THE PRINCIPLE OF LIMITING ABSORPTION FOR THE NON-SELFADJOINT THE PRINCIPLE OF LIMITING ABSORPTION FOR THE NON-SELFADJOINT THE PRINCIPLE OF LIMITING ABSORPTION FOR THE NON-SELFADJOINT
SCHRÖDINGER OPERATOR WITH ENERGY DEPENDENT POTENTIAL SCHRÖDINGER OPERATOR WITH ENERGY DEPENDENT POTENTIAL SCHRÖDINGER OPERATOR WITH ENERGY DEPENDENT POTENTIAL

Hideo Nakazawa

Department of Mathematics, Tokyo Metropolitan University

2000. 1.29

1.Introduction

In this paper we shall show the principle of limiting absorption for the quadratic operator pencil

(1.1)  L(κ) = −Δ − iκB − κ^{2}

in the N–dimensional euclidean space ℝ^{N} with N ≠ 2. Here Δ is the N–dimensional Laplacian, κ ∈ ℂ and B denotes the multiplication operator by a real-valued function b(x) on ℝ^{N}. If we consider the solutions of the form w(x, t) = u(x)e^{−iκt} in (1.1), then u(x) satisfies

L(κ)u(x) = (−Δ − iκb(x) − κ^{2})u(x) = 0.

Throughout the paper we require the following condition:

(1.2)

There exist constants b_{0} and δ ∈ (0, 1] such that 0 < b_{0} < \frac{2 − \sqrt{2}}{2}δ
and |b(x)| ≤ b_{0}(1 + r)^{−1−2δ} for any x ∈ ℝ^{N}, where r = |x|.

The operator L(κ) is derived from the wave equation

(1.3)  w_{tt}(x, t) − Δw(x, t) + b(x)w_{t}(x, t) = 0

where x ∈ ℝ^{N}, t ≥ 0, w_{t} = \frac{∂w}{∂t}, w_{tt} = \frac{∂^{2}w}{∂t^{2}} and b(x) ∈ C^{1}(ℝ^{N}). If b(x) ≥ 0, b(x)w_{t} represents a friction of viscous type. As is known in Mochizuki [14], under a more general assumption than (1.2) with b(x) ≥ 0, the solutions of (1.3) are asymptotically equal for t → +∞ to those of free wave equations

(1.4)  w_{0tt}(x, t) − Δw_{0}(x, t) = 0.

More precisely, he proved

Assume N ≠ 2. If b(x) satisfies 0 ≤ b(x) ≤ b_{0}(1+r)^{−1−δ} for some δ > 0, then the following statements hold:

(1) The Møller Weve operators exist:

∃W_{+} = lim_{t→+∞} U(t)U_{0}(−t), ^{∃}W_{−} = lim_{t→+∞} U_{0}(−t)U(t),

where {U_{0}(t)}_{t∈ℝ} and {U(t)}_{t≥0} represent a one–parameter group of unitary operators and semi–group of contraction operators, respectively (lim means the strong limit in uniform operator topology).

(2) They both are not identically vanishing contraction operators in H ≡ Ḣ^{1}(ℝ^{N}) × L^{2}(ℝ^{N}).

(3) U_{0}(t) and U(t) are intertwined by W_{±}, i.e.,

W_{+}U_{0}(t) = U(t)W_{+}, W_{−}U(t) = U_{0}(t)W_{−} for any t ≥ 0.

(4) The scattering operator, defined by S = W_{−}W_{+}, commutes with U_{0}(t):

SU_{0}(t) = U_{0}(t)S for any t ∈ ℝ.

If we consider the solutions of the form w(x, t) = u(x)e^{−iκt} in (1.3), then u(x) satisfies

(1.5)  L(κ)u(x) = (−Δ − iκb(x) − κ^{2})u(x) = 0.

The principle of limiting absorption states that there exists two Hilbert spaces

L^{2,(1+δ)/2}(ℝ^{N}) and L^{2,−(1+δ)/2}(ℝ^{N})

such that

L^{2,(1+δ)/2}(ℝ^{N}) ⊂ L^{2}(ℝ^{N}) ⊂ L^{2,−(1+δ)/2}(ℝ^{N})

(the definition of these spaces is given in section 3), and each of R ((σ ± iτ)^{2}) ≡ L(σ±iτ)^{−1} is continuously extended to τ = 0 as an operator from L^{2,(1+δ)/2}(ℝ^{N}) to L^{2,−(1+δ)/2}(ℝ^{N}). More precisely,

Main Theorem. Assume N ≠ 2 and (1.2). Let

K_{+}^{+} = {κ = σ + iτ ∈ ℂ \ {0} | σ ∈ I = (a_{1}, a_{2}), τ ∈ (0, 1)}

with 0 < a_{1} < a_{2} < ∞. For σ + iτ ∈ K_{+}^{+}, and f ∈ L^{2,(1+δ)/2}(ℝ^{N}) (0 < δ ≤ 1), we define

u(σ, τ) = R ((σ + iτ)^{2}) f.

