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 ∈ L1

loc(R), |r| > 0 a.e. on R.

Define the maximal operator L := 1 r(x)

• − d2

dx2 + q(x)

• ,

D(L) = Dmin, (2) If r(x) > 0 a.e on R, then L is symmetric in a Hilbert space L2(R, r(x)dx). 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 ∈ L1

loc(R), |r| > 0 a.e. on R.

Define the maximal operator L := 1 r(x)

• − d2

dx2 + q(x)

• ,

D(L) = Dmin, (2) If r(x) > 0 a.e on R, then L is symmetric in a Hilbert space L2(R, r(x)dx). 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 ∈ L1

loc(R), |r| > 0 a.e. on R.

Define the maximal operator L := 1 r(x)

• − d2

dx2 + q(x)

• ,

D(L) = Dmin, (2) If r(x) > 0 a.e on R, then L is symmetric in a Hilbert space L2(R, r(x)dx). 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 ∈ L1

loc(R), |r| > 0 a.e. on R.

Define the maximal operator L := 1 r(x)

• − d2

dx2 + q(x)

• ,

D(L) = Dmin, (2) If r(x) > 0 a.e on R, then L is symmetric in a Hilbert space L2(R, r(x)dx). 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 L2(R, |r|), consider the operator A := 1 |r(x)|

• − d2

dx2 + q(x)

• ,

D(A) = Dmin, (3) A = JL, (Jf)(x) := (sgn x)f(x), J = J∗ = J−1. Hypothesis: r(x) = (sgn x)|r(x)|, A = A∗ ≥ 0 i.e., L is J-nonnegative and J-self-adjoint in L2(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 L2(R, |r|), consider the operator A := 1 |r(x)|

• − d2

dx2 + q(x)

• ,

D(A) = Dmin, (3) A = JL, (Jf)(x) := (sgn x)f(x), J = J∗ = J−1. Hypothesis: r(x) = (sgn x)|r(x)|, A = A∗ ≥ 0 i.e., L is J-nonnegative and J-self-adjoint in L2(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 L2(R, |r|), consider the operator A := 1 |r(x)|

• − d2

dx2 + q(x)

• ,

D(A) = Dmin, (3) A = JL, (Jf)(x) := (sgn x)f(x), J = J∗ = J−1. Hypothesis: r(x) = (sgn x)|r(x)|, A = A∗ ≥ 0 i.e., L is J-nonnegative and J-self-adjoint in L2(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 L2(R, |r|), consider the operator A := 1 |r(x)|

• − d2

dx2 + q(x)

• ,

D(A) = Dmin, (3) A = JL, (Jf)(x) := (sgn x)f(x), J = J∗ = J−1. Hypothesis: r(x) = (sgn x)|r(x)|, A = A∗ ≥ 0 i.e., L is J-nonnegative and J-self-adjoint in L2(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 L2(R, |r|), consider the operator A := 1 |r(x)|

• − d2

dx2 + q(x)

• ,

D(A) = Dmin, (3) A = JL, (Jf)(x) := (sgn x)f(x), J = J∗ = J−1. Hypothesis: r(x) = (sgn x)|r(x)|, A = A∗ ≥ 0 i.e., L is J-nonnegative and J-self-adjoint in L2(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 L2(R, |r|), consider the operator A := 1 |r(x)|

• − d2

dx2 + q(x)

• ,

D(A) = Dmin, (3) A = JL, (Jf)(x) := (sgn x)f(x), J = J∗ = J−1. Hypothesis: r(x) = (sgn x)|r(x)|, A = A∗ ≥ 0 i.e., L is J-nonnegative and J-self-adjoint in L2(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, ...]

• perators 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+ ∈ C1[0, δ], p− ∈ C1[−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, ...]

• perators 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+ ∈ C1[0, δ], p− ∈ C1[−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, ...]

• perators 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+ ∈ C1[0, δ], p− ∈ C1[−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, ...]

• perators 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+ ∈ C1[0, δ], p− ∈ C1[−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, ...]

• perators 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+ ∈ C1[0, δ], p− ∈ C1[−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, ...]

• perators 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+ ∈ C1[0, δ], p− ∈ C1[−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, ...]

• perators 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+ ∈ C1[0, δ], p− ∈ C1[−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 0 Abstract criteria for J-nonnegative operators [Veseli´ c’ 1972], [Akopyan’ 1980], [ ´ Curgus’ 1985] L := 1 r(x) d2 dx2 , 0 is regular if r(x) = sgn x ´ Curgus, Najman, Proc. AMS, 1995 r(x) = (sgn x)|x|α, α > −1 Fleige, Najman’ Oper. Theory:

• Adv. Appl. 102, 1998

higher order operators, partial differential operators ´ Curgus, Najman’ 1996, ...

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 0 Abstract criteria for J-nonnegative operators [Veseli´ c’ 1972], [Akopyan’ 1980], [ ´ Curgus’ 1985] L := 1 r(x) d2 dx2 , 0 is regular if r(x) = sgn x ´ Curgus, Najman, Proc. AMS, 1995 r(x) = (sgn x)|x|α, α > −1 Fleige, Najman’ Oper. Theory:

• Adv. Appl. 102, 1998

higher order operators, partial differential operators ´ Curgus, Najman’ 1996, ...

