Sturm-Liouville Completeness Proof
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I am looking for a basic but rigorous introduction to Sturm-Liouville theory. In particular, I would like to see a proof that the eigenfunction solutions of Sturm-Liouville problems are complete.
Some books seem to suggest that this follows from the completeness of eigenvectors of finite-dimensional Hermitian operators in linear algebra, but that can't be right. All the linear algebra proofs I can find use the fact that the (finite) number of eigenvectors of a Hermitian matrix equals the (finite) dimension of the matrix, and this result cannot carry over into infinite-dimensional function space.
differential-equations sturm-liouville
asked Nov 26 at 23:56
WillG
42738
add a comment |
up vote
0
down vote
favorite
I am looking for a basic but rigorous introduction to Sturm-Liouville theory. In particular, I would like to see a proof that the eigenfunction solutions of Sturm-Liouville problems are complete.
Some books seem to suggest that this follows from the completeness of eigenvectors of finite-dimensional Hermitian operators in linear algebra, but that can't be right. All the linear algebra proofs I can find use the fact that the (finite) number of eigenvectors of a Hermitian matrix equals the (finite) dimension of the matrix, and this result cannot carry over into infinite-dimensional function space.
differential-equations sturm-liouville
asked Nov 26 at 23:56
WillG
42738
I assume you are working with regular Sturm-Liouville problems on a finite interval. Otherwise, the completeness can be trickier because the expansions may require continuous components resembling Fourier integral expansions.
– DisintegratingByParts
Nov 27 at 0:53
Sure let's say the interval is finite, though now I'm curious about the continuous case too—is there a similar theorem for that case?
– WillG
Nov 27 at 2:38
The nicest proofs use Complex Analysis. Do you have any background with contour integrals and residues? And, yes, the singular cases have expansions which may include fourier integral components as well as series components.
– DisintegratingByParts
Nov 27 at 3:07
Yes, I'm all about Complex Analysis. Specific references to these proofs / books would be welcome.
– WillG
Nov 27 at 4:46
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am looking for a basic but rigorous introduction to Sturm-Liouville theory. In particular, I would like to see a proof that the eigenfunction solutions of Sturm-Liouville problems are complete.
Some books seem to suggest that this follows from the completeness of eigenvectors of finite-dimensional Hermitian operators in linear algebra, but that can't be right. All the linear algebra proofs I can find use the fact that the (finite) number of eigenvectors of a Hermitian matrix equals the (finite) dimension of the matrix, and this result cannot carry over into infinite-dimensional function space.
differential-equations sturm-liouville
asked Nov 26 at 23:56
WillG
42738
I am looking for a basic but rigorous introduction to Sturm-Liouville theory. In particular, I would like to see a proof that the eigenfunction solutions of Sturm-Liouville problems are complete.
Some books seem to suggest that this follows from the completeness of eigenvectors of finite-dimensional Hermitian operators in linear algebra, but that can't be right. All the linear algebra proofs I can find use the fact that the (finite) number of eigenvectors of a Hermitian matrix equals the (finite) dimension of the matrix, and this result cannot carry over into infinite-dimensional function space.
differential-equations sturm-liouville
differential-equations sturm-liouville
asked Nov 26 at 23:56
WillG
42738
asked Nov 26 at 23:56
WillG
42738
asked Nov 26 at 23:56
WillG
42738
asked Nov 26 at 23:56
WillG
42738
asked Nov 26 at 23:56
WillG
42738
42738
I assume you are working with regular Sturm-Liouville problems on a finite interval. Otherwise, the completeness can be trickier because the expansions may require continuous components resembling Fourier integral expansions.
– DisintegratingByParts
Nov 27 at 0:53
Sure let's say the interval is finite, though now I'm curious about the continuous case too—is there a similar theorem for that case?
– WillG
Nov 27 at 2:38
The nicest proofs use Complex Analysis. Do you have any background with contour integrals and residues? And, yes, the singular cases have expansions which may include fourier integral components as well as series components.
