The similarity problem for J-nonnegative Sturm–Liouville operators

Abstract

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 conditions for the regularity of the critical points 0 and ∞ of J -nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r(x) = sgn x and q ∈ L 1 (R, (1 + |x|) dx), we prove that A is similar to a self-adjoint operator if and only if A is J -nonnegative. The latter condition on q is sharp, i.e., we construct q ∈ γ <1 L 1 (R, (1 + |x|) γ dx) such that A is J -nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J -positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q ≡ 0, we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for "forward-backward" diffusion equations. (I.M. Karabash), duzer80@mail.ru (A.S. Kostenko), mmm@telenet.dn.ua (M.M. Malamud).

On Similarity of Indefinite Sturm‐Liouville Operators

“…Moreover, the proof above can be extended to a more general class of indefinite Sturm-Liouville operators, where the indefinite weight function sgn is replaced by a function r with r, 1/r ∈ L ∞ (R) which has exactly one sign change. Theorem 1 complements the results of [2].…”

“…there exists a bijection V in L 2 (R) such that V (JA)V −1 is self-adjoint in L 2 (R). This property has been investigated in [1] for general n−th order differential expressions, in [2] for p ≡ 1, and in [3] for p ≡ 1, q ≡ 0. The proof of our main result uses techniques established in [1, Proofs of Lemma 3.2 and Theorem 3.5].…”

Non-real eigenvalues of singular indefinite Sturm-Liouville operators

“…the real spectrum of A necessarily accumulates to +∞ and −∞ and A may have non-real eigenvalues which possibly accumulate to the real axis (see [3,4,9,14,16,20]). For further indefinite Sturm-Liouville problems, applications and references, see, e.g., [2,6,7,11,13,15,23,26].…”

A Functional Model, Eigenvalues, and Finite Singular Critical Points for Indefinite Sturm-Liouville Operators

“…If 0 and ∞ are not singular critical points of A than the algebraic multiplicity of 0 affects proper settings of boundary value problems for the corresponding diffusion equation (see [6,5,32]). If 0 is a singular critical point of A, as in the examples constructed in [42,44] and Section 5, the existence and uniqueness theory for corresponding diffusion equations is not well-understood (see [48,Section 1], [14,59,40]). …”

“…The necessary similarity condition given in [42,Theorem 3.4] is of independent interest since it provides a criterion of similarity to a self-adjoint operator for operators (sgn x)(−d 2 /dx 2 + q) with finite-zone potentials (see [42,Remark 3.7]). And it is unknown whether the condition of [42,Theorem 3.4] provides a criterion of similarity to a self-adjoint operator for the general operator Using Proposition 5.1, a large class of operators with the singular critical point 0 similar to that of [42,44] can be constructed. In the next theorem, we characterize the case described in Proposition 5.1 among the operators A r := − sgn x |r(x)| d 2 dx 2 that have the limit point case both at ±∞ (and act in L 2 (R; |r(x)|dx)).…”