Liouville Operator Research Articles

Traces and inverse nodal problem for Sturm–Liouville operators with frozen argument

In this paper, Sturm–Liouville operators with frozen argument are studied. We derive the regularized trace formulae for these operators. Also we provide asymptotic expressions for the nodal points and give a constructive procedure for solving an inverse nodal problem.

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

Abstract A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm–Liouville problems, differential models with involution, fractional Sturm–Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a “cooling function”, and how the involution normally slows down the cooling speed of the rod.

Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues

Abstract In this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on ( 0 , π ) {(0,\pi)} with the Robin boundary conditions and the jump conditions at the point π 2 {\frac{\pi}{2}} . We prove that the potential M ⁢ ( x ) {M(x)} on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential M ⁢ ( x ) {M(x)} is given on ( 0 , ( 1 + α ) ⁢ π 4 ) {(0,\frac{(1+\alpha)\pi}{4})} ; (ii) the potential M ⁢ ( x ) {M(x)} is given on ( ( 1 + α ) ⁢ π 4 , π ) {(\frac{(1+\alpha)\pi}{4},\pi)} , where 0 < α < 1 {0<\alpha<1} , respectively. It is also shown that the potential and all the parameters can be uniquely recovered by one spectrum and some information on the eigenfunctions at some interior point.

On the Separation Property of the Sturm–Liouville Operator in Weighted Spaces of Multiplicators

In this paper, we prove the separation theorem for the Sturm–Liouville operator in terms of point multiplicators in weighted Sobolev spaces. The research method is based on local estimates on intervals of characteristic length.

On the Localization Conditions for the Spectrum of a Non-Self-Adjoint Sturm–Liouville Operator with Slowly Growing Potential

We consider the Sturm–Liouville operator T0 on the semi-axis (0,+∞) with the potential eiθq, where 0 < θ < π and q is a real-valued function that may have arbitrarily slow growth at infinity. This operator does not meet any condition of the Keldysh theorem: T0 is non-self-adjoint and its resolvent does not belong to the Neumann–Schatten class for any p < ∞. We find conditions for q and perturbations of V under which the localization or the asymptotics of its spectrum is preserved.

On the Riesz basis property of the root functions of a discontinuous boundary problem

In this paper, we consider the nonself‐adjoint discontinuous Sturm Liouville operator with periodic (antiperiodic) boundary condition and compatibility conditions. Asymptotic formulas of eigenvalues and eigenfunctions of the operator are obtained. Using these accurate asymptotic formulas for eigenvalues and eigenfunctions, we prove the basisness of the root functions of the boundary value problem.

Investigation of the spectrum of singular Sturm–Liouville operators on unbounded time scales

In this article, we consider the self-adjoint singular operators associated with the Sturm–Liouville expression $$\begin{aligned} Ly:=-\left[ p\left( t\right) y^{\Delta }\left( t\right) \right] ^{\nabla }+q\left( t\right) y\left( t\right) ,\ t\in (-\infty ,\infty )_{\mathbb {T}}. \end{aligned}$$on time scale $$\mathbb {T}$$. Some conditions are given for this operator to have a discrete spectrum. Further, we investigate the continuous spectrum of this operator. We also prove that the regular Sturm–Liouville operator on time scale is semi-bounded from below which is not studied in literature yet.

Riemann–Liouville Operator in Weighted Lp Spaces via the Jacobi Series Expansion

In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann–Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces.

Correction to: On Borg’s method for non-selfadjoint Sturm–Liouville operators

The original version of the article unfortunately contained few errors under Preliminaries section. The corrected text is given below.

Spectral Analysis of Singular Matrix-Valued Sturm–Liouville Operators

In the Hilbert space $$L_{W}^{2}([a,b);E)$$ ( $$-\infty<a<b\le +\infty ,$$ $$\dim E=N<+\infty ,$$ $$W>0$$ ) a space of boundary values of the symmetric singular matrix-valued Sturm–Liouville operator with maximal deficiency indices (2N, 2N) (in limit-circle case at singular end point b) is constructed. With the help of the boundary conditions at a and b, all maximal dissipative, maximal accumulative and self-adjoint extensions of such a symmetric operator are established. In particular, the maximal dissipative operators with separated boundary conditions, called ‘dissipative at a’ and ‘self-adjoint at b’ are investigated. A self-adjoint dilation of the dissipative operator is constructed and then its incoming and outgoing spectral representations are determined. This representation allows us to determine the scattering matrix of the dilation with the help of the Weyl matrix-valued function of a self-adjoint matrix-valued Sturm–Liouville operator. Further a functional model of the dissipative operator is determined and its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl function) is established. Finally, a theorem on completeness of the system of root vectors of the dissipative operator is proved.

Interrelationships Between Marichev–Saigo–Maeda Fractional Integral Operators, the Laplace Transform and the $${\overline{H}}$$-Function

In this article, we first evaluate the Laplace transform of Marichev–Saigo–Maeda (M–S–M) fractional integral operators whose kernel is the Appell function $$F_{3}$$ and point out its six special cases (Saigo, Erdelyi–Kober, Riemann–Liouville and Weyl fractional integral operators). Certain new and known results can be obtained as special cases of our key findings. Next, we find the image of $${\overline{H}}$$ -Function under the operators of our study. Some interesting special cases of our key result are also considered and demonstrated to be connected with certain existing ones.

