- Email: [email protected]
Spectral representation of the weighted Laplace transform
Journal Pre-proof Spectral representation of the weighted Laplace transform
Gisele Ruiz Goldstein, Jerome A. Goldstein, Giorgio Metafune, Luigi Negro
PII: DOI: Reference:
S0893-9659(19)30460-4 https://doi.org/10.1016/j.aml.2019.106136 AML 106136
To appear in:
Applied Mathematics Letters
Received date : 12 September 2019 Accepted date : 8 November 2019 Please cite this article as: G.R. Goldstein, J.A. Goldstein, G. Metafune et al., Spectral representation of the weighted Laplace transform, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106136. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Journal Pre-proof
Spectral representation of the weighted Laplace transform Gisele Ruiz Goldstein, Jerome A. Goldstein, Giorgio Metafune, and Luigi Negro
na lP repr oo f
. Abstract. We find the spectral representation of the selfadjoint operators T Z ∞ T f (λ) := K(λt)f (t) dt, 0
in L2 (]0, ∞[). More precisely (see Theorem 4.1) for these operators which include the Laplace transform as a special case, the spectrum of T is a compact interval [−κ, κ], and we find explicitly a unitary operator U : L2 (]0, ∞[) → L2 (R) and a continuous real function α on R such that U T U −1 is the operator of multiplication by α.
1. Introduction
ur
The spectral theory of unbounded selfadjoint operators on Hilbert space started in the 19th century, even if the concepts were not formally defined until the 20th century. Starting with Fourier, Sturm-Liouville theory and the associated eigenfunction expansions developed with the aid of special functions. But the creation of quantum mechanics in 1925-26 underlined the need for a rigorous theory, so, independently around 1926-27, John von Neumann and Marshall Stone defined Hilbert space (which for them was a complex separable infinite dimensional Hilbert space) and defined unbounded selfadjoint operators on it. They proved the Spectral Theorem, which said that every selfadjoint operator S on a Hilbert space H is unitarily equivalent to an operator of multiplication by a real function on a concrete L2 space. More precisely, there is a measure space (Ω, Σ, µ), a corresponding Hilbert space L2 (Ω, Σ, µ), a Σ-measurable function m on Ω, and a unitary operator U from H onto L2 (Ω, Σ, µ) such that
Jo
S = U −1 Mm U where Mm f = mf for f ∈ D (Mm ) = g : g, mg ∈ L2 (Ω, Σ, µ) . Sometimes we write Mm(x) to emphasize the role of x ∈ Ω. In case the operator S has a compact resolvent, then σ(S) ⊂ R consists of the (necessarily real) eigenvalues of S together with 0, (that is, the spectrum of S) so that L2 (Ω, Σ, µ) is an l2 sequence space, and knowing all the eigenvalues 2010 Mathematics Subject Classification. 44A10, 45P05, 47B38, 47G10 . Key words and phrases. Laplace Transform, Integral Operator, Spectral analysis. 1
Journal Pre-proof
2
G. R. GOLDSTEIN, J. A. GOLDSTEIN, G. METAFUNE, AND L. NEGRO
(including multiplicities) and the corresponding eigenfunctions provides an explicit way to define L2 (Ω, Σ, µ), M and U . This easily extends to the discrete spectrum case, when H has an orthonormal basis of eigenvectors of S. For F any real Borel on σ(S), F (S) = U −1 MF (m) U and F 7→ F (S) is an algebra homomorphism from the real Borel functions on σ(S) to the selfadjoint operators on H. The case of selfadjoint operators without a basis of eigenvectors is much more complicated. But one special case is well understood. The Fourier transform F is a unitary operator on L2 (RN ). Moreprecisely, using obvious notation, the Fourier transform of a function f ∈ Cc∞ RN is defined by Z N e−ix·ξ f (x) dx (Ff ) (ξ) = (2π)− 2
na lP repr oo f
RN
2 N and extends uniquely by closure to a unitary operator F from L2 (RN x ) to L (Rξ ). P P Moreover, if P (x) = |α|≤k aα xα is a polynomial and P (D) = |α|≤k aα Dα where α1 αN 1 ∂ 1 ∂ αN α1 α α ··· x = (x1 , . . . , xN ), D = i ∂x1 i ∂xN
for
x = (x1 , . . . , xN ),
then
α = (α1 , . . . , αN ) ∈ N0N ,
P (D) = F −1 MP (ξ) F.
As a special case, D2 = −∆ corresponds to P (ξ) = |ξ|2 = ξ · ξ. Thus the solution of the heat equation ∂u = ∆u, u(x, 0) = f (x) ∂t has its Fourier transform u ˆ = Fu for fixed t given by 2 u ˆ(ξ, t) = e−t|ξ| fˆ(ξ),
and the solution of the wave equation
ur
∂2u = ∆u, u(x, 0) = f (x), ∂t2 has its Fourier transform given by
Jo
(and note that
∂u (x, 0) = g(x) ∂t
sin(t|ξ|) u ˆ(ξ, t) = cos(t|ξ|)fˆ(ξ) + gˆ(ξ), |ξ| sin(t|ξ|) =t ξ→0 |ξ| lim
so that the Fourier transform of the solution is not singular at ξ = 0). Let ϕ(x, ξ) = eix·ξ ,
for
x, ξ ∈ RN .
