On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators

Rösler, Frank

doi:10.1007/s00020-019-2555-x

Cited by 11 publications

(9 citation statements)

References 10 publications

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“…We finally note that recent years have seen a flurry of activity in this direction with many results classifying various problems within the SCI Hierarchy. We point out [11,12] where some of the theory of spectral computations has been further developed; [32] where this has been applied to certain classes of unbounded operators; [2] where solutions of PDEs were considered; [6,7] where we considered resonance problems; and [13] where the authors give further examples of how to perform certain spectral computations with error bounds.…”

Section: Previous Resultsmentioning

confidence: 99%

Universal algorithms for computing spectra of periodic operators

2022

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Schrödinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a ‘one-size-fits-all’ algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schrödinger operators with periodic potentials that are sufficiently smooth. In both cases implementable algorithms are provided, along with examples. For certain Schrödinger operators whose potentials may diverge at a single point (but are otherwise well-behaved) it is shown that there does not exist such an algorithm, though it is shown that the computation is possible if one allows for two successive limits.

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Section: Previous Resultsmentioning

confidence: 99%

Universal algorithms for computing spectra of periodic operators

2022

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“…Research related to this theory has gathered pace in recent years. We point out [13,12] where some of the theory of spectral computations has been further developed; [27] where this has been applied to certain classes of unbounded operators; [4] where solutions of PDEs were considered; [9] where we considered periodic spectral problems; [8,7] where we considered resonance problems; and [14] where the authors give further examples of how to perform certain spectral computations with error bounds. Let us summarize the main definitions of the SCI theory.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

On the complexity of the inverse Sturm-Liouville problem

Ben-Artzi¹,

Marletta²,

Rösler³

2022

Preprint

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This paper explores the complexity associated with solving the inverse Sturm-Liouville problem with Robin boundary conditions: given a sequence of eigenvalues and a sequence of norming constants, how many limits does a universal algorithm require to return the potential and boundary conditions? It is shown that if all but finitely many of the eigenvalues and norming constants coincide with those for the zero potential then the number of limits is zero, i.e. it is possible to retrieve the potential and boundary conditions precisely in finitely many steps. Otherwise, it is shown that this problem requires a single limit; moreover, if one has a priori control over how much the eigenvalues and norming constants differ from those of the zero-potential problem, and one knows that the average of the potential is zero, then the computation can be performed with complete error control. This is done in the spirit of the Solvability Complexity Index. All algorithms are provided explicitly along with numerical examples.

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“…Recent years have seen a flurry of activity in this direction. We point out [11,10] where some of the theory of spectral computations has been further developed; [31] where this has been applied to certain classes of unbounded operators; [2] where solutions of PDEs were considered; [6,7] where we considered resonance problems; and [12] where the authors give further examples of how to perform certain spectral computations with error bounds.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Universal algorithms for computing spectra of periodic operators

Ben-Artzi¹,

Marletta²,

Rösler³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Schrödinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a 'one-sizefits-all' algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schrödinger operators with periodic potentials that are sufficiently smooth. In both cases implementable algorithms are provided, along with examples. For certain Schrödinger operators whose potentials may diverge at a single point (but are otherwise well-behaved) it is shown that there does not exist such an algorithm, though it is shown that the computation is possible if one allows for two successive limits.

show abstract

On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators

Cited by 11 publications

References 10 publications

Universal algorithms for computing spectra of periodic operators

Universal algorithms for computing spectra of periodic operators

On the complexity of the inverse Sturm-Liouville problem

Universal algorithms for computing spectra of periodic operators

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