Similarity problem for indefinite SturmLiouville operators Mark - PowerPoint PPT Presentation

Similarity problem for indefinite Sturm–Liouville operators Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Institute of Applied Mathematics and Mechanics, Donetsk, Ukarine Dubrovnik 11 May, 2009 Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Consider the differential expression − y ′′ ( x ) + q ( x ) y ( x ) = λ r ( x ) y ( x ) , x ∈ R . (1) q , r are real functions, q , r ∈ L 1 l oc ( R ) , | r | > 0 a.e. on R . Define the maximal operator � � − d 2 1 L := d x 2 + q ( x ) , D ( L ) = D min , (2) r ( x ) If r ( x ) > 0 a.e on R , then L is symmetric in a Hilbert space L 2 ( R , r ( x ) d x ) . If r changes its sign, then L is not symmetric in a Hilbert space, but it is symmetric a Krein space Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Consider the differential expression − y ′′ ( x ) + q ( x ) y ( x ) = λ r ( x ) y ( x ) , x ∈ R . (1) q , r are real functions, q , r ∈ L 1 l oc ( R ) , | r | > 0 a.e. on R . Define the maximal operator � � − d 2 1 L := d x 2 + q ( x ) , D ( L ) = D min , (2) r ( x ) If r ( x ) > 0 a.e on R , then L is symmetric in a Hilbert space L 2 ( R , r ( x ) d x ) . If r changes its sign, then L is not symmetric in a Hilbert space, but it is symmetric a Krein space Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Consider the differential expression − y ′′ ( x ) + q ( x ) y ( x ) = λ r ( x ) y ( x ) , x ∈ R . (1) q , r are real functions, q , r ∈ L 1 l oc ( R ) , | r | > 0 a.e. on R . Define the maximal operator � � − d 2 1 L := d x 2 + q ( x ) , D ( L ) = D min , (2) r ( x ) If r ( x ) > 0 a.e on R , then L is symmetric in a Hilbert space L 2 ( R , r ( x ) d x ) . If r changes its sign, then L is not symmetric in a Hilbert space, but it is symmetric a Krein space Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Consider the differential expression − y ′′ ( x ) + q ( x ) y ( x ) = λ r ( x ) y ( x ) , x ∈ R . (1) q , r are real functions, q , r ∈ L 1 l oc ( R ) , | r | > 0 a.e. on R . Define the maximal operator � � − d 2 1 L := d x 2 + q ( x ) , D ( L ) = D min , (2) r ( x ) If r ( x ) > 0 a.e on R , then L is symmetric in a Hilbert space L 2 ( R , r ( x ) d x ) . If r changes its sign, then L is not symmetric in a Hilbert space, but it is symmetric a Krein space Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

J-nonnegative operators In L 2 ( R , | r | ) , consider the operator � � − d 2 1 A := d x 2 + q ( x ) , D ( A ) = D min , (3) | r ( x ) | A = JL , ( Jf )( x ) := ( sgn x ) f ( x ) , J = J ∗ = J − 1 . Hypothesis: A = A ∗ ≥ 0 r ( x ) = ( sgn x ) | r ( x ) | , i.e., L is J -nonnegative and J -self-adjoint in L 2 ( R , | r | ) Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

J-nonnegative operators In L 2 ( R , | r | ) , consider the operator � � − d 2 1 A := d x 2 + q ( x ) , D ( A ) = D min , (3) | r ( x ) | A = JL , ( Jf )( x ) := ( sgn x ) f ( x ) , J = J ∗ = J − 1 . Hypothesis: A = A ∗ ≥ 0 r ( x ) = ( sgn x ) | r ( x ) | , i.e., L is J -nonnegative and J -self-adjoint in L 2 ( R , | r | ) Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

J-nonnegative operators In L 2 ( R , | r | ) , consider the operator � � − d 2 1 A := d x 2 + q ( x ) , D ( A ) = D min , (3) | r ( x ) | A = JL , ( Jf )( x ) := ( sgn x ) f ( x ) , J = J ∗ = J − 1 . Hypothesis: A = A ∗ ≥ 0 r ( x ) = ( sgn x ) | r ( x ) | , i.e., L is J -nonnegative and J -self-adjoint in L 2 ( R , | r | ) Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

J-nonnegative operators In L 2 ( R , | r | ) , consider the operator � � − d 2 1 A := d x 2 + q ( x ) , D ( A ) = D min , (3) | r ( x ) | A = JL , ( Jf )( x ) := ( sgn x ) f ( x ) , J = J ∗ = J − 1 . Hypothesis: A = A ∗ ≥ 0 r ( x ) = ( sgn x ) | r ( x ) | , i.e., L is J -nonnegative and J -self-adjoint in L 2 ( R , | r | ) Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