Journal Pre-proof
SPECTRAL REPRESENTATION OF THE WEIGHTED LAPLACE TRANSFORM
3
Then ∆x ϕ(x, ξ) = −|ξ|2 ϕ(x, ξ) holds, so that x 7→ ϕ(x, ξ) is an eigenfunction of the distributional Laplacian ∆ for fixed ξ with eigenvalue −|ξ|2 . But ϕ( · , ξ), an L∞ function, is not in L2 (RN x ), so view it as a generalized function. Then we have the Fourier transform and its and fˆ = Ff ∈ L2 RN inverse given by, for f ∈ C 2 RN x ξ , Z N fˆ(ξ) = (2π)− 2 ϕ(x, ξ)f (x) dx, N ZR N f (x) = (2π)− 2 ϕ(x, ξ)fˆ(ξ) dξ. RN
F : f 7→ fˆ is unitary from L
2
(RN x )
2
to L (RN ξ ) and
so that
na lP repr oo f
F −1 (−∆)F = M|ξ|2
Z N F −1 (−∆)fˆ (ξ) = (2π)− 2
RN
|ξ|2 ϕ(x, ξ)fˆ(ξ) dξ.
N
The ”normalized” eigenfunction is (2π)− 2 ϕ(x, ξ). This is the explicit form of the Spectral Theorem for −∆ on L2 RN , written as a generalized eigenfunction expansion, and this is what led Paul Dirac to write Z (2π)−N ϕ(x, ξ)ϕ(y, ξ) dξ = δ(x − y). RN
This is the Dirac delta function which L. Schwartz explained rigorously in the 1930s. It seems that the founders of quantum mechanics (and mathematical analysts today) wanted to answer the following question. If S is selfadjoint, can one find (Ω, Σ, ν) , m, U
explicitly and in a usable form? This remains an open question, and we are not close to a solution. But what about specific operators that arise in the applications? The answer is sometimes yes, and clearly Dirac thought a lot about this in the late 1920s.
Jo
ur
More or less simultaneously with Stone and von Neumann, Dirac created his own version of Hilbert space and the Spectral Theorem. He started with brackets. For Dirac, hf | is a ”bra”, |gi is a ”ket”, and hf |gi is a bracket. In mathematical notation, f and g are vectors in H and Dirac’s hf |gi corresponds to the inner product hg, f i since each is linear in g and conjuqate linear in f . Thus |gi is a vector and hf | is a covector or linear functional. The only selfadjoint operators that concerned Dirac at first were Schr¨ odinger operators, of the form S = −∆ + V (x) = −∆ + MV ,
on L2 (RN ),
where V (x) is the potential energy function on RN . V often vanishes at ∞ and can a have mild local singularities like the Coulomb potential |x| on L2 (R3 ). Here various physical constants such as Planck’s constant, the speed of light, and the mass have been normalized for notational simplicity. Dirac’s idea was to write S in the form
Journal Pre-proof
4
G. R. GOLDSTEIN, J. A. GOLDSTEIN, G. METAFUNE, AND L. NEGRO
Sf =
X j∈J
λj hf, ψj iψj +
Z
RN
ϕ± (x, ξ)f (x) dx
involving a discrete eigenvalue expansion part and an integral involving the spectrally continuous part of S. To see Dirac’s brilliant ideas see his 1930 masterpiece book. Now consider the free Schroedinger equation ( i.e. V = 0) i
∂u = Su, ∂t
u(x, 0) = f (x)
which has N u(x, t) = e−itS f (x) = (2π)− 2
Z
RN
2
e−it|ξ| ϕ(x, ξ)fˆ(ξ) dξ
where
na lP repr oo f
as its unique solution. Again let S = −∆ + V and let Sac be the absolutely continuous part of S. T. Ikebe and T. Kato gave a modern interpretation of Dirac’s theory. Let ψ ± (x, ξ) solve −∆ψ ± (x, ξ) + V (x)ψ ± (x, ξ) = |ξ|2 ψ ± (x, ξ), ψ ± (x, ξ) ≈
e±ix·ξ N
(2π) 2
as
|x| → ∞.
Thus assuming V = V1 + V2 , where V1 ∈ Lp (RN ) with p ≥ 2 and p > N2 , V2 ∈ L∞ (RN ) and |V (x)| ≤ |x|c1+ for some > 0 and all |x| ≥ R, then Sac = W (−∆)W −1 and the unitary operator W from the orthogonal complement of the space of eigenvectors of S onto L2 (RN ) is based on the analogue G of the Fourier transform F. Then Z (Gf ) (ξ) = ψ − (x, ξ)f (x) dx, RN Z f (x) = ψ + (x, ξ) (Gf ) (ξ) dξ, RN
G −1 SG = −∆.