Then u(σ, τ) ∈ L^{2,−(1+δ)/2}(ℝ^{N}) converges in L^{2,−(1+δ)/2}(ℝ^{N}) as τ ↓ 0. Moreover this convergence is uniform with respect to σ ∈ I. If we denote this limit by u(σ), then u(σ) is a solution of L(σ)u(x) = f(x) and is uniformly continuous with respect to σ ∈ I in L^{2,−(1+δ)/2}(ℝ^{N}).

Remark. In this theorem, we can replace K_{+}^{+} by K_{−}^{+}, K_{+}^{−} or K_{−}^{−}:

K_{−}^{+} = {κ = σ − iτ ∈ ℂ \ {0} | σ ∈ I = (a_{1}, a_{2}), τ ∈ (0, 1)} ,
K_{+}^{−} = {κ = σ + iτ ∈ ℂ \ {0} | σ ∈ I = (−a_{2}, −a_{1}), τ ∈ (0, 1)} ,
K_{−}^{−} = {κ = σ − iτ ∈ ℂ \ {0} | σ ∈ I = (−a_{2}, −a_{1}), τ ∈ (0, 1)} .

The principle of limiting absorption has been studied for the Schrödinger equation

(1.6)  (−Δ + V (x) − κ^{2}) u(x) = f(x) in ℝ^{N}.

After the pioneering work of Éĭdus [2], Agmon [1], Ikebe–Saito [4] and Mochizuki [11] extended it to a wider class of real valued potentials. The spectral theory is also developed for complex potentials by Kato [6], Mochizuki [9], [10] and Saito [14], [15], and Saito extended the principle there to complex potentials.

The methods developed in these works are not directly applied to our problem since iκb(x) depends on the energy κ. In order to overcome the difficulty we suppose the smallness condition (1.2). Then Mochizuki’s method [12] directly applies to obtain the key a-priori estimate ( Theorem 3.4 (3.5) ). From this follows the uniqueness of solutions for L(κ)u(x) = f(x) (Imκ ⋛ 0) which plays an important role in the proof of Main Theorem (Corollary 3.5 (1)).

2. Spectral Structure for L(κ)

In this section, for the sake of simplicity, we assume b(x) ≥ 0. We define the linear operator H_{0} by 𝔇(H_{0}) = H^{2}(ℝ^{N}), H_{0} = −Δ. As is well known, H_{0} is a selfadjoint operator and it holds that ρ(H_{0}) = ℂ \ [0, ∞), σ(H_{0}) = σ_{ess}(H_{0}) = σ_{c}(H_{0}) = [0, ∞), σ_{r}(H_{0}) = σ_{p}(H_{0}) = ∅, where ρ(H_{0}), σ(H_{0}), σ_{ess}(H_{0}), σ_{c}(H_{0}), σ_{r}(H_{0}) and σ_{p}(H_{0}) denote the resolvent set, spectrum, essential spectrum, continuous spectrum, residual spectrum, and point spectrum, respectively.

For κ ∈ ℂ \ {0}, we define the linear operator L(κ) by L(κ) = H_{0} − iκB − κ^{2}, 𝔇(L(κ)) = H^{2}(ℝ^{N}). This is a non-selfadjoint but closed operator. The adjoint operator L^{∗}(κ) of L(κ) is defind by L^{∗}(κ) = H_{0} + i\overline{κ}B − \overline{κ}^{2}, 𝔇(L^{∗}(κ)) = H^{2}(ℝ^{N}).

The resolvent set, the spectrum, the essential spectrum, the continuous spectrum, the residual spectrum and the point spectrum of L(κ) are defined as follows ( cf. Markus [7] pp.56, Saito [14] pp.406 ).

ρ (L(κ)) = \{ κ^{2} ∈ ℂ | Ker(L(κ)) = \{0\}, \overline{ℜ(L(κ))} = L^{2}(ℝ^{N}),
(L(κ))^{−1} ∈ 𝔏 (L^{2}(ℝ^{N})) },
σ (L(κ)) = ℂ \ ρ (L(κ)) ,
σ_{ess} (L(κ)) = { κ^{2} ∈ σ (L(κ)) | ^{∃}{f_{n}} ⊂ 𝔇(L(κ)) s.t ∥f_{n}∥_{L^{2}} = 1,

L(κ)f_{n} →_{S} 0 (n → ∞), f_{n} →_{w} 0 (n → ∞) },

(where →_{S} and →_{w} denote the strong and weak convergence in L^{2} respectively),

σ_{c} (L(κ)) = \{ κ^{2} ∈ σ (L(κ)) | Ker (L(κ)) = \{0\}, \overline{ℜ (L(κ))} = L^{2}(ℝ^{N}),

(L(κ))^{−1} ∉ 𝔏 (L^{2}(ℝ^{N})) },

σ_{r} (L(κ)) = \{ κ^{2} ∈ σ (L(κ)) | Ker (L(κ)) = \{0\}, \overline{ℜ (L(κ))} ≠ L^{2}(ℝ^{N}) \} ,

σ_{p} (L(κ)) = { κ^{2} ∈ σ (L(κ)) | Ker (L(κ)) ≠ {0} } ,

where Ker (L(κ)) = \{ f ∈ 𝔇 (L(κ)) | L(κ)f = 0 \}, ℜ (L(κ)) is the range of the operator L(κ), \overline{ℜ (L(κ))} denotes the closure of ℜ (L(κ)) in L^{2}(ℝ^{N}), and 𝔏 (L^{2}(ℝ^{N})) is the set of the bounded linear operator on L^{2}(ℝ^{N}). For κ^{2} ∈ ρ (L(κ)), we put R(κ^{2}) = (L(κ))^{−1} and call it resolvent of L(κ). We can define the spectrum sets for L^{∗}(κ) similarly as above. As in the case of the usual definition, we see

(2.1)  κ^{2} ∈ σ_{r}(L(κ)) ⇔ \overline{κ}^{2} ∈ σ_{p}(L^{∗}(κ)), κ^{2} ∉ σ_{p}(L(κ)).

From this we obtain the absence of the residual spectrum of L(κ).

In these definitions, noting that iκB is H_{0}– and (H_{0} − iκB)–compact, we can follow the same line of proof as Schechter [16] Chapter 3 to show the stability theorem of essential spectrum.

Moreover, if we assume N ≠ 2 and the following condition:

(2.2)

There exist positive constants b_{0} and δ such that 0 < b_{0} < δ ≤ 1

and 0 ≤ b(x) ≤ b_{0}(1 + r)^{−1−2δ} for any x ∈ ℝ^{N},

then the absence of positive eigenvalue of L(κ) is obtaind by the modifications of the proof of (3.5) of Theorem 3.4 below. Therefore, we obtain

Theorem 2.1. Assume that N ≠ 2 and (2.2). Then the following assertions hold:

σ(L(κ)) = σ_{ess}(L(κ)) = σ_{c}(L(κ)) = [0, ∞),

σ_{p}(L(κ)) = σ_{r}(L(κ)) = ∅, ρ(L(κ)) = ℂ \ [0, ∞).

3. Some Inequalities-Resolvent Estimate

In this section we shall show some inequalities for L^{2}-solutions of the Schrödinger equation of the form

(3.1)  −Δu − iκb(x)u − κ^{2}u = f(x)  in ℝ^{N},

where κ = σ + iτ (σ(≠ 0), τ ∈ ℝ) ∈ ℂ \ {0} and f(x) ∈ L^{2,(1+δ)/2}(ℝ^{N}), where

L^{2,α}(G) = \{ f(x) | ∥f∥_{α,G} = (∫_{G}(1 + |x|)^{2α}|f(x)|^{2}dx)^{\frac{1}{2}} < ∞ \} .

If G = ℝ^{N}, we shall omit the second symbol G.

We define two operators D^{±} and D_{r}^{±} as follows ( cf. Ikebe–Saito [4], Mochizuki [11], [12], [13], Saito [14], [15] ):

D^{±}u = ∇u + \frac{N − 1}{2r}u\frac{x}{r} ∓ iκu\frac{x}{r} (Imκ ⋛ 0) ,

D_{r}^{±}u = \frac{x}{r} · D^{±}u = u_{r} + \frac{N − 1}{2r}u ∓ iκu (Imκ ⋛ 0) .