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 0 Abstract criteria for J-nonnegative operators [Veseli´ c’ 1972], [Akopyan’ 1980], [ ´ Curgus’ 1985] L := 1 r(x) d2 dx2 , 0 is regular if r(x) = sgn x ´ Curgus, Najman, Proc. AMS, 1995 r(x) = (sgn x)|x|α, α > −1 Fleige, Najman’ Oper. Theory:

• Adv. Appl. 102, 1998

higher order operators, partial differential operators ´ Curgus, Najman’ 1996, ...

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 0 Abstract criteria for J-nonnegative operators [Veseli´ c’ 1972], [Akopyan’ 1980], [ ´ Curgus’ 1985] L := 1 r(x) d2 dx2 , 0 is regular if r(x) = sgn x ´ Curgus, Najman, Proc. AMS, 1995 r(x) = (sgn x)|x|α, α > −1 Fleige, Najman’ Oper. Theory:

• Adv. Appl. 102, 1998

higher order operators, partial differential operators ´ Curgus, Najman’ 1996, ...

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 0 Abstract criteria for J-nonnegative operators [Veseli´ c’ 1972], [Akopyan’ 1980], [ ´ Curgus’ 1985] L := 1 r(x) d2 dx2 , 0 is regular if r(x) = sgn x ´ Curgus, Najman, Proc. AMS, 1995 r(x) = (sgn x)|x|α, α > −1 Fleige, Najman’ Oper. Theory:

• Adv. Appl. 102, 1998

higher order operators, partial differential operators ´ Curgus, Najman’ 1996, ...

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 0 Abstract criteria for J-nonnegative operators [Veseli´ c’ 1972], [Akopyan’ 1980], [ ´ Curgus’ 1985] L := 1 r(x) d2 dx2 , 0 is regular if r(x) = sgn x ´ Curgus, Najman, Proc. AMS, 1995 r(x) = (sgn x)|x|α, α > −1 Fleige, Najman’ Oper. Theory:

• Adv. Appl. 102, 1998

higher order operators, partial differential operators ´ Curgus, Najman’ 1996, ...

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Abstract resolvent criteria for arbitrary operators [van Casteren’ 1983], [Naboko’1984], [Malamud’1985] r(x) = sgn x, q ≡ C, then 0 is regular iff C ≥ 0 Karabash’ 1998 r(x) = sgn x,

• R |q(x)|(1 + x2)dx < ∞, q ≥ 0

Faddeev, Shterenberg’ 2000 r(x) = sgn x, ∃ q ∈ L2(R) such that 0 is singular Karabash, Kostenko’ 2008

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Abstract resolvent criteria for arbitrary operators [van Casteren’ 1983], [Naboko’1984], [Malamud’1985] r(x) = sgn x, q ≡ C, then 0 is regular iff C ≥ 0 Karabash’ 1998 r(x) = sgn x,

• R |q(x)|(1 + x2)dx < ∞, q ≥ 0

Faddeev, Shterenberg’ 2000 r(x) = sgn x, ∃ q ∈ L2(R) such that 0 is singular Karabash, Kostenko’ 2008

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Abstract resolvent criteria for arbitrary operators [van Casteren’ 1983], [Naboko’1984], [Malamud’1985] r(x) = sgn x, q ≡ C, then 0 is regular iff C ≥ 0 Karabash’ 1998 r(x) = sgn x,

• R |q(x)|(1 + x2)dx < ∞, q ≥ 0

Faddeev, Shterenberg’ 2000 r(x) = sgn x, ∃ q ∈ L2(R) such that 0 is singular Karabash, Kostenko’ 2008

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Abstract resolvent criteria for arbitrary operators [van Casteren’ 1983], [Naboko’1984], [Malamud’1985] r(x) = sgn x, q ≡ C, then 0 is regular iff C ≥ 0 Karabash’ 1998 r(x) = sgn x,

• R |q(x)|(1 + x2)dx < ∞, q ≥ 0

Faddeev, Shterenberg’ 2000 r(x) = sgn x, ∃ q ∈ L2(R) such that 0 is singular Karabash, Kostenko’ 2008

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 1 [KKM’09] Assume A = A∗ ≥ 0. If r(x) = sgn x

• R

(1 + |x|)|q(x)|dx < ∞, (4) then 0 is regular and L is similar to a s-a in L2(R). Theorem 2 [KKM’09] If r(x) = sgn x and (??) holds, then the following statements are equivalent: (i) L is similar to a self-adjoint operator, (ii) L is J-nonnegative (i.e., L ≥ 0), (iii) the spectrum of L is real.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 1 [KKM’09] Assume A = A∗ ≥ 0. If r(x) = sgn x

