Eigenfunction expansions of a quadratic pencil of differential operator with periodic generalized potential (original) (raw)
ON MULTIPLE EIGENFUNCTION EXPANSION OF AN OPERATOR PENCIL WITH COMPLEX ALMOST PERIODIC POTENTIALS
Stochastic Modelling and Computational Sciences, 2023
In the paper, the multiple eigenfunction expansion of the operator pencil on a whole axis is considered. The coefficients of the considered operators are taken as complex and almost periodic. The eigenfunction expansion of the resolvent is obtained in terms of continuous spectrum eigenfunction and the multiple expansion of arbitrary test functions.
A note on Kellogg's eigenfunctions of a periodic Sturm-Liouville system
Applied Mathematics Letters, 1988
In this note, we shall supplement Kellogg's theory of interface problems, and provide complete eigenfunctions for the Sturm-Liouville system concerned. The complete eigenfunctions of interface problems are essential not only for theoretical research but also for numerical methods.
2015
We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then by using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite. Copyright q 2009 O. A. Veliev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let LP2, P3,..., Pn be the differential operator generated in the space Lm2 −∞,∞ of vector-valued functions by the differential expression −inynx −in−2P2xyn−2x n∑ v3 Pvxyn−vx, 1 where n is an integer greater than 1 and Pkx, for k 2, 3,..., n, is the m × m matrix with the complex-valued summable entries pk,i,jx satisfying pk,i,jx ...
On the point spectrum of some perturbed differential operators with periodic coefficients
2007
Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the resolvent function through continuous spectrum. In the second part, the abstract result is applied to differential operators which can be represented as a differential operator with periodic coefficients perturbed by an arbitrary subordinated differential operator.
Half-eigenvalues of periodic Sturm–Liouville problems
Journal of Differential Equations, 2004
We consider the nonlinear Sturm-Liouville problem Àðpu 0 Þ 0 þ qu ¼ au þ À bu À þ lu; in ð0; 2pÞ; ð1Þ uð0Þ ¼ uð2pÞ; ðpuÞ 0 ð0Þ ¼ ðpuÞ 0 ð2pÞ; ð2Þ where 1=p; qAL 1 ð0; 2pÞ; with p40 a.e. on ð0; 2pÞ; a; bAL 1 ð0; 2pÞ; l is a real parameter, and u 7 ðtÞ ¼ maxf7uðtÞ; 0g for tA½0; 2p: Values of l for which (1)-(2) has a non-trivial solution u will be called half-eigenvalues while the corresponding solutions u will be called halfeigenfunctions. The set of half-eigenvalues will be denoted by S H : We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties associated with S H : These properties yield results on the existence and non-existence of solutions of the problem Àðpu 0 Þ 0 þ qu ¼ f ðt; uÞ þ h; in ð0; 2pÞ ð 3Þ (together with (2)), where hAL 1 ð0; 2pÞ; f : ½0;
Schr "odinger operator with a junction of two 1-dimensional periodic potentials
Asymptotic Analysis, 2005
The spectral properties of the Schrödinger operator Tty=−y′′+qtyT_ty= -y''+q_tyTty=−y′′+qty in L2(R)L^2(\R)L2(R) are studied, with a potential qt(x)=p1(x),x<0,q_t(x)=p_1(x), x<0, qt(x)=p1(x),x<0, and qt(x)=p(x+t),x>0,q_t(x)=p(x+t), x>0, qt(x)=p(x+t),x>0, where p1,pp_1, pp1,p are periodic potentials and tinRt\in \RtinR is a parameter of dislocation. Under some conditions there exist simultaneously gaps in the continuous spectrum of T0T_0T0 and eigenvalues in these gaps. The main goal of this paper is to study the discrete spectrum and the resonances of TtT_tTt. The following results are obtained: i) In any gap of TtT_tTt there exist 0,10,10,1 or 2 eigenvalues. Potentials with 0,1 or 2 eigenvalues in the gap are constructed. ii) The dislocation, i.e. the case p1=pp_1=pp1=p is studied. If tto0t\to 0tto0, then in any gap in the spectrum there exist both eigenvalues ($ \le 2 )andresonances() and resonances ()andresonances( \le 2 )of) of )ofT_t$ which belong to a gap on the second sheet and their asymptotics as tto0t\to 0 tto0 are determined. iii) The eigenvalues of the half-solid, i.e. p1=rmconstantp_1={\rm constant}p1=rmconstant, are also studied. iv) We prove that for any even 1-periodic potential ppp and any sequences dn1iy\{d_n\}_1^{\iy}dn1iy, where dn=1d_n=1dn=1 or dn=0d_n=0dn=0 there exists a unique even 1-periodic potential p1p_1p1 with the same gaps and dnd_ndn eigenvalues of T_0T_0T_0 in the n-th gap for each nge1.n\ge 1.nge1.
The Finite Spectrum of Sturm-Liouville Operator With δ-Interactions 1
The goal of this paper is to study the finite spectrum of Sturm-Liouville operator with δinteractions. Such an equation gives us a Sturm-Liouville boundary value problem which has n transmission conditions. We show that for any positive numbers m j (j = 0, 1, ..., n) that are related to number of partition of the intervals between two successive interaction points, we can construct a Sturm-Liouville equations with δ-interactions, which have exactly d eigenvalues. Where d is the sum of m j 's.