Jo
ur
Thus S = −∆+V and the restriction Sac of S to its absolutely continuous subspace is unitarily equivalent to the operator −∆. Later S. Agmon made a substantial extension of this technique work in a much broader context. There are Schroedinger operators having nontrivial singular continuous subspaces. There are no known ways of finding the Spectral Theorem in explicit form for these operators. In this paper we find the desired form of the Spectral Theorem for a class of operators including the Laplace transform on L2 spaces on the half line. 2. Formulation of the problem
The spectral theorem says that every bounded selfadjoint operator in a Hilbert space H is unitarily equivalent to a multiplication operator by a real bounded function on a function space L2 (Ω, µ). This means that given T = T ∗ bounded in H, one can find a measure space (Ω, µ), a unitary operator U : H → L2 (Ω, µ) and a real measurable bounded function m defined in Ω such that U T U −1 g = mg for
Journal Pre-proof
SPECTRAL REPRESENTATION OF THE WEIGHTED LAPLACE TRANSFORM
5
every g ∈ L2 (Ω, µ). Finding Ω, µ, m can be a difficult task in concrete situation. The starting point of this paper is the following result due to Gilliam, Schlunberger and Lund [2] concerning the Laplace transform Z ∞ Lf (λ) = e−λt f (t) dt, f ∈ L2 (0, ∞). 0
Theorem 2.1. The Laplace transform √ √ L is unitarily equivalent to the multipli2 cation operator x → 7 xg(x) in L ([− π, π]; |x| dx). It follows that L has spectrum √ √ σ(L) = [− π, π], and that L is spectrally absolutely continuous of multiplicity one. We shall give a different proof of this result which generalizes to operators of the form Z ∞ T f (λ) = K(λt)f (t) dt. 0
na lP repr oo f
Notation. The Fourier transform is conveniently redefined in the following way. For f ∈ L2 (R), Z ˆ f (ξ) = f (t)e−2πiξt dt. R
With this choice the Fourier transform is unitary on L2 (R) and, moreover, now the Fourier transform of a convolution is the product of the Fourier transforms. 3. An auxiliary operator
This section is devoted to the spectral analysis of the operator S defined on L2 (R) by (3.1)
Sf (ξ) = m(ξ)f (−ξ),
f ∈ L2 (R), ξ ∈ R,
where m ∈ L∞ (R) satisfies m(−ξ) = m(ξ) for every ξ ∈ R. Clearly S is a bounded operator and it is selfadjoint since, for every f, g ∈ L2 (R), Z Z (Sf, g) = m(ξ)f (−ξ)g(ξ) dξ = m(−η)f (η)g(−η) dη R ZR = f (η)m(η)g(−η) dη = (f, Sg) . R
Finally, S 2 f = |m|2 f since
S 2 f (ξ) = m(ξ) Sf (−ξ) = m(ξ)m(−ξ)f (ξ) = m(ξ)m(ξ)f (ξ) = |m(ξ)|2 f (ξ).
ur
We write now, m = |m|eih , where the argument function h : R → R satisfies h(−ξ) = −h(ξ) for every ξ ∈ R. We start by studying the case |m| = 1. Then S 2 = I and σ(S) ⊂ {−1, 1} (actually, σ(S) = {−1, 1}, since S 6= ±I). Next we note that
Jo
Sf = f
Sg = −g Let us define
is equivalent to
is equivalent to
f (−ξ) = e−ih(ξ) f (ξ), g(−ξ) = −e
−ih(ξ)
g(ξ),
for all ξ ∈ R,
for all ξ ∈ R.
Hhe = {f ∈ L2 (R) : f (−ξ) = e−ih(ξ) f (ξ)} = Ker(I − S)
Hho = {g ∈ L2 (R) : g(−ξ) = −e−ih(ξ) g(ξ)} = Ker(I + S).
Journal Pre-proof
6
G. R. GOLDSTEIN, J. A. GOLDSTEIN, G. METAFUNE, AND L. NEGRO
These subspaces are orthogonal since they are the eigenspaces corresponding to different eigenvalues. For f ∈ L2 (R) let
f (ξ) + eih(ξ) f (−ξ) f (ξ) − eih(ξ) f (−ξ) , fho (ξ) = . 2 2 Since, as it is immediate to check, fhe ∈ Hhe , fho ∈ Hho and f = fhe + fho , it follows that L2 (R) = Hhe ⊕ Hho e o and that fh , fh are the orthogonal projections of f onto Hhe , Hho , respectively. fhe (ξ) =
Lemma 3.1. The map U : L2 (R) → L2 (R) defined by e ξ ≥ 0, √ fh (ξ) U f (ξ) = 2 o fh (ξ) ξ < 0,
na lP repr oo f
is a unitary operator whose inverse is given by U −1 g = g1 + g2 where g(ξ) ξ ≥ 0, 1 g1 (ξ) = √ 2 eih(ξ) g(−ξ) ξ < 0, g(ξ) ξ < 0, 1 g2 (ξ) = √ 2 −eih(ξ) g(−ξ) ξ ≥ 0.
√ √ Proof. Note that U f (ξ) = 2 fhe (ξ)χ[0,∞[ (ξ) + 2 fho (ξ)χ]−∞,0[ (ξ). Then Z Z ∞ Z 0 Z 2 2 2 o 2 |U f (ξ)| dξ = 2 |fh (ξ)| dξ + 2 |fh (ξ)| dξ = (|fh2 (ξ)|2 + |fho (ξ)|2 ) dξ R 0 −∞ R Z 2 = |f (ξ)| dξ, R
fhe , fho
since are orthogonal and |fhe (−ξ)| = |fhe (ξ)|, |fho (−ξ)| = |fho (ξ)|. This shows that U is an isometry. Given g ∈ L2 (R), let us define g1 , g2 as in the statement. It is elementary to check that g1 ∈ Hhe and g2 ∈ Hho . If f = g1 + g2 , then fhe = g1 and fho = g2 and, consequently, U f = g. Proposition 3.2. If U is the unitary map of Lemma 3.1, then for every g ∈ L2 (R), U SU −1 g(ξ) = sign(ξ)|m(ξ)| g(ξ), ξ ∈ R.
ur
Proof. We write, for ξ ∈ R, m(ξ) = |m(ξ)|eih(ξ) and recall that |m(−ξ)| = |m(ξ)|, h(−ξ) = −h(ξ). Accordingly we factor S = M S0 where S0 f (ξ) = eih(ξ) f (−ξ),
M f (ξ) = |m(ξ)|f (ξ).
−1
Jo
Then, using the notation of Lemma 3.1, U g = g1 + g2 and S0 g1 = g1 , S0 g2 = −g2 since g1 ∈ Hhe , g2 ∈ Hho . Moreover, since |m(−ξ)| = |m(ξ)|, then |m|g1 ∈ Hhe and |m|g2 ∈ Hho . If follows that U SU −1 g
=
=
U M S0 (g1 + g2 ) = U M (g1 − g2 ) = U |m|g1 − U |m|g2 √ √ 2 |m|g1 χ[0,∞[ − 2 |m|g2 χ]−∞,0[ = sign(·)|m|g.
Journal Pre-proof
SPECTRAL REPRESENTATION OF THE WEIGHTED LAPLACE TRANSFORM
7
Note that the spectral representation of the operator Sf (ξ) = f (−ξ), which corresponds to m = 1, is given by U SU −1 g = sign(·)g. 4. Spectral representation of dilation kernels Here we consider selfadjoint operators of the form Z ∞ (4.1) T f (λ) = K(λt)f (t) dt 0
where 0 ≤ K ∈ L1loc (0, ∞) and f ∈ L2 (I), I = (0, ∞). The special case K(t) = e−t corresponds to the Laplace transform. Is is known, see [4], that T is bounded in L2 (I) if and only if (a condition we shall always assume) Z ∞ K(t) √ dt < ∞ (4.2) κ= t 0
na lP repr oo f
and in this case the norm of T√ as an operator from L2 (I) to itself is κ. In the case of the Laplace tranform κ = π. We refer to [4] for weighted Lp properties of the operator T . Let us consider the map V : L2 (I) → L2 (R),
x
V f (x) = e 2 f (ex ).
Then V is unitary and V −1 g(y) = √1y g(log y). Here f ∈ L2 (I) and g ∈ L2 (R). By a straightforward computation we obtain for g ∈ L2 (R) and λ ∈ R Z ∞ Z λ+s λ K(eλ t) −1 2 √ g(log t) dt = V T V g(λ) = e K(eλ+s )e 2 g(s) ds. t 0 R t
Setting H(t) = K(et )e 2 , then Z Z (4.3) V T V −1 g(λ) = H(λ + s)g(s) ds = H(λ − s)g(−s) ds = H ∗ T0 g(λ), R
R
where we defined T0 g(s) = g(−s). Note that H ≥ 0 and Z Z ∞ K(s) √ ds = κ < ∞. H(t) dt = s R 0 Here is the main result.
Theorem 4.1. Let m be the Fourier transform of the L1 function H(t) = t K(e )e 2 . Then T is unitarily equivalent to the multiplication operator defined on L2 (R) by
ur
t
g 7→ α g = sign(·)|m| g
Jo
Proof. First we note that m(−ξ) = m(ξ), since H is real-valued. The Fourier transform of V T V −1 g is m(ξ)g(−ξ) and hence T is unitarily equivalent to the operator Sg(ξ) = m(ξ)g(−ξ) on L2 (R), which was studied in the preceding section. The result then follows from Proposition 3.2. Remark 4.2. The proof shows that any convolution-type operator Z S1 g(λ) = H(λ + s)g(s) ds, λ ∈ R R
Journal Pre-proof
8
G. R. GOLDSTEIN, J. A. GOLDSTEIN, G. METAFUNE, AND L. NEGRO
is unitarily equivalent to the multiplication by sign(·)|m|. Moreover, the convolution operator Z S2 g(λ) =
R
λ∈R
H(λ − s)g(s) ds,
is unitarily equivalent to multiplication by m.
The spectrum of T is easily computed from the spectral representation. Proposition 4.3. The spectrum of T is the closed interval [−κ, κ], where κ = kT k is defined in 4.2.
na lP repr oo f
Proof. By Theorem 4.1, T is unitarily equivalent to multiplication by sign(·)|m|, t where m is the Fourier transform of H(t) = K(et )e 2 . By the Riemann-Lebesgue lemma, m(ξ) → 0 as |ξ| → ∞. Moreover, since H is non-negative Z Z ∞ K(t) √ dt = κ. |m(ξ)| ≤ m(0) = H(s) ds = t R 0 Since m is continuous, the (essential) range of sign(·)|m| is the closed interval [−κ, κ] and the proof follows. The multiplicity function of T can be determined, once m is known, using the results of [1] or [7] where the multiplicity function for a multiplication operator is computed. In our situation the multiplier is sign(·)|m|. Since |m(0)| = κ and |m(ξ)| → 0 as ξ → +∞, we see that sign(·)|m| is injective whenever |m| is strictly decreasing on [0, ∞[. In this case T has multiplicity 1.
Let us compute m in the case of the Laplace transform, that is, when K(t) = e−t . In this case t t H(t) = e−e e 2 and
m(ξ) =
Z
R
e
−et
t 2
e e
−2πiξt
dt =
Z
∞
e
−s − 21 −2πiξ log s
s
e
ds = Γ
0
Then, using the complement formula, 1 1 2 |m(ξ)| = Γ − 2πiξ Γ + 2πiξ = 2 2 sin π
1 − 2πiξ . 2
π 2π = 2π2 ξ , e + e−2π2 ξ − 2πiξ
1 2
ur
and |m(ξ)| is decreasing from [0, ∞[ onto ]0, π]. The preceding discussion therefore provides a different proof of the main result in [2], namely 4.4. The Laplace transform has absolutely continuous spectrum √Proposition √ [− π, π] and multiplicity 1.
