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Register a new userOn majorants of eigenvalues of Sturm-Liouville problems with potentials from balls of weighted spaces
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A.A. Vladimirov
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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.
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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.
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.
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 $q$.
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.
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