– DisintegratingByParts
Nov 27 at 3:07
Yes, I'm all about Complex Analysis. Specific references to these proofs / books would be welcome.
– WillG
Nov 27 at 4:46
add a comment |
I assume you are working with regular Sturm-Liouville problems on a finite interval. Otherwise, the completeness can be trickier because the expansions may require continuous components resembling Fourier integral expansions.
– DisintegratingByParts
Nov 27 at 0:53
Sure let's say the interval is finite, though now I'm curious about the continuous case too—is there a similar theorem for that case?
– WillG
Nov 27 at 2:38
The nicest proofs use Complex Analysis. Do you have any background with contour integrals and residues? And, yes, the singular cases have expansions which may include fourier integral components as well as series components.
– DisintegratingByParts
Nov 27 at 3:07
Yes, I'm all about Complex Analysis. Specific references to these proofs / books would be welcome.
– WillG
Nov 27 at 4:46
I assume you are working with regular Sturm-Liouville problems on a finite interval. Otherwise, the completeness can be trickier because the expansions may require continuous components resembling Fourier integral expansions.
– DisintegratingByParts
Nov 27 at 0:53
I assume you are working with regular Sturm-Liouville problems on a finite interval. Otherwise, the completeness can be trickier because the expansions may require continuous components resembling Fourier integral expansions.
– DisintegratingByParts
Nov 27 at 0:53
Sure let's say the interval is finite, though now I'm curious about the continuous case too—is there a similar theorem for that case?
– WillG
Nov 27 at 2:38
Sure let's say the interval is finite, though now I'm curious about the continuous case too—is there a similar theorem for that case?
– WillG
Nov 27 at 2:38
The nicest proofs use Complex Analysis. Do you have any background with contour integrals and residues? And, yes, the singular cases have expansions which may include fourier integral components as well as series components.
– DisintegratingByParts
Nov 27 at 3:07
The nicest proofs use Complex Analysis. Do you have any background with contour integrals and residues? And, yes, the singular cases have expansions which may include fourier integral components as well as series components.
– DisintegratingByParts
Nov 27 at 3:07
Yes, I'm all about Complex Analysis. Specific references to these proofs / books would be welcome.
– WillG
Nov 27 at 4:46
Yes, I'm all about Complex Analysis. Specific references to these proofs / books would be welcome.
– WillG
Nov 27 at 4:46
add a comment |
1 Answer
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2
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The classic book on this subject is E. C. Titchmarsh's 1942 text, Eigenfunction Expansions Associated with Second Order Differential Equations -- Part I. This was written by a master who studied under G. H. Hardy. Titchmarsh was the "Savilian Professor of Geometry in the University of Oxford."
Titchmarsh starts from basic Complex Analysis and Advanced Calculus, and by page 13 he has stated the completeness theorem for regular expansions. In the next few pages, he proves that the regular expansions are complete by showing that they are asymptotically the same as ordinary Fourier expansions, with specific bounds. He makes this topic look easy, which it is not. He also deals with the singular problems, the spectrum, and spectral measures. The heart of Titchmarsh's analysis relies on Complex Analysis, of which he was a master. Much of what is found in this text is his original research. Titchmarsh is still referenced by those in the field.
answered Nov 27 at 5:03
DisintegratingByParts
58.3k42579
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
The classic book on this subject is E. C. Titchmarsh's 1942 text, Eigenfunction Expansions Associated with Second Order Differential Equations -- Part I. This was written by a master who studied under G. H. Hardy. Titchmarsh was the "Savilian Professor of Geometry in the University of Oxford."
Titchmarsh starts from basic Complex Analysis and Advanced Calculus, and by page 13 he has stated the completeness theorem for regular expansions. In the next few pages, he proves that the regular expansions are complete by showing that they are asymptotically the same as ordinary Fourier expansions, with specific bounds. He makes this topic look easy, which it is not. He also deals with the singular problems, the spectrum, and spectral measures. The heart of Titchmarsh's analysis relies on Complex Analysis, of which he was a master. Much of what is found in this text is his original research. Titchmarsh is still referenced by those in the field.
answered Nov 27 at 5:03
DisintegratingByParts
58.3k42579
add a comment |
up vote
2
down vote
accepted
The classic book on this subject is E. C. Titchmarsh's 1942 text, Eigenfunction Expansions Associated with Second Order Differential Equations -- Part I. This was written by a master who studied under G. H. Hardy. Titchmarsh was the "Savilian Professor of Geometry in the University of Oxford."
Titchmarsh starts from basic Complex Analysis and Advanced Calculus, and by page 13 he has stated the completeness theorem for regular expansions. In the next few pages, he proves that the regular expansions are complete by showing that they are asymptotically the same as ordinary Fourier expansions, with specific bounds. He makes this topic look easy, which it is not. He also deals with the singular problems, the spectrum, and spectral measures. The heart of Titchmarsh's analysis relies on Complex Analysis, of which he was a master. Much of what is found in this text is his original research. Titchmarsh is still referenced by those in the field.
answered Nov 27 at 5:03
DisintegratingByParts
58.3k42579
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
The classic book on this subject is E. C. Titchmarsh's 1942 text, Eigenfunction Expansions Associated with Second Order Differential Equations -- Part I. This was written by a master who studied under G. H. Hardy. Titchmarsh was the "Savilian Professor of Geometry in the University of Oxford."
Titchmarsh starts from basic Complex Analysis and Advanced Calculus, and by page 13 he has stated the completeness theorem for regular expansions. In the next few pages, he proves that the regular expansions are complete by showing that they are asymptotically the same as ordinary Fourier expansions, with specific bounds. He makes this topic look easy, which it is not. He also deals with the singular problems, the spectrum, and spectral measures. The heart of Titchmarsh's analysis relies on Complex Analysis, of which he was a master. Much of what is found in this text is his original research. Titchmarsh is still referenced by those in the field.
answered Nov 27 at 5:03
DisintegratingByParts
58.3k42579
The classic book on this subject is E. C. Titchmarsh's 1942 text, Eigenfunction Expansions Associated with Second Order Differential Equations -- Part I. This was written by a master who studied under G. H. Hardy. Titchmarsh was the "Savilian Professor of Geometry in the University of Oxford."
Titchmarsh starts from basic Complex Analysis and Advanced Calculus, and by page 13 he has stated the completeness theorem for regular expansions. In the next few pages, he proves that the regular expansions are complete by showing that they are asymptotically the same as ordinary Fourier expansions, with specific bounds. He makes this topic look easy, which it is not. He also deals with the singular problems, the spectrum, and spectral measures. The heart of Titchmarsh's analysis relies on Complex Analysis, of which he was a master. Much of what is found in this text is his original research. Titchmarsh is still referenced by those in the field.
answered Nov 27 at 5:03
DisintegratingByParts
58.3k42579
edited Nov 27 at 5:45
answered Nov 27 at 5:03
DisintegratingByParts
58.3k42579
answered Nov 27 at 5:03
DisintegratingByParts
58.3k42579
answered Nov 27 at 5:03
DisintegratingByParts
58.3k42579
58.3k42579
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I assume you are working with regular Sturm-Liouville problems on a finite interval. Otherwise, the completeness can be trickier because the expansions may require continuous components resembling Fourier integral expansions.
– DisintegratingByParts
Nov 27 at 0:53
Sure let's say the interval is finite, though now I'm curious about the continuous case too—is there a similar theorem for that case?
– WillG
Nov 27 at 2:38
The nicest proofs use Complex Analysis. Do you have any background with contour integrals and residues? And, yes, the singular cases have expansions which may include fourier integral components as well as series components.
– DisintegratingByParts
Nov 27 at 3:07
Yes, I'm all about Complex Analysis. Specific references to these proofs / books would be welcome.
– WillG
Nov 27 at 4:46