Jo
The results above can be easily generalized to weighted spaces. Let Z ∞ (4.4) Tα f (λ) := K(λt)f (t)tα dt 2
0
α
where f ∈ L (I; t dt) and α is any real number. Using the isometry α
J : L2 (I) → L2 (I; tα dt), Jf 7→ f t− 2
Journal Pre-proof
SPECTRAL REPRESENTATION OF THE WEIGHTED LAPLACE TRANSFORM
we find that Tα is unitarily equivalent to Z ∞ Z α α J −1 Tα Jf (λ) = λ 2 K(λt)f (t)t− 2 tα dt = 0
∞
9
Kα (λt)f (t) dt
0
α
in the unweighted L2 (I), with Kα (s) = s 2 K(s). Theorem 4.1 and Proposition 4.3 can be applied with Kα instead of K. References 1. M. B. Abrahamse, T. L. Kriete, The spectral multiplicity of a multiplication operator, Indiana U. Math. Jour. 22 (1973), 845-857. 2. D. S. Gilliam, J. R. Schulenberger, J. R. Lund, Spectral representation of the Laplace and Stieltjes transforms, Mat. Apl. Comput. 7 (1988), no. 2, 101-107.
na lP repr oo f
3. G. R. Goldstein, Semigroups of Linear Operators and Applications, Oxford U. Press, New York and Oxford, 1985. 4. G. R. Goldstein, J. A. Goldstein, G. Metafune, L. Negro, The weighted Laplace transform, Functional Analysis and Geometry, ed. by P. Kuchment and E. Semenov, CONM Series of the American Mathematical Society, 733 (2019), 175-185. 5. T. Ikebe, Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory, Arch. Rat. Mech. Anal., 5 (1960), 1-34. 6. K. Yosida, Lectures on Differential and Integral Equations, Interscience, New York, 1960. 7. T. L. Kriete An Elementary approach to the multiplicity theory of multiplication operators, Rocky Mountain J. Math., 16 (1986), 23-32.
Department of Mathematical Sciences, 373 Dunn Hall, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected] Department of Mathematical Sciences, 373 Dunn Hall, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected] ` del Salento, C.P.193, Dipartimento di Matematica “Ennio De Giorgi”, Universita 73100, Lecce, Italy E-mail address: [email protected]
Jo
ur
` del Salento, C.P.193, Dipartimento di Matematica “Ennio De Giorgi”, Universita 73100, Lecce, Italy E-mail address: [email protected]
Gisele Ruiz Goldstein, Jerome A. Goldstein, Giorgio Metafune, Luigi Negro
PII: DOI: Reference:
S0893-9659(19)30460-4 https://doi.org/10.1016/j.aml.2019.106136 AML 106136
To appear in:
Applied Mathematics Letters
Received date : 12 September 2019 Accepted date : 8 November 2019 Please cite this article as: G.R. Goldstein, J.A. Goldstein, G. Metafune et al., Spectral representation of the weighted Laplace transform, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106136. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Journal Pre-proof
Spectral representation of the weighted Laplace transform Gisele Ruiz Goldstein, Jerome A. Goldstein, Giorgio Metafune, and Luigi Negro
na lP repr oo f
. Abstract. We find the spectral representation of the selfadjoint operators T Z ∞ T f (λ) := K(λt)f (t) dt, 0
in L2 (]0, ∞[). More precisely (see Theorem 4.1) for these operators which include the Laplace transform as a special case, the spectrum of T is a compact interval [−κ, κ], and we find explicitly a unitary operator U : L2 (]0, ∞[) → L2 (R) and a continuous real function α on R such that U T U −1 is the operator of multiplication by α.
1. Introduction
ur
The spectral theory of unbounded selfadjoint operators on Hilbert space started in the 19th century, even if the concepts were not formally defined until the 20th century. Starting with Fourier, Sturm-Liouville theory and the associated eigenfunction expansions developed with the aid of special functions. But the creation of quantum mechanics in 1925-26 underlined the need for a rigorous theory, so, independently around 1926-27, John von Neumann and Marshall Stone defined Hilbert space (which for them was a complex separable infinite dimensional Hilbert space) and defined unbounded selfadjoint operators on it. They proved the Spectral Theorem, which said that every selfadjoint operator S on a Hilbert space H is unitarily equivalent to an operator of multiplication by a real function on a concrete L2 space. More precisely, there is a measure space (Ω, Σ, µ), a corresponding Hilbert space L2 (Ω, Σ, µ), a Σ-measurable function m on Ω, and a unitary operator U from H onto L2 (Ω, Σ, µ) such that
Jo
S = U −1 Mm U where Mm f = mf for f ∈ D (Mm ) = g : g, mg ∈ L2 (Ω, Σ, µ) . Sometimes we write Mm(x) to emphasize the role of x ∈ Ω. In case the operator S has a compact resolvent, then σ(S) ⊂ R consists of the (necessarily real) eigenvalues of S together with 0, (that is, the spectrum of S) so that L2 (Ω, Σ, µ) is an l2 sequence space, and knowing all the eigenvalues 2010 Mathematics Subject Classification. 44A10, 45P05, 47B38, 47G10 . Key words and phrases. Laplace Transform, Integral Operator, Spectral analysis. 1
Journal Pre-proof
2
G. R. GOLDSTEIN, J. A. GOLDSTEIN, G. METAFUNE, AND L. NEGRO
(including multiplicities) and the corresponding eigenfunctions provides an explicit way to define L2 (Ω, Σ, µ), M and U . This easily extends to the discrete spectrum case, when H has an orthonormal basis of eigenvectors of S. For F any real Borel on σ(S), F (S) = U −1 MF (m) U and F 7→ F (S) is an algebra homomorphism from the real Borel functions on σ(S) to the selfadjoint operators on H. The case of selfadjoint operators without a basis of eigenvectors is much more complicated. But one special case is well understood. The Fourier transform F is a unitary operator on L2 (RN ). Moreprecisely, using obvious notation, the Fourier transform of a function f ∈ Cc∞ RN is defined by Z N e−ix·ξ f (x) dx (Ff ) (ξ) = (2π)− 2
na lP repr oo f
RN
2 N and extends uniquely by closure to a unitary operator F from L2 (RN x ) to L (Rξ ). P P Moreover, if P (x) = |α|≤k aα xα is a polynomial and P (D) = |α|≤k aα Dα where α1 αN 1 ∂ 1 ∂ αN α1 α α ··· x = (x1 , . . . , xN ), D = i ∂x1 i ∂xN
for
x = (x1 , . . . , xN ),
then
α = (α1 , . . . , αN ) ∈ N0N ,
P (D) = F −1 MP (ξ) F.