Proposition 3.1. If u is the solution of (3.1), then the following inequality holds:

(3.2)  \begin{matrix}∫_{ℝ^{N}} (1 + r)^{−1−δ} |u_{r} + \frac{N − 1}{2r}u|^{2} dx + |κ|^{2} ∫_{ℝ^{N}} \{(1 + r)^{−1−δ} − \frac{2|b(x)|}{δ}\} |u|^{2}dx \\ ≦ ∫_{ℝ^{N}} (1 + r)^{−1−δ}|D_{r}^{±}u|^{2}dx + \frac{2}{δ} ∫_{ℝ^{N}} |f\overline{iκu}|dx  (Imκ ⋛ 0) . \end{matrix}

Lemma 3.2. Let φ = φ(r) be a real-valued C^{1} –function. If u is a solution of (3.1), then the following identity holds:

(3.3)  \begin{matrix}∫_{ℝ^{N}} \{(±τφ + \frac{φ_{r}}{2} ) |D^{±}u|^{2} + (\frac{φ}{r} − φ_{r}) (|D^{±}u|^{2} − |D_{r}^{±}u|^{2})\} dx \\ ± ∫_{ℝ^{N}} \frac{b(x)φ}{2} (|D_{r}^{±}u|^{2} + |κ|^{2}|u|^{2} − |u_{r} + \frac{N − 1}{2r}u|^{2}) dx \\ + ∫_{ℝ^{N}} \{−\frac{(N − 1)(N − 3)}{8} (\frac{φ}{r^{2}} )_{r} |u|^{2} ± \frac{τ(N − 1)(N − 3)}{4r^{2}}φ|u|^{2}\} dx \\ = Re ∫_{ℝ^{N}} fφ\overline{D_{r}^{±}u}dx (Imκ ⋛ 0) . \end{matrix}

From this, we obtain

Proposition 3.3. Assume that N ≠ 2. If u is a solution of (3.1), then the following inequality holds:

(3.4)  \begin{matrix}∫_{ℝ^{N}} (1 + r)^{−1+δ}|D^{±}u|^{2}dx \\ − \frac{1}{δ} ∫_{ℝ^{N}} |b(x)|(1 + r)^{δ} (|D^{±}u|^{2} + |κ|^{2}|u|^{2} + |u_{r} + \frac{N − 1}{2r}u|^{2}) dx \\ ≤ \frac{2}{δ} ∫_{ℝ^{N}} (1 + r)^{δ}|f\overline{D_{r}^{±}u}|dx  (Imκ ⋛ 0) . \end{matrix}

By Proposition 3.1 and 3.3, we have Theorem 3.4. Assume that N ≠ 2 and (1.2). If u is a solution of (3.1) then there exist positive constants C_{1} and C_{2} independent of κ such that the following inequalities hold:

(3.5)  |κ|∥u∥_{−(1+δ)/2} ≤ C_{1}∥f∥_{(1+δ)/2} (Imκ ⋛ 0) ,

(3.6)  ∥D^{±}u∥_{(−1+δ)/2} ≤ C_{2}∥f∥_{(1+δ)/2} (Imκ ⋛ 0) .

From these, we find

Corollary 3.5. Assume that N ≠ 2 and (1.2).

(1) There exists a unique solution u of (3.1) in L^{2,−(1+δ)/2}(ℝ^{N}) for Imκ ⋛ 0.

(2) There exists a positive constant C_{3} independent of κ such that the following inequality holds:

(3.7)  |κ|^{2}∥u∥_{−(1+δ)/2,E_{R}}^{2} ≤ C_{3}(1 + R)^{−δ}∥f∥_{(1+δ)/2}^{2}  (Imκ ⋛ 0) ,

where E_{R} = {x ∈ ℝ^{N}||x| > R} for R > 0.

From Theorem3.4, Corollary 3.5 and the elliptic estimate for (3.1), we can show Main Theorem.

4. The Principle of Limiting Absorption for i ( \begin{matrix}0 & 1 \\ Δ & −B \end{matrix} ).

First of all, note that (1.3) with initial data w(x, 0) = w_{1}, w_{t}(x, 0) = w_{2} is equivalent to

(4.1)  i ( \begin{matrix}w \\ w_{t} \end{matrix} )_{t} − i ( \begin{matrix}0 & 1 \\ Δ & −B \end{matrix} ) ( \begin{matrix}w \\ w_{t} \end{matrix} ) = ( \begin{matrix}0 \\ 0 \end{matrix} ) , ( \begin{matrix}w(0) \\ w_{t}(0) \end{matrix} ) = ( \begin{matrix}w_{1} \\ w_{2} \end{matrix} ) .

We define the Hilbert space H ≡ Ḣ^{1}(ℝ^{N}) × L^{2}(ℝ^{N}) ( Ḣ^{1}(ℝ^{N}) being the Beppo-Levi space ) with inner product

(f̃, g̃)_{H} = ∫_{ℝ^{N}} (∇f_{1} · \overline{∇g_{1}} + f_{2} · \overline{g_{2}}) dx

for f̃ = ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) , g̃ = ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) ∈ H.

If b(x) ≥ 0 then the solution of (4.1) is given by

( \begin{matrix}w \\ w_{t} \end{matrix} ) = e^{−iW t} ( \begin{matrix}w_{1} \\ w_{2} \end{matrix} )

where W = i ( \begin{matrix}0 & 1 \\ Δ & −B \end{matrix} ) is the generator of the contraction semi-group \{e^{−iW t}\}_{t≥0} on H (cf. Mochizuki

[12] pp.384).

The limiting absorption principle for W can be stated as follows:

Theorem 4.1. Assume that N ≠ 2, (1.2) and f̃ = ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) ∈ H_{1} ≡ Ḣ^{1}(ℝ^{N}) ∩ L^{2,(1+δ)/2}(ℝ^{N}) ×

L^{2,(1+δ)/2}(ℝ^{N}) (⊂ H). If ũ = ũ_{σ±iτ} = ( \begin{matrix}u_{1,σ±iτ} \\ u_{2,σ±iτ} \end{matrix} ) is a solution of the equation:

(4.2)  W ũ − (σ ± iτ)ũ = f̃

(σ ∈ ℝ \ {0}, τ > 0), then there exists the limit

lim_{τ↓0} ũ_{σ±iτ} = ũ_{±}

in H_{2} ≡ H^{1,−(1+δ)/2}(ℝ^{N}) × L^{2,−(1+δ)/2}(ℝ^{N}) (⊃ H) and ũ_{±} solves the equation

W ũ_{±} − σũ_{±} = f̃,

where

H^{1,−(1+δ)/2}(ℝ^{N}) = { f(x) | ∫_{ℝ^{N}} (1 + r)^{−1−δ} (|∇f(x)|^{2} + |f(x)|^{2}) dx < ∞ } .

References

1. S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola. Norm. Sup. pisa(4). 2 (1975), 151–218.

2. D. M. Éĭdus, The principle of limiting absorption, Math. Sb. 57 (99) (1962), 13–44 = Amer. Math. Soc. Transl. (2) 47 (1965), 157–191.

3. T. Ikebe, A spectral theory for reduced wave equation with a complex refractive index, Publ. RIMS Kyoto Univ. 8 (1972/73), 579–606.

4. T. Ikebe and Y. Saito, Limiting absorption method and absolute continuity for the Schrödinger operators, J. Math. Kyoto. Univ. 12 (1972), 513–542.

5. N. Iwasaki, On the principle of limiting amplitude, Publ. RIMS Kyoto Univ. 3 (1968), 373–392.

6. T. Kato, Wave operators and similality for some non-selfadjoint operators, Math. Ann. 162 (1966), 258–279.

7. A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, vol.71, A.M.S. Soc. Providence, Rhode Island, 1988.

8. S. Mizohata and K. Mochizuki, On the principle of limiting amplitude for dissipative wave equations, J. Math. Kyoto. Univ 6 (1966), 109–127.

9. K. Mochizuki, On the large perturbation by a class of non-selfadjoint operators, J. Math. Soc. Japan 19 (1967), 123–158.

10. , Eigenfunction expansions associated with the Schrödinger operator with a complex potential and the scattering theory, Publ. RIMS Kyoto Univ 4 (1968), 419–466.

11. , Spectral and Scattering Theory for Second Order Elliptic Differential Operators in an Exterior Domain, Lecture Notes Univ Utah. Winter and Spring, 1972.

12. , Scattering theory for wave equations with dissipative terms, Publ. RIMS Kyoto Univ 12 (1976), 383–390.

13. , Scatteing Theory for Wave Equations, Kinokuniya, Tokyo, 1984. (japanese)

14. Y. Saito, The principle of limiting absorption for the non-selfadjoint Schrödinger operator in ℝ^{N} (N ≠ 2), Publ. RIMS Kyoto Univ 9 (1974), 397–428.

15. , The principle of limiting absorption for the non-selfadjoint Schrödinger operator in ℝ^{2}, Osaka. J. Math 11 (1974), 295–306.

16. M. Schechter, Operator Methods in Quantum Mechanics, North Holland, New York, 1981.