Spectral partitions for Sturm–Liouville problems

AbstractWe look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm–Liouville problems. Via Γ-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm–Liouville operator.

Inverse Problems for the Sturm—Liouville Operator with Discontinuity Conditions

Inverse Problems for the Sturm—Liouville Operator with Discontinuity Conditions

Sturm – Liouville integrable operators, generated by generalized Dunkl operators

Based on the generalized intertwining relation of Dunkl operator with operator of differentiation, eigenfunctions for certain Sturm – Liouville operators were obtained.

Qualitative Spectral Analysis of Singular q-Sturm–Liouville Operators

This paper considers properties of the spectrum of q-Sturm–Liouville operator derived from the q-Sturm–Liouville expression $$\begin{aligned} Ly:=-\frac{1}{q}D_{q^{-1}}\left( p(x)D_{q}y(x)\right) +r\left( x\right) y(x),\quad 0<x<a\le \infty . \end{aligned}$$We prove that the regular symmetric q-Sturm–Liouville operator is semi-bounded from below. Using splittings technique, we will give some conditions for the self-adjoint operator associated with the singular q-Sturm–Liouville expression to have a discrete spectrum. We also investigate the continuous spectrum of this operator.

An inverse problem for Sturm–Liouville operators on the half-line with complex weights

Abstract Second order differential operators on the half-line with complex-valued weights are considered. Properties of spectral characteristics are established, and the inverse problem of recovering operator’s coefficients from the given Weyl-type function is studied. The uniqueness theorem is proved for this class of nonlinear inverse problems, and a number of examples are provided.

Inverse Spectral Problems for Sturm–Liouville Operators with a Constant Delay Less than Half the Length of the Interval and Robin Boundary Conditions

The topic of this paper are non-self-adjoint second-order differential operators with a constant delay, which is less than half of the length of the interval. We consider the case when a delay is from $$\tau \in [\frac{2\pi }{5},\frac{\pi }{2})$$ , and the potential is a real-valued function which satisfy $$q\in L^{2}[0,\pi ]$$ . The inverse spectral problem of recovering the potential from the spectra of two boundary value problems with Robin boundary conditions has been studied. We have proved that the delay and the potential are uniquely determined by two spectra of boundary spectral problem, one with boundary conditions $$y'(0)-hy(0)=0$$ , $$y'(\pi )+H_{1} y(\pi )=0$$ and the other with boundary conditions $$y'(0)-hy(0)=0$$ , $$y'(\pi )+H_{2} y(\pi )=0$$ .

Optomechanical damping basis

We present a closed-form analytical solution to the eigenvalue problem of the Liouville operator generating the dissipative dynamics of the standard optomechanical system. The corresponding Lindblad master equation describes the dynamics of a single-mode field inside an optical cavity coupled by radiation pressure to its moving mirror. The optical field and the mirror are in contact with separate environments, which are assumed at zero and finite temperature, respectively. The optomechanical damping basis refers to the exact set of eigenvectors of the generator that, together with the exact eigenvalues, are explicitly derived. Both the weak- and the strong-coupling regime, which includes combined decay mechanisms, are solved in this work.

Spectral bounds for indefinite singular Sturm–Liouville operators with uniformly locally integrable potentials

The non-real spectrum of a singular indefinite Sturm–Liouville operatorA=1r(−ddxpddx+q) with a sign changing weight function r consists (under suitable additional assumptions on the real coefficients 1/p,q,r∈Lloc1(R)) of isolated eigenvalues with finite algebraic multiplicity which are symmetric with respect to the real line. In this paper bounds on the absolute values and the imaginary parts of the non-real eigenvalues of A are proved for uniformly locally integrable potentials q and potentials q∈Ls(R) for some s∈[1,∞]. The bounds depend on the negative part of q, on the norm of 1/p, and in an implicit way on the sign changes and zeros of the weight function.

Wigner equation for general electromagnetic fields: The Weyl-Stratonovich transform

Gauge-invariant Wigner theories are formulated in terms of the kinetic momentum, which---being a physical quantity---is conserved after a change of the gauge. These theories rely on a transform of the density matrix, originally introduced by Stratonovich, which generalizes the Weyl transform by involving the vector potential. We thus present an alternative derivation of the Weyl-Stratonovich transform, which bridges the concepts and notions used by the different, available gauge-invariant approaches and thus links physically intuitive with formal mathematical viewpoints. Furthermore, an explicit form of the Wigner equation, suitable for numerical analysis and corresponding to general, inhomogeneous, and time-dependent electromagnetic conditions, is obtained. For a constant magnetic field, the equation reduces to two models: in the case of a constant electric field, this is the ballistic Boltzmann equation, where classical particles are driven by local forces. The second model, derived for general electrostatic conditions, involves novel physics, where the magnetic field acts locally via the Liouville operator, while the electrostatics is determined by the manifestly nonlocal Wigner potential. A significant consequence of our work is the fact that now the constant magnetic field case can be treated with existing numerical approaches developed for the standard, scalar potential Wigner theory. Therefore, in order to demonstrate the feasibility of the approach, a stochastic method is applied to simulate a physically intuitive evolution problem.