Sturm-Liouville Equations with Besicovitch Almost-Periodicity (original) (raw)
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Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum
Commentarii Mathematici Helvetici, 1995
~' 1995 Birkh'/iuser Verlag, Basel C(0, 2) = S'(0, 2) = 1, C'(0, 2) = S(0, 2) = 0. By virtue of the classical Weyl theorem (see, for example, Titchmarsh [25, Ch. 2]), for each nonreal 2 Equation (1.1.1) has solutions ~b_+(x, 2) = C(x, 2) + m• 2), such that r177 e L2(R• This work was partially supported by ISF Grant no. U2Z000. 9 639 640 MILKHAIL SODIN AND PETER YI.;I)ITSKll The functions m__ are holomorphic outside the real axis, m_+(Z)= m~(2) and ~m.~().)/~2 >0, ~m. ().)/~). <0. The functions m• are called the Weylfunctions; they are defined uniquely by virtue of the boundedness from below of the potential q(x). We denote by g(x, y; 2) the Green function of L[q] which is defined as the kernel of the resolvent Re. = (L[q] -2) ~. Then (see Titchmarsh [25, Ch. 2])
ON A QUESTION IN THE THEORY OF ALMOST PERIODIC DIFFERENTIAL EQUATIONS
We show that there exists a real homogeneous differential equation of order n with classical almost periodic coefficients such that all solutions are uniformly bounded on the real line yet no non-trivial solution is almost periodic. This now appears to make the search for a Floquet theory of such equations a futile enterprise.
Mathematical Notes, 2009
The paper deals with the Sturm-Liouville operator Ly = −y ′′ + q(x)y, x ∈ [0, 1], generated in the space L 2 = L 2 [0, 1] by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to Sobolev space W p 1 [0, 1] with some integer p ≥ 0 and satisfy the conditions q (k) (0) = q (k) (1) = 0 for 0 ≤ k ≤ s − 1, where s≤ p. Let the functions Q and S be defined by the equalities Q(x) = x 0 q(t) dt, S(x) = Q 2 (x) and let q n , Q n , S n be the Fourier coefficients of q, Q, S with respect to the trigonometric system {e 2πinx } ∞ −∞. Assume that the sequence q 2n − S 2n + 2Q 0 Q 2n decreases not faster than the powers n −s−2. Then the system of eigen and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L 2 [0, 1] (provided that the eigenfunctions are normalized) if and only if the condition q 2n − S 2n + Q 0 Q 2n ≍ q −2n − S −2n + 2Q 0 Q −2n , n > 1, holds.
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;
Almost periodic Schr�dinger operators
Communications in Mathematical Physics, 1983
We discuss the absolutely continuous spectrum of H =-d 2 / d x 2 + V(x) with Valmost periodic and its discrete analog (hu)(n) = u(n + 1) + u (n-1) + V(n)u(n). Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential support of the a.c. part of the spectral measure. We prove for a.e. Vin the hull and a.e. E in A, H and h have continuum eigenfunctions, u, with [u[ almost periodic. In the discrete case, we prove that IAI < 4 with equality only if V=const. If k is the integrated density of states, we prove that on A, 2kdk/dE>rc-2 in the continuum case and that 2nsinrckdk/dE> 1 in the discrete case. We also provide a new proof of the Pastur-Ishii theorem and that the multiplicity of the absolutely continuous spectrum is 2.
Relative oscillation theory for Sturm-Liouville operators
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(1) for functions r, p, qa Sturm–Liouville operator. Sturm–Liouville operators arise for example when considering the radial part of the Laplacian of a rotation symmetric problem in any dimension. Sturm–Liouville equations of the type− f (x)+ q (x) f (x)= λf (x) arise in quantum mechanics and are called onedimensional Schrödinger equations. Periodic Sturm–Liouville equations are for example used as one dimensional crystal models (eg the Kronig-Penney model).
Almost periodic Schrödinger operators II. The integrated density of states
Duke Mathematical Journal, 1983
1. Introduction. In this paper, we will study Schr6dingeroperators, -A + V, on L2(R) where V is an almost periodic function on R". We will be especially interested in the case t, where we will also consider the finite difference analog on/2(Z) ... (MU)n= u,,+, + u,,_ + V(n)u
On a spectral criterion for almost periodicity of solutions of periodic evolution equations
Electronic Journal of Qualitative Theory of Differential Equations, 1999
This paper is concerned with equations of the form: u = A(t)u + f (t) , where A(t) is (unbounded) periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations which says that if u is a bounded uniformly continuous mild solution and P is the monodromy operator, then their spectra satisfy e isp AP (u) ⊂ σ(P ) ∩ S 1 , where S 1 is the unit circle. This result is then applied to find almost periodic solutions to the above-mentioned equations. In particular, parabolic and functional differential equations are considered. Existence conditions for almost periodic and quasi-periodic solutions are discussed.