J-nonnegative operators In L 2 ( R , | r | ) , consider the operator � � − d 2 1 A := d x 2 + q ( x ) , D ( A ) = D min , (3) | r ( x ) | A = JL , ( Jf )( x ) := ( sgn x ) f ( x ) , J = J ∗ = J − 1 . Hypothesis: A = A ∗ ≥ 0 r ( x ) = ( sgn x ) | r ( x ) | , i.e., L is J -nonnegative and J -self-adjoint in L 2 ( R , | r | ) Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

J-nonnegative operators In L 2 ( R , | r | ) , consider the operator � � − d 2 1 A := d x 2 + q ( x ) , D ( A ) = D min , (3) | r ( x ) | A = JL , ( Jf )( x ) := ( sgn x ) f ( x ) , J = J ∗ = J − 1 . Hypothesis: A = A ∗ ≥ 0 r ( x ) = ( sgn x ) | r ( x ) | , i.e., L is J -nonnegative and J -self-adjoint in L 2 ( R , | r | ) Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Critical point ∞ σ ( L ) is discrete: [Beals’ 1977, 1985], [Pyatkov’ 1984, 1989, ...] operators with continuous spectra: [ ´ Curgus, Langer’ 1989] ∞ is regular if r ( x ) = ( sgn x ) p ± ( x ) | x | β ± , ± x ∈ ( 0 , δ ) , with some δ > 0, β ± > − 1, and positive functions p + ∈ C 1 [ 0 , δ ] , p − ∈ C 1 [ − δ, 0 ] ∞ may be singular [Volkmer’ 1996] examples [Fleige’ 1996,1998], [Abasheeva, Pyatkov’ 1997], [Parfenov’ 2003] Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Critical point ∞ σ ( L ) is discrete: [Beals’ 1977, 1985], [Pyatkov’ 1984, 1989, ...] operators with continuous spectra: [ ´ Curgus, Langer’ 1989] ∞ is regular if r ( x ) = ( sgn x ) p ± ( x ) | x | β ± , ± x ∈ ( 0 , δ ) , with some δ > 0, β ± > − 1, and positive functions p + ∈ C 1 [ 0 , δ ] , p − ∈ C 1 [ − δ, 0 ] ∞ may be singular [Volkmer’ 1996] examples [Fleige’ 1996,1998], [Abasheeva, Pyatkov’ 1997], [Parfenov’ 2003] Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Critical point ∞ σ ( L ) is discrete: [Beals’ 1977, 1985], [Pyatkov’ 1984, 1989, ...] operators with continuous spectra: [ ´ Curgus, Langer’ 1989] ∞ is regular if r ( x ) = ( sgn x ) p ± ( x ) | x | β ± , ± x ∈ ( 0 , δ ) , with some δ > 0, β ± > − 1, and positive functions p + ∈ C 1 [ 0 , δ ] , p − ∈ C 1 [ − δ, 0 ] ∞ may be singular [Volkmer’ 1996] examples [Fleige’ 1996,1998], [Abasheeva, Pyatkov’ 1997], [Parfenov’ 2003] Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Critical point ∞ σ ( L ) is discrete: [Beals’ 1977, 1985], [Pyatkov’ 1984, 1989, ...] operators with continuous spectra: [ ´ Curgus, Langer’ 1989] ∞ is regular if r ( x ) = ( sgn x ) p ± ( x ) | x | β ± , ± x ∈ ( 0 , δ ) , with some δ > 0, β ± > − 1, and positive functions p + ∈ C 1 [ 0 , δ ] , p − ∈ C 1 [ − δ, 0 ] ∞ may be singular [Volkmer’ 1996] examples [Fleige’ 1996,1998], [Abasheeva, Pyatkov’ 1997], [Parfenov’ 2003] Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Critical point ∞ σ ( L ) is discrete: [Beals’ 1977, 1985], [Pyatkov’ 1984, 1989, ...] operators with continuous spectra: [ ´ Curgus, Langer’ 1989] ∞ is regular if r ( x ) = ( sgn x ) p ± ( x ) | x | β ± , ± x ∈ ( 0 , δ ) , with some δ > 0, β ± > − 1, and positive functions p + ∈ C 1 [ 0 , δ ] , p − ∈ C 1 [ − δ, 0 ] ∞ may be singular [Volkmer’ 1996] examples [Fleige’ 1996,1998], [Abasheeva, Pyatkov’ 1997], [Parfenov’ 2003] Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Critical point ∞ σ ( L ) is discrete: [Beals’ 1977, 1985], [Pyatkov’ 1984, 1989, ...] operators with continuous spectra: [ ´ Curgus, Langer’ 1989] ∞ is regular if r ( x ) = ( sgn x ) p ± ( x ) | x | β ± , ± x ∈ ( 0 , δ ) , with some δ > 0, β ± > − 1, and positive functions p + ∈ C 1 [ 0 , δ ] , p − ∈ C 1 [ − δ, 0 ] ∞ may be singular [Volkmer’ 1996] examples [Fleige’ 1996,1998], [Abasheeva, Pyatkov’ 1997], [Parfenov’ 2003] Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators