficult to construct. Such class of self-adjoint operators emerge from theory of random processes on ultrametric spaces. In this paper we collect known material on spectral properties of the generators of random processes on \( \mathbb{S}_B \) an ultrametric space of sequences. (The set of p-adic numbers is a subset of \( \mathbb{S}_B \).) Then we discuss structure of the eigenspaces of the generators.
Generalized Lebesgue points for Sobolev functions
2017
In this article, we show that a function $f\in M^{s,p}(X),$ $0<s\leq 1,$ $0<p<1,$ where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal{H}^h$-Hausdorff measure zero for a suitable gauge function $h.$
VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE
2003
Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Kothe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions ‖f‖X : Ω → [0,∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the pvariation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE(X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X))∗ = VE′ (X ∗), that VE(X) can be identified with space of cone absolutely summing operators fr…
Some remarks on few recent results on the damped quantum harmonic oscillator
2020
Abstract In a recent paper, Deguchi et al. (2019), the authors proposed an analysis of the damped quantum harmonic oscillator in terms of ladder operators. This approach was shown to be partly incorrect in Bagarello et al. (2019), via a simple no-go theorem. More recently, (Deguchi and Fujiwara, 2019), Deguchi and Fujiwara claimed that our results in Bagarello et al. (2019) are wrong, and compute what they claim is the square integrable vacuum of their annihilation operators. In this brief note, we show that their vacuum is indeed not a vacuum, and we try to explain what is behind their mistakes in Deguchi et al. (2019) and Deguchi and Fujiwara (2019). We also propose a very simple example …
THE SPACE OF STRING CONFIGURATIONS IN STRING FIELD THEORY
1990
In this paper we consider the set of maps from the interval [0, π] which constitute the argument of the functionals of a String Field Theory. We show that in order to correctly reproduce results of the dual model one has to include all square integrable functions in the functional integral, or Ω0 in terms of Sobolev spaces.
On Scattering and Bound States for a Singular Potential
1970
To understand the origin of the difficulties in the determination of the physical wavefunc tion for an attractive inverse square potential, we study a model in which the singularity at the origin is substituted by a repulsive core. The structure of the spectrum of energy levels is discussed in some detail. The physical interpretation of the solutions of the Schrodinger equation for a potential of the form - (-h 2 /2m) 11/ r 2 presents difficulties, which occur for 11 larger than (l + 1/2)\ where l is the angular momentum. The difficulties are due to the fact that the condition of square integrability usually imposed on the wavefunction is not sufficient in this case to determine phase shif…
A no-go result for the quantum damped harmonic oscillator
2019
Abstract In this letter we show that it is not possible to set up a canonical quantization for the damped harmonic oscillator using the Bateman Lagrangian. In particular, we prove that no square integrable vacuum exists for the natural ladder operators of the system, and that the only vacua can be found as distributions. This implies that the procedure proposed by some authors is only formally correct, and requires a much deeper analysis to be made rigorous.
An implicit non-linear time dependent equation has a solution
1991
has a solution (u, u, w). The operators &s(l) and a(t) are maximal monotone from a real Hilbert space V to its dual such that &(r) + 9?(r) are V-coercive and a(r) are not degenerate. A linear compact injection i embeds V to a real Banach space W and each d(r) is the strongly monotone subdifferential of a continuous convex function #(I, ) on W. The function f is square integrable. The functions W(r): V+ W* are Lipschitzian as V*-valued functions. Section 3 contains the theorems. The main result is Theorem 2. Theorems 3 and 4 demonstrate the smoothing effect on the initial condition. Their proofs are given in Section 4. They exploit the methods of di Benedetto and Showalter, [4], who studied …
Hilbert-Schmidt Hankel operators on the Segal-Bargmann space
2004
This paper considers Hankel operators on the Segal-Bargmann space of holomorphic functions onCn\mathbb {C}^nthat are square integrable with respect to the Gaussian measure. It is shown that in the case of a bounded symbolg∈L∞(Cn)g \in L^{\infty }(\mathbb {C}^n)the Hankel operatorHgH_gis of the Hilbert-Schmidt class if and only ifHg¯H_{\bar {g}}is Hilbert-Schmidt. In the case where the symbol is square integrable with respect to the Lebesgue measure it is known that the Hilbert-Schmidt norms of the Hankel operatorsHgH_gandHg¯H_{\bar {g}}coincide. But, in general, if we deal with bounded symbols, only the inequality‖Hg‖HS≤2‖Hg¯‖HS\|H_g\|_{HS}\leq 2\|H_{\bar {g}}\|_{HS}can be proved. The resul…
Product and moment formulas for iterated stochastic integrals (associated with Lévy processes)
2019
In this paper, we obtain explicit product and moment formulas for products of iterated integrals generated by families of square integrable martingales associated with an arbitrary Levy process. We...