arXiv subjects
Publications and source records attributed to Michael Ruzhansky.
We consider the Schrödinger equation with singular position dependent effective mass and prove that it is very weakly well posed. A uniqueness result is proved in an appropriate sense, moreover, we prove the consistency with the classical theory. In particular, this allows one to consider Delta-like or more singular masses.
In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on $\mathbb{R}^n$ with an explicit expression for the constant. Namely, we show that for all $0 0$ we have the inequality \[ \int_{\mathbb{R}^n}|f(x)|^2 \log \left( \frac{|f(x)|^2}{\|f\|^{2}_{L^2(\mathbb{R}^n)}}\right)\,dx+\frac{n}{s}(1+\log a)\|f\|_{L^2(\mathbb{R}^n)}^{2}\leq C(n,s,a)\|(-Δ)^{s/2}f\|^{2}_{L^2(\mathbb{R}^n)} \] with an explicit $C(n,s,a)$ depending on $a$, the order $s$, and the dimension $n$, and investigate the behaviour of $C(n,s,a)$ for large $n$. Notably, for large $n$ and when $s=1$, the constant $C(n,1,a)$ is asymptotically the same as the sharp constant of Lieb and Loss. Moreover, we prove a similar type inequality for functions $f \in L^q(\mathbb{R}^n)\cap W^{1,p}(\mathbb{R}^n)$ whenever $1<p<n$ and $p<q\leq \frac{p(n-1)}{n-p}$.
We establish the Kato-type smoothing property, i.e., global-in-time smoothing estimates with homogeneous weights, for the Schrödinger equation on Riemannian symmetric spaces of non-compact type and general rank. These form a rich class of manifolds with nonpositive sectional curvature and exponential volume growth at infinity, e.g., hyperbolic spaces. We achieve it by proving the Stein-Weiss inequality and the resolvent estimate of the corresponding Fourier multiplier, which are of independent interest. Moreover, we extend the comparison principles to symmetric spaces and deduce different types of smoothing properties for the wave equation, the Klein-Gordon equation, the relativistic and general orders Schrödinger equations. In particular, we observe that some smoothing properties, which are known to fail on the Euclidean plane, hold on the hyperbolic plane.
In this work we investigate a class of degenerate Schrödinger equations associated to degenerate elliptic operators with irregular potentials on $\Ran$ by introducing a suitable Hörmander metric $g$ and a $g$-weight $m$. We establish the well-posedness for the corresponding degenerate Schrödinger and degenerate parabolic equations. When the subelliticity is available on the degenerate elliptic operator we deduce spectral properties for a class of degenerate Hamiltonians. We also study the $L^p$ mapping properties for operators with symbols in the $S(m^{-β},g)$ classes in the spirit of classical Fefferman's $L^p$-bounds for the $(ρ, δ)$ calculus. Finally, within our $S(m,g)$-classes, sharp $L^p$-estimates and Schatten properties for Schrödinger operators for Hörmander sums of squares are also investigated.
In recent years, deep learning methods have achieved great success in various fields due to their strong performance in practical applications. In this paper, we present a light-weight neural network for Parkinson's disease diagnostics, in which a series of hand-drawn data are collected to distinguish Parkinson's disease patients from healthy control subjects. The proposed model consists of a convolution neural network (CNN) cascading to long-short-term memory (LSTM) to adapt the characteristics of collected time-series signals. To make full use of their advantages, a multilayered LSTM model is firstly used to enrich features which are then concatenated with raw data and fed into a shallow one-dimensional (1D) CNN model for efficient classification. Experimental results show that the proposed model achieves a high-quality diagnostic result over multiple evaluation metrics with much fewer parameters and operations, outperforming conventional methods such as support vector machine (SVM), random forest (RF), lightgbm (LGB) and CNN-based methods.
The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. In the present paper we prove the following estimate for the $q$-Mittag-Leffler function: \begin{eqnarray*} \frac{1}{1+Γ_q\left(1-α\right)z}\leq e_{α,1}\left(-z;q\right)\leq\frac{1}{1+Γ_q\left(α+1\right)^{-1}z}. \end{eqnarray*} for all $0 < α< 1$ and $z>0$. Moreover, we apply it to investigate the solvability results for direct and inverse problems for time-fractional pseudo-parabolic equations in quantum calculus for a large class of positive operators with discrete spectrum.
This note gives a wide-ranging update on the multiplier theorems by Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The proofs of the latter crucially rely on $L^p$-$L^q$ norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians and Rockland operators. By relating spectral projectors to heat kernels, explicit estimates of the $L^p$-$L^q$ norms can be immediately exploited for a much wider range of (connected unimodular) Lie groups and operators than previously known. The comparison with previously established bounds by authors show that the heat kernel estimates are sharp. As an application, it is shown that several consequences of the multiplier theorems, such as time asymptotics for the $L^p$-$L^q$ norms of the heat kernels and Sobolev-type embeddings, are then automatic for the considered operators.
We study heat and wave type equations on a separable Hilbert space $\mathcal{H}$ by considering non-local operators in time with any positive densely defined linear operator with discrete spectrum. We show the explicit representation of the solution and analyse the time-decay rate in a scale of suitable Sobolev space. We perform similar analysis on multi-term heat and multi-wave type equations. The main tool here is the Fourier analysis which can be developed in a separable Hilbert space based on the linear operator involved. As an application, the same Cauchy problems are considered and analysed in the setting of a graded Lie group. In this case our analysis relies on the group Fourier analysis. An extra ingredient in this framework allows, in the case of heat type equations, to establish $L^p$-$L^q$ estimates for $1\leqslant p\leqslant 2\leqslant q<+\infty$ for the solutions on graded Lie group groups. Examples and applications of the developed theory are given, either in terms of self-adjoint operators on compact or non-compact manifolds, or in the case of particular settings of graded Lie groups. The results of this paper significantly extend in different directions the results of Part I, where operators on compact Lie groups were considered. We note that the results obtained in this paper are also new already in the Euclidean setting of $\mathbb{R}^n$.