• R

(1 + |x|)|q(x)|dx < ∞, (4) then 0 is regular and L is similar to a s-a in L2(R). Theorem 2 [KKM’09] If r(x) = sgn x and (??) holds, then the following statements are equivalent: (i) L is similar to a self-adjoint operator, (ii) L is J-nonnegative (i.e., L ≥ 0), (iii) the spectrum of L is real.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 1 [KKM’09] Assume A = A∗ ≥ 0. If r(x) = sgn x

• R

(1 + |x|)|q(x)|dx < ∞, (4) then 0 is regular and L is similar to a s-a in L2(R). Theorem 2 [KKM’09] If r(x) = sgn x and (??) holds, then the following statements are equivalent: (i) L is similar to a self-adjoint operator, (ii) L is J-nonnegative (i.e., L ≥ 0), (iii) the spectrum of L is real.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 3 [KKM’09] Let r(x) = sgn x and q0(x) = −χ[0,π/4](x) + 2 χ(π/4,+∞)(x) (1 + x − π/4)2 , x ∈ R+. (5) Then (i) L is J-nonnegative and σ(A) ⊂ R. (ii) 0 is a simple eigenvalue of L. (iii) 0 is a singular critical point of L. (iv) L is not similar to a self-adjoint operator. Remark If

• q0(x) = 2 χ(π/4,+∞)(x)

(1 + x − π/4)2 , then 0 is a regular critical point of L and L is similar to s.-a.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 3 [KKM’09] Let r(x) = sgn x and q0(x) = −χ[0,π/4](x) + 2 χ(π/4,+∞)(x) (1 + x − π/4)2 , x ∈ R+. (5) Then (i) L is J-nonnegative and σ(A) ⊂ R. (ii) 0 is a simple eigenvalue of L. (iii) 0 is a singular critical point of L. (iv) L is not similar to a self-adjoint operator. Remark If

• q0(x) = 2 χ(π/4,+∞)(x)

(1 + x − π/4)2 , then 0 is a regular critical point of L and L is similar to s.-a.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 3 [KKM’09] Let r(x) = sgn x and q0(x) = −χ[0,π/4](x) + 2 χ(π/4,+∞)(x) (1 + x − π/4)2 , x ∈ R+. (5) Then (i) L is J-nonnegative and σ(A) ⊂ R. (ii) 0 is a simple eigenvalue of L. (iii) 0 is a singular critical point of L. (iv) L is not similar to a self-adjoint operator. Remark If

• q0(x) = 2 χ(π/4,+∞)(x)

(1 + x − π/4)2 , then 0 is a regular critical point of L and L is similar to s.-a.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 3 [KKM’09] Let r(x) = sgn x and q0(x) = −χ[0,π/4](x) + 2 χ(π/4,+∞)(x) (1 + x − π/4)2 , x ∈ R+. (5) Then (i) L is J-nonnegative and σ(A) ⊂ R. (ii) 0 is a simple eigenvalue of L. (iii) 0 is a singular critical point of L. (iv) L is not similar to a self-adjoint operator. Remark If

• q0(x) = 2 χ(π/4,+∞)(x)

(1 + x − π/4)2 , then 0 is a regular critical point of L and L is similar to s.-a.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 3 [KKM’09] Let r(x) = sgn x and q0(x) = −χ[0,π/4](x) + 2 χ(π/4,+∞)(x) (1 + x − π/4)2 , x ∈ R+. (5) Then (i) L is J-nonnegative and σ(A) ⊂ R. (ii) 0 is a simple eigenvalue of L. (iii) 0 is a singular critical point of L. (iv) L is not similar to a self-adjoint operator. Remark If

• q0(x) = 2 χ(π/4,+∞)(x)

(1 + x − π/4)2 , then 0 is a regular critical point of L and L is similar to s.-a.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 3 [KKM’09] Let r(x) = sgn x and q0(x) = −χ[0,π/4](x) + 2 χ(π/4,+∞)(x) (1 + x − π/4)2 , x ∈ R+. (5) Then (i) L is J-nonnegative and σ(A) ⊂ R. (ii) 0 is a simple eigenvalue of L. (iii) 0 is a singular critical point of L. (iv) L is not similar to a self-adjoint operator. Remark If

• q0(x) = 2 χ(π/4,+∞)(x)

(1 + x − π/4)2 , then 0 is a regular critical point of L and L is similar to s.-a.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators

Theorem 3 [KKM’09] Let r(x) = sgn x and q0(x) = −χ[0,π/4](x) + 2 χ(π/4,+∞)(x) (1 + x − π/4)2 , x ∈ R+. (5) Then (i) L is J-nonnegative and σ(A) ⊂ R. (ii) 0 is a simple eigenvalue of L. (iii) 0 is a singular critical point of L. (iv) L is not similar to a self-adjoint operator. Remark If

• q0(x) = 2 χ(π/4,+∞)(x)

(1 + x − π/4)2 , then 0 is a regular critical point of L and L is similar to s.-a.

Mark Malamud (joint work with I.Karabash and A.Kostenko J. Diff. Eqs. 246 (2009), 964–997) Similarity problem for indefinite SL operators