A criterion for Hill operators to be spectral operators of scalar type

Gesztesy, Fritz; Trachenko, Vadim

doi:10.1007/s11854-009-0012-5

Cited by 55 publications

(77 citation statements)

References 62 publications

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“…Another step is to consider e.g. on Hill operators, where a criterion on being spectral operator of scalar type has been obtained in [16] and recently extended in [12]. On the other hand, the structure of similarity transformations for operators with continuous spectrum as well as for multidimensional Schrödinger operators is almost unexplored and constitutes thus a challenging open problem.…”

Section: Bounded Perturbationsmentioning

confidence: 99%

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

Krejčiřı́k

Siegl

Železný

2013

Complex Anal. Oper. Theory

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We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated "charge conjugation" operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

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Section: Bounded Perturbationsmentioning

confidence: 99%

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

Krejčiřı́k

Siegl

Železný

2013

Complex Anal. Oper. Theory

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“…However we did not proved yet the completeness of the corresponding basis. An obstacle for that can be found in the works [9], [10], [11], [12]. The Darboux-Crum transformations play only technical role here.…”

Section: Appendixmentioning

confidence: 99%

Singular soliton operators and indefinite metrics

Grinevich

Novikov

2013

Bull Braz Math Soc, New Series

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Abstract. We consider singular real second order 1D Schrödinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebro-geometrical potentials (i.e. "singular finite-gap" and "singular solitons") satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functions with singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ∼ g(g + 1)/x 2 . Here g is the genus of spectral curve.

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“…(Spectral operators of scalar type comprise, for instance, the normal operators or the generic one-dimensional periodic Schrödinger operators (Remark 8.7 of [22]). ) In both cases the weakly associated projections are just given by P A ({λ}) where P A denotes the spectral measure -or, in the terminology of [18], the spectral resolution -of the spectral operator A (Definition XV.2.5 and XVIII.2.1 of [18]).…”

Section: Spectral Theory Uniform and Non-uniform Spectral Gaps Regumentioning

confidence: 99%

“…In view of the estimates (4.7), (4.9), (4.14) and 22) we would now like to find functions ε → δ 1 ε , . .…”

Section: ∞ * (I L(x Y )) In Particular T → A(t)p (T) Is Sot-conmentioning

confidence: 99%

“…We now specialize to the case of spectral operators of scalar type which -by what has been remarked in Section 2.1 -automatically have weakly semisimple eigenvalues and which arise as the generic one-dimensional periodic Schrödinger operators (Remark 8.7 of [22]). As a simple corollary of the adiabatic theorem above, we obtain the following quantitative adiabatic theorem tailored to scalar type spectral operators A(t) whose spectral measures P A(t) are -in some sense -Hölder continuous in t. It generalizes a result for skew self-adjoint A(t) of Avron and Elgart (Corollary 1 in [7]) and a refinement of it due to Teufel (Remark 1 in [52]) and slightly improves the rates of convergence given there.…”

Section: A Quantitative Adiabatic Theorem Without Spectral Gap Conditmentioning

confidence: 99%

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Adiabatic theorems with and without spectral gap condition for non-semisimple spectral values

Schmid

2014

Mathematical Results in Quantum Mechanics

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We establish adiabatic theorems with and without spectral gap condition for general operators A(t) : D(A(t)) ⊂ X → X with possibly time-dependent domains in a Banach space X. We first prove adiabatic theorems with uniform and non-uniform spectral gap condition (including a slightly extended adiabatic theorem of higher order). In these adiabatic theorems the considered spectral subsets σ(t) have only to be compact -in particular, they need not consist of eigenvalues. We then prove an adiabatic theorem without spectral gap condition for not necessarily (weakly) semisimple eigenvalues: in essence, it is only required there that the considered spectral subsets σ(t) = {λ(t)} consist of eigenvalues λ(t) ∈ ∂σ(A(t)) and that there exist projections P (t) reducing A(t) such that A(t)| P (t)D(A(t)) − λ(t) is nilpotent and A(t)| (1−P (t))D(A(t)) − λ(t) is injective with dense range in (1 − P (t))X for almost every t. In all these theorems, the regularity conditions imposed on t → A(t), σ(t), P (t) are fairly mild. We explore the strength of the presented adiabatic theorems in numerous examples. And finally, we apply the adiabatic theorems for time-dependent domains to obtain -in a very simple way -adiabatic theorems for operators A(t) defined by symmetric sesquilinear forms.

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A criterion for Hill operators to be spectral operators of scalar type

Cited by 55 publications

References 62 publications

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

Singular soliton operators and indefinite metrics

Adiabatic theorems with and without spectral gap condition for non-semisimple spectral values

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