Let $A$ and $B$ be invariant linear operators with respect to a decomposition $\{H_{j}\}_{j\in \mathbb{N}}$ of a Hilbert space $\mathcal{H}$ in subspaces of finite dimension. We give necessary and sufficient conditions for the controllability of the Cauchy problem $$ u_t=Au+Bv,\,\,u(0)=u_0,$$ in terms of the (global) matrix-valued symbols $σ_A$ and $σ_B$ of $A$ and $B,$ respectively, associated to the decomposition $\{H_{j}\}_{j\in \mathbb{N}}$. Then, we present some applications including the controllability of the Cauchy problem on compact manifolds for elliptic operators and the controllability of fractional diffusion models for Hörmander sub-Laplacians on compact Lie groups. We also give conditions for the controllibility of wave and Schrödinger equations in these settings.
We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty$. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat type equation as well.
We study the Lane-Emden system involving the logarithmic Laplacian: $$ \begin{cases} \ \mathcal{L}_Δu(x)=v^{p}(x) ,& x\in\mathbb{R}^{n},\\ \ \mathcal{L}_Δv(x)=u^{q}(x) ,& x\in\mathbb{R}^{n}, \end{cases} $$ where $p,q>1$ and $\mathcal{L}_Δ$ denotes the Logarithmic Laplacian arising as a formal derivative $\partial_s|_{s=0}(-Δ)^s$ of fractional Laplacians at $s=0.$ By using a direct method of moving planes for the logarithmic Laplacian, we obtain the symmetry and monotonicity of the positive solutions to the Lane-Emden system. We also establish some key ingredients needed in order to apply the method of moving planes such as the maximum principle for anti-symmetric functions, the narrow region principle, and decay at infinity. Further, we discuss such results for a generalized system of the Lane-Emden type involving the logarithmic Laplacian.
In this note we prove the anisotropic version of the Shannon inequality. This can be conveniently realised in the setting of Folland and Stein's homogeneous groups. We give two proofs: one giving the best constant, and another one using the Kubo-Ogawa-Suguro inequality.
In this paper we consider the problem of estimation of oscillatory integrals with Mittag-Leffler functions in two variables. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals.
Let $G$ be a compact Lie group of dimension $n.$ In this work we characterise the membership of classical pseudo-differential operators on $G$ in the trace class ideal $S_{1}(L^2(G)),$ as well as in the setting of the Schatten ideals $S_{r}(L^2(G)),$ for all $r>0.$ In particular, we deduce Schatten characterisations of elliptic pseudo-differential operators of $(ρ,δ)$-type for the large range $0\leq δ<ρ\leq 1.$ Additional necessary and sufficient conditions are given in terms of the matrix-valued symbols of the operators, which are global functions on the phase space $G\times \widehat{G},$ with the momentum variables belonging to the unitary dual $\widehat{G}$ of $G$. In terms of the parameters $(ρ,δ),$ on the torus $\mathbb{T}^n,$ we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class $Ψ^{-\varkappa}_{0,0}(\mathbb{T}^n),$ $\varkappa>0,$ belonging to all the Schatten ideals. Additional order criteria are given in the setting of classical pseudo-differential operators. We present also some open problems in this setting.
This paper presents a novel intrinsic image transfer (IIT) algorithm for illumination manipulation, which creates a local image translation between two illumination surfaces. This model is built on an optimization-based framework consisting of three photo-realistic losses defined on the sub-layers factorized by an intrinsic image decomposition. We illustrate that all losses can be reduced without the necessity of taking an intrinsic image decomposition under the well-known spatial-varying illumination illumination-invariant reflectance prior knowledge. Moreover, with a series of relaxations, all of them can be directly defined on images, giving a closed-form solution for image illumination manipulation. This new paradigm differs from the prevailing Retinex-based algorithms, as it provides an implicit way to deal with the per-pixel image illumination. We finally demonstrate its versatility and benefits to the illumination-related tasks such as illumination compensation, image enhancement, and high dynamic range (HDR) image compression, and show the high-quality results on natural image datasets.
In this paper, we have introduced the Prabhakar fractional $q$-integral and $q$-differential operators. We first study the semi-group property of the Prabhakar fractional $q$-integral operator, which allowed us to introduce the corresponding $q$-differential operator. Formulas for compositions of $q$-integral and $q$-differential operators are also presented. We show the boundedness of the Prabhakar fractional $q$-integral operator in the class of $q$-integrable functions.
In this paper, we study a critical exponent to the semilinear heat equation with forcing term on Heisenberg group. Our technique of proof is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg group. Surprisingly, the critical exponent will be bounded for all dimensions of the Heisenberg group, in contrast to the Euclidean case which in the 1D and 2D cases will be infinite. In addition, we give an upper bound estimate on the lifespan of local solutions.
In this paper we explore the weak solutions of the Cauchy problem and an inverse source problem for the heat equation in the quantum calculus, formulated in abstract Hilbert spaces. For this we use the Fourier series expansions. Moreover, we prove the existence, uniqueness and stability of the weak solution of the inverse problem with a final determination condition. We give some examples such as the q-Sturm-Liouville problem, the q-Bessel operator, the q-deformed Hamiltonian, the fractional Sturm-Liouville operator, and the restricted fractional Laplacian, covered by our analysis.