As a special case, D2 = −∆ corresponds to P (ξ) = |ξ|2 = ξ · ξ. Thus the solution of the heat equation ∂u = ∆u, u(x, 0) = f (x) ∂t has its Fourier transform u ˆ = Fu for fixed t given by 2 u ˆ(ξ, t) = e−t|ξ| fˆ(ξ),
and the solution of the wave equation
ur
∂2u = ∆u, u(x, 0) = f (x), ∂t2 has its Fourier transform given by
Jo
(and note that
∂u (x, 0) = g(x) ∂t
sin(t|ξ|) u ˆ(ξ, t) = cos(t|ξ|)fˆ(ξ) + gˆ(ξ), |ξ| sin(t|ξ|) =t ξ→0 |ξ| lim
so that the Fourier transform of the solution is not singular at ξ = 0). Let ϕ(x, ξ) = eix·ξ ,
for
x, ξ ∈ RN .
Journal Pre-proof
SPECTRAL REPRESENTATION OF THE WEIGHTED LAPLACE TRANSFORM
3
Then ∆x ϕ(x, ξ) = −|ξ|2 ϕ(x, ξ) holds, so that x 7→ ϕ(x, ξ) is an eigenfunction of the distributional Laplacian ∆ for fixed ξ with eigenvalue −|ξ|2 . But ϕ( · , ξ), an L∞ function, is not in L2 (RN x ), so view it as a generalized function. Then we have the Fourier transform and its and fˆ = Ff ∈ L2 RN inverse given by, for f ∈ C 2 RN x ξ , Z N fˆ(ξ) = (2π)− 2 ϕ(x, ξ)f (x) dx, N ZR N f (x) = (2π)− 2 ϕ(x, ξ)fˆ(ξ) dξ. RN
F : f 7→ fˆ is unitary from L
2
(RN x )
2
to L (RN ξ ) and
so that
na lP repr oo f
F −1 (−∆)F = M|ξ|2
Z N F −1 (−∆)fˆ (ξ) = (2π)− 2
RN
|ξ|2 ϕ(x, ξ)fˆ(ξ) dξ.
N
The ”normalized” eigenfunction is (2π)− 2 ϕ(x, ξ). This is the explicit form of the Spectral Theorem for −∆ on L2 RN , written as a generalized eigenfunction expansion, and this is what led Paul Dirac to write Z (2π)−N ϕ(x, ξ)ϕ(y, ξ) dξ = δ(x − y). RN
This is the Dirac delta function which L. Schwartz explained rigorously in the 1930s. It seems that the founders of quantum mechanics (and mathematical analysts today) wanted to answer the following question. If S is selfadjoint, can one find (Ω, Σ, ν) , m, U
explicitly and in a usable form? This remains an open question, and we are not close to a solution. But what about specific operators that arise in the applications? The answer is sometimes yes, and clearly Dirac thought a lot about this in the late 1920s.
Jo
ur
More or less simultaneously with Stone and von Neumann, Dirac created his own version of Hilbert space and the Spectral Theorem. He started with brackets. For Dirac, hf | is a ”bra”, |gi is a ”ket”, and hf |gi is a bracket. In mathematical notation, f and g are vectors in H and Dirac’s hf |gi corresponds to the inner product hg, f i since each is linear in g and conjuqate linear in f . Thus |gi is a vector and hf | is a covector or linear functional. The only selfadjoint operators that concerned Dirac at first were Schr¨ odinger operators, of the form S = −∆ + V (x) = −∆ + MV ,
on L2 (RN ),
where V (x) is the potential energy function on RN . V often vanishes at ∞ and can a have mild local singularities like the Coulomb potential |x| on L2 (R3 ). Here various physical constants such as Planck’s constant, the speed of light, and the mass have been normalized for notational simplicity. Dirac’s idea was to write S in the form
Journal Pre-proof
4
G. R. GOLDSTEIN, J. A. GOLDSTEIN, G. METAFUNE, AND L. NEGRO
Sf =
X j∈J
λj hf, ψj iψj +
Z
RN
ϕ± (x, ξ)f (x) dx
involving a discrete eigenvalue expansion part and an integral involving the spectrally continuous part of S. To see Dirac’s brilliant ideas see his 1930 masterpiece book. Now consider the free Schroedinger equation ( i.e. V = 0) i
∂u = Su, ∂t
u(x, 0) = f (x)
which has N u(x, t) = e−itS f (x) = (2π)− 2
Z
RN
2
e−it|ξ| ϕ(x, ξ)fˆ(ξ) dξ
where
na lP repr oo f
as its unique solution. Again let S = −∆ + V and let Sac be the absolutely continuous part of S. T. Ikebe and T. Kato gave a modern interpretation of Dirac’s theory. Let ψ ± (x, ξ) solve −∆ψ ± (x, ξ) + V (x)ψ ± (x, ξ) = |ξ|2 ψ ± (x, ξ), ψ ± (x, ξ) ≈
e±ix·ξ N
(2π) 2
as
|x| → ∞.
Thus assuming V = V1 + V2 , where V1 ∈ Lp (RN ) with p ≥ 2 and p > N2 , V2 ∈ L∞ (RN ) and |V (x)| ≤ |x|c1+ for some > 0 and all |x| ≥ R, then Sac = W (−∆)W −1 and the unitary operator W from the orthogonal complement of the space of eigenvectors of S onto L2 (RN ) is based on the analogue G of the Fourier transform F. Then Z (Gf ) (ξ) = ψ − (x, ξ)f (x) dx, RN Z f (x) = ψ + (x, ξ) (Gf ) (ξ) dξ, RN
G −1 SG = −∆.
Jo
ur
Thus S = −∆+V and the restriction Sac of S to its absolutely continuous subspace is unitarily equivalent to the operator −∆. Later S. Agmon made a substantial extension of this technique work in a much broader context. There are Schroedinger operators having nontrivial singular continuous subspaces. There are no known ways of finding the Spectral Theorem in explicit form for these operators. In this paper we find the desired form of the Spectral Theorem for a class of operators including the Laplace transform on L2 spaces on the half line. 2. Formulation of the problem
The spectral theorem says that every bounded selfadjoint operator in a Hilbert space H is unitarily equivalent to a multiplication operator by a real bounded function on a function space L2 (Ω, µ). This means that given T = T ∗ bounded in H, one can find a measure space (Ω, µ), a unitary operator U : H → L2 (Ω, µ) and a real measurable bounded function m defined in Ω such that U T U −1 g = mg for
Journal Pre-proof
SPECTRAL REPRESENTATION OF THE WEIGHTED LAPLACE TRANSFORM
5
every g ∈ L2 (Ω, µ). Finding Ω, µ, m can be a difficult task in concrete situation. The starting point of this paper is the following result due to Gilliam, Schlunberger and Lund [2] concerning the Laplace transform Z ∞ Lf (λ) = e−λt f (t) dt, f ∈ L2 (0, ∞). 0
Theorem 2.1. The Laplace transform √ √ L is unitarily equivalent to the multipli2 cation operator x → 7 xg(x) in L ([− π, π]; |x| dx). It follows that L has spectrum √ √ σ(L) = [− π, π], and that L is spectrally absolutely continuous of multiplicity one. We shall give a different proof of this result which generalizes to operators of the form Z ∞ T f (λ) = K(λt)f (t) dt. 0
na lP repr oo f
Notation. The Fourier transform is conveniently redefined in the following way. For f ∈ L2 (R), Z ˆ f (ξ) = f (t)e−2πiξt dt. R
With this choice the Fourier transform is unitary on L2 (R) and, moreover, now the Fourier transform of a convolution is the product of the Fourier transforms. 3. An auxiliary operator
This section is devoted to the spectral analysis of the operator S defined on L2 (R) by (3.1)
Sf (ξ) = m(ξ)f (−ξ),
f ∈ L2 (R), ξ ∈ R,
where m ∈ L∞ (R) satisfies m(−ξ) = m(ξ) for every ξ ∈ R. Clearly S is a bounded operator and it is selfadjoint since, for every f, g ∈ L2 (R), Z Z (Sf, g) = m(ξ)f (−ξ)g(ξ) dξ = m(−η)f (η)g(−η) dη R ZR = f (η)m(η)g(−η) dη = (f, Sg) . R
Finally, S 2 f = |m|2 f since
S 2 f (ξ) = m(ξ) Sf (−ξ) = m(ξ)m(−ξ)f (ξ) = m(ξ)m(ξ)f (ξ) = |m(ξ)|2 f (ξ).
ur
We write now, m = |m|eih , where the argument function h : R → R satisfies h(−ξ) = −h(ξ) for every ξ ∈ R. We start by studying the case |m| = 1. Then S 2 = I and σ(S) ⊂ {−1, 1} (actually, σ(S) = {−1, 1}, since S 6= ±I). Next we note that
Jo
Sf = f
Sg = −g Let us define
is equivalent to
is equivalent to
f (−ξ) = e−ih(ξ) f (ξ), g(−ξ) = −e
−ih(ξ)
g(ξ),
for all ξ ∈ R,
for all ξ ∈ R.
Hhe = {f ∈ L2 (R) : f (−ξ) = e−ih(ξ) f (ξ)} = Ker(I − S)
Hho = {g ∈ L2 (R) : g(−ξ) = −e−ih(ξ) g(ξ)} = Ker(I + S).
Journal Pre-proof
6
G. R. GOLDSTEIN, J. A. GOLDSTEIN, G. METAFUNE, AND L. NEGRO
These subspaces are orthogonal since they are the eigenspaces corresponding to different eigenvalues. For f ∈ L2 (R) let
f (ξ) + eih(ξ) f (−ξ) f (ξ) − eih(ξ) f (−ξ) , fho (ξ) = . 2 2 Since, as it is immediate to check, fhe ∈ Hhe , fho ∈ Hho and f = fhe + fho , it follows that L2 (R) = Hhe ⊕ Hho e o and that fh , fh are the orthogonal projections of f onto Hhe , Hho , respectively. fhe (ξ) =
Lemma 3.1. The map U : L2 (R) → L2 (R) defined by e ξ ≥ 0, √ fh (ξ) U f (ξ) = 2 o fh (ξ) ξ < 0,
na lP repr oo f
is a unitary operator whose inverse is given by U −1 g = g1 + g2 where g(ξ) ξ ≥ 0, 1 g1 (ξ) = √ 2 eih(ξ) g(−ξ) ξ < 0, g(ξ) ξ < 0, 1 g2 (ξ) = √ 2 −eih(ξ) g(−ξ) ξ ≥ 0.
