Related papers: Nonlinear Sturm Oscillation: from the interval to …
We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm-Liouville problems. Via \Gamma-convergence theory, we study the asymptotic distribution of the minimizers as the…
A survey of estimates on the number $N_-(\BM_{\a G})$ of negative eigenvalues (bound states) of the Sturm-Liouville operator $\BM_{\a G}u=-u"-\a G$ on the half-line, as depending on the properties of the function $G$ and the value of the…
We study the instability of standing waves for nonlinear Schr\"{o}dinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a…
We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[\tau f = \frac{1}{r} \left(- \big(p[f' + s…
We study the scattering problem, the Sturm-Liouville problem and the spectral problem with periodic or skew-periodic boundary conditions for the one-dimensional Schr\"odinger equation with an $n$-cell (finite periodic) potential. We give…
Sufficient conditions for the similarity of the operator $A := 1/r(x) (-d^2/dx^2 +q(x))$ with an indefinite weight $r(x)=(\sgn x)|r(x)|$ are obtained. These conditions are formulated in terms of Titchmarsh-Weyl $m$-coefficients. Sufficient…
In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\lambda r)y=0$, $(U-1)y^{\vee}+i(U+1)y^{\wedge}=0$, where function $p\in L_{\infty}[0,1]$ is uniformly positive, generalized functions $q,r\in W_2^{-1}[0,1]$…
We consider the nonlinear Schr\"odinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the graph is compact, for every value of the mass there is a…
This paper studies a Sturm--Liouville boundary value problem in which one of the boundary conditions depends bilinearly on the spectral parameter. The differential equation is considered on the interval $(0,1)$ with a classical boundary…
We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue…
We study the nonlinear Schr\"odinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate an orbital…
The linearized operator for non-radial oscillations of spherically symmetric self-gravitating gaseous stars is analyzed in view of the functional analysis. The evolution of the star is supposed to be governed by the Euler-Poisson equations…
In this paper we examine the asymptotic structure of the pseudospectrum of the singular Sturm-Liouville operator $L=\partial_x(f\partial_x)+\partial_x$ subject to periodic boundary conditions on a symmetric interval, where the coefficient…
Boundary value problems on hedgehog-type graphs for Sturm-Liouville differential operators with general matching conditions are studied. We investigate inverse spectral problems of recovering the coefficients of the differential equation…
We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with some regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root…
In this paper we study the noncompact star-type graph with perturbed radial Schrodinger equation on each ray and the matching conditions of some special form at the vertex. The results include the uniqueness theorem and constructive…
We study the inverse problem of recovering Sturm-Liouville operators on the half-line with a Bessel-type singularity inside the interval from the given Weyl function. The corresponding uniqueness theorem is proved, a constructive procedure…
The limit distribution of the discrete spectrum of the Sturm-Liouville problem with complex-valued polynomial potential on an interval, on a half-axis, and on the entire axis is studied. It is shown that at large parameter values, the…
We consider standard subordinacy theory for general Sturm--Liouville operators and give criteria when boundedness of solutions implies that no subordinate solutions exist. As applications, we prove a Weidmann-type result for general…
We study standing waves for a nonlinear Schr\"odinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength…