The Experts below are selected from a list of 114 Experts worldwide ranked by ideXlab platform
Saad Zagloul Rida - One of the best experts on this subject based on the ideXlab platform.
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explicit formulae for the powers of a schrodinger like Ordinary Differential Operator ii
Chaos Solitons & Fractals, 2002Co-Authors: Rainer Schimming, Saad Zagloul RidaAbstract:Abstract We derive some new formulae in the Schur pseudoDifferential Operator calculus with respect to one real variable x , namely for (i) the inverse of a Schrodinger-like Operator D r + u , where D =d/dx, r is a positive integer, and u = u ( x ) is a smooth function; (ii) arbitrary integer powers of D + u ; (iii) the square root of the Schrodinger Operator D 2 + u . The result (iii) is closely connected with the theory of the KdV hierarchy.
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explicit formulae for the powers of a schrodinger like Ordinary Differential Operator
Journal of Physics A, 1998Co-Authors: Saad Zagloul RidaAbstract:We call a Schrodinger-like Differential Operator and derive some explicit formulae for the powers or iterations of it.
V M Kurbanov - One of the best experts on this subject based on the ideXlab platform.
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on an analog of the riesz theorem and the basis property of the system of root functions of a Differential Operator in l p ii
Differential Equations, 2013Co-Authors: V M KurbanovAbstract:We consider an Ordinary Differential Operator of arbitrary order, obtain necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of the system of root functions of the given Operator in Lp.
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analog of the riesz theorem and the basis property in l p of a system of root functions of a Differential Operator i
Differential Equations, 2013Co-Authors: V M KurbanovAbstract:An Ordinary Differential Operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this Operator in Lp.
Michael Unser - One of the best experts on this subject based on the ideXlab platform.
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B-Spline-Based Exact Discretization of Continuous-Domain Inverse Problems With Generalized TV Regularization
IEEE Transactions on Information Theory, 2019Co-Authors: Thomas Debarre, Harshit Gupta, Julien Fageot, Michael UnserAbstract:We study continuous-domain linear inverse problems with generalized total-variation (gTV) regularization, expressed in terms of a regularization Operator L. It has recently been proved that such inverse problems have sparse spline solutions, with fewer jumps than the number of measurements. Moreover, the type of spline solely depends on L (L-splines) and is independent of the measurements. The continuous-domain inverse problem can be recast in an exact way as a finite-dimensional problem by restricting the search space to splines with knots on a uniform finite grid. However, expressing the L-spline coefficients in the dictionary basis of the Green’s function of L is ill-suited for practical problems due to its infinite support. Instead, we propose to formulate the problem in the B-spline dictionary basis, which leads to better-conditioned problems. As we make the grid finer, we show that a solution of the continuous-domain problem can be approached arbitrarily closely with functions of this search space. This result motivates our proposed multiresolution algorithm, which computes sparse solutions of our inverse problem. We demonstrate that this algorithm is computationally feasible for 1D signals when L is an Ordinary Differential Operator.
Vadim Mogilevskii - One of the best experts on this subject based on the ideXlab platform.
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unitary equivalence of proper extensions of a symmetric Operator and the weyl function
Integral Equations and Operator Theory, 2013Co-Authors: Seppo Hassi, M M Malamud, Vadim MogilevskiiAbstract:Let A be a densely defined simple symmetric Operator in \({\mathfrak{H}}\), let \({\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}\) be a boundary triplet for A * and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A 0}, uniquely up to the unitary similarity. Here \({A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}\). At the same time the Weyl function corresponding to a boundary triplet for a dual pair of Operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric \({A \subset A^*}\) and a special boundary triplet \({\widetilde{\Pi}}\) for{A, A} such that the corresponding Weyl function is \({\widetilde{M}(z) = K^*(B-M(z))^{-1} K}\), where B is a non-self-adjoint bounded Operator in \({\mathcal{H}}\). We are interested in the problem whether the result on the unitary similarity remains valid for \({\widetilde{M}(\cdot)}\) in place of M(·). We indicate some sufficient conditions in terms of the Operators A 0 and \({A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}\), which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order Ordinary Differential Operator A in \({L^2(\mathbb{R}_+)}\), we show that \({\widetilde{M}(\cdot)}\) defined in \({\Omega_+ \subset \mathbb{C}_+}\)determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac Operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function \({\widetilde{M}(\cdot)}\) does not determine A B even up to the similarity.
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minimal spectral functions of an Ordinary Differential Operator
Proceedings of the Edinburgh Mathematical Society (Series 2), 2012Co-Authors: Vadim MogilevskiiAbstract:Let $l[y]$ be a formally selfadjoint Differential expression of an even order on the interval $[0,b> \;(b\leq \infty)$ and let $L_0$ be the corresponding minimal Operator. By using the concept of a decomposing boundary triplet we consider the boundary problem formed by the equation $l[y]-ly=f\;(f\in L_2[0,b>)$ and the Nevanlinna $l$-depending boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the $m$-function, which in the case of selfadjoint decomposing boundary conditions coincides with the classical characteristic (Titchmarsh-Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e., all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indices $n_\pm(L_0)$ and not decomposing boundary conditions) the known estimate of the spectral multiplicity of the (exit space) selfadjoint extension $\wt A\supset L_0$. The results of the paper are obtained for expressions $l[y]$ with Operator valued coefficients and arbitrary (equal or unequal) deficiency indices $n_\pm(L_0)$.
Oksana Guba - One of the best experts on this subject based on the ideXlab platform.
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using root functions for eigenvalue problems of Ordinary Differential Operators
Numerical Functional Analysis and Optimization, 2009Co-Authors: Oksana GubaAbstract:We study eigenvalue problems for an Ordinary Differential Operator L acting on L 2(ℝ)-spaces (Problem 1) and on L 2(J)-spaces (Problem 2). Here J is a bounded but large interval. Assuming that in Problem 1 the spectral parameter s lies in the set of normal points of L, we show that the structure of eigenspaces for both problems is similar to the structure of finite complex-valued matrices. In the case of a finite matrix, the geometry of eigenspaces is described by the Jordan form. In the case of Ordinary Differential Operators, the corresponding geometry is described by a sequence of root functions. Therefore, the main tool of our studies is root functions for complex-valued analytical matrix functions.