Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Second-order Frechet Derivatives of Eigenvalues of Sturm-Liouville Problems in Potentials
88
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The works of V. A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm-Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and paly an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.
rate research
Read More
It is constructively proved that for class $A_{r,gamma}={qin L_{1,loc}(0,1): qleq 0, int_0^1 rq^gamma,dxleqslant 1}$, where $rin C[0,1]$ is uniformly positive weight and $gamma>1$, there exists a unique potential $hat qin A_{r,gamma}$ such that minimal eigenvalue $lambda_0(hat q)$ of boundary problem $$-y+hat qy=lambda y, y(0)=y(1)=0 $$ is equal to $M_{r,gamma}=sup_{qin A_{r,gamma}}lambda_0(q)$. For case $gamma=1$ we obtain that there exists a unique potential $hat qinGamma_{r,gamma}$ with analogous property. Here $Gamma_{r,gamma}$ is a closure of $A_{r,gamma}$ in the space $W_{2,loc}^{-1}(0,1)$ of generalized functions.
Sturm-Liouville spectral problem for equation $-(y/r)+qy=lambda py$ with generalized functions $rge 0$, $q$ and $p$ is considered. It is shown that the problem may be reduced to analogous problem with $requiv 1$. The case of $q=0$ and self-similar $r$ and $p$ is considered as an example.
For a particular family of long-range potentials $V$, we prove that the eigenvalues of the indefinite Sturm--Liouville operator $A = mathrm{sign}(x)(-Delta + V(x))$ accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.
We study perturbations of the self-adjoint periodic Sturm--Liouville operator [ A_0 = frac{1}{r_0}left(-frac{mathrm d}{mathrm dx} p_0 frac{mathrm d}{mathrm dx} + q_0right) ] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.
The spectrum of the singular indefinite Sturm-Liouville operator $$A=text{rm sgn}(cdot)bigl(-tfrac{d^2}{dx^2}+qbigr)$$ with a real potential $qin L^1(mathbb R)$ covers the whole real line and, in addition, non-real eigenvalues may appear if the potential $q$ assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction were obtained. In this paper the bound $$|lambda|leq |q|_{L^1}^2$$ on the absolute values of the non-real eigenvalues $lambda$ of $A$ is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the $L^1$-norm of the negative part of $q$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