√ √ Proof. Note that U f (ξ) = 2 fhe (ξ)χ[0,∞[ (ξ) + 2 fho (ξ)χ]−∞,0[ (ξ). Then Z Z ∞ Z 0 Z 2 2 2 o 2 |U f (ξ)| dξ = 2 |fh (ξ)| dξ + 2 |fh (ξ)| dξ = (|fh2 (ξ)|2 + |fho (ξ)|2 ) dξ R 0 −∞ R Z 2 = |f (ξ)| dξ, R
fhe , fho
since are orthogonal and |fhe (−ξ)| = |fhe (ξ)|, |fho (−ξ)| = |fho (ξ)|. This shows that U is an isometry. Given g ∈ L2 (R), let us define g1 , g2 as in the statement. It is elementary to check that g1 ∈ Hhe and g2 ∈ Hho . If f = g1 + g2 , then fhe = g1 and fho = g2 and, consequently, U f = g. Proposition 3.2. If U is the unitary map of Lemma 3.1, then for every g ∈ L2 (R), U SU −1 g(ξ) = sign(ξ)|m(ξ)| g(ξ), ξ ∈ R.
ur
Proof. We write, for ξ ∈ R, m(ξ) = |m(ξ)|eih(ξ) and recall that |m(−ξ)| = |m(ξ)|, h(−ξ) = −h(ξ). Accordingly we factor S = M S0 where S0 f (ξ) = eih(ξ) f (−ξ),
M f (ξ) = |m(ξ)|f (ξ).
−1
Jo
Then, using the notation of Lemma 3.1, U g = g1 + g2 and S0 g1 = g1 , S0 g2 = −g2 since g1 ∈ Hhe , g2 ∈ Hho . Moreover, since |m(−ξ)| = |m(ξ)|, then |m|g1 ∈ Hhe and |m|g2 ∈ Hho . If follows that U SU −1 g
=
=
U M S0 (g1 + g2 ) = U M (g1 − g2 ) = U |m|g1 − U |m|g2 √ √ 2 |m|g1 χ[0,∞[ − 2 |m|g2 χ]−∞,0[ = sign(·)|m|g.
Journal Pre-proof
SPECTRAL REPRESENTATION OF THE WEIGHTED LAPLACE TRANSFORM
7
Note that the spectral representation of the operator Sf (ξ) = f (−ξ), which corresponds to m = 1, is given by U SU −1 g = sign(·)g. 4. Spectral representation of dilation kernels Here we consider selfadjoint operators of the form Z ∞ (4.1) T f (λ) = K(λt)f (t) dt 0
where 0 ≤ K ∈ L1loc (0, ∞) and f ∈ L2 (I), I = (0, ∞). The special case K(t) = e−t corresponds to the Laplace transform. Is is known, see [4], that T is bounded in L2 (I) if and only if (a condition we shall always assume) Z ∞ K(t) √ dt < ∞ (4.2) κ= t 0
na lP repr oo f
and in this case the norm of T√ as an operator from L2 (I) to itself is κ. In the case of the Laplace tranform κ = π. We refer to [4] for weighted Lp properties of the operator T . Let us consider the map V : L2 (I) → L2 (R),
x
V f (x) = e 2 f (ex ).
Then V is unitary and V −1 g(y) = √1y g(log y). Here f ∈ L2 (I) and g ∈ L2 (R). By a straightforward computation we obtain for g ∈ L2 (R) and λ ∈ R Z ∞ Z λ+s λ K(eλ t) −1 2 √ g(log t) dt = V T V g(λ) = e K(eλ+s )e 2 g(s) ds. t 0 R t
Setting H(t) = K(et )e 2 , then Z Z (4.3) V T V −1 g(λ) = H(λ + s)g(s) ds = H(λ − s)g(−s) ds = H ∗ T0 g(λ), R
R
where we defined T0 g(s) = g(−s). Note that H ≥ 0 and Z Z ∞ K(s) √ ds = κ < ∞. H(t) dt = s R 0 Here is the main result.
Theorem 4.1. Let m be the Fourier transform of the L1 function H(t) = t K(e )e 2 . Then T is unitarily equivalent to the multiplication operator defined on L2 (R) by
ur
t
g 7→ α g = sign(·)|m| g
Jo
Proof. First we note that m(−ξ) = m(ξ), since H is real-valued. The Fourier transform of V T V −1 g is m(ξ)g(−ξ) and hence T is unitarily equivalent to the operator Sg(ξ) = m(ξ)g(−ξ) on L2 (R), which was studied in the preceding section. The result then follows from Proposition 3.2. Remark 4.2. The proof shows that any convolution-type operator Z S1 g(λ) = H(λ + s)g(s) ds, λ ∈ R R
Journal Pre-proof
8
G. R. GOLDSTEIN, J. A. GOLDSTEIN, G. METAFUNE, AND L. NEGRO
is unitarily equivalent to the multiplication by sign(·)|m|. Moreover, the convolution operator Z S2 g(λ) =
R
λ∈R
H(λ − s)g(s) ds,
is unitarily equivalent to multiplication by m.
The spectrum of T is easily computed from the spectral representation. Proposition 4.3. The spectrum of T is the closed interval [−κ, κ], where κ = kT k is defined in 4.2.
na lP repr oo f
Proof. By Theorem 4.1, T is unitarily equivalent to multiplication by sign(·)|m|, t where m is the Fourier transform of H(t) = K(et )e 2 . By the Riemann-Lebesgue lemma, m(ξ) → 0 as |ξ| → ∞. Moreover, since H is non-negative Z Z ∞ K(t) √ dt = κ. |m(ξ)| ≤ m(0) = H(s) ds = t R 0 Since m is continuous, the (essential) range of sign(·)|m| is the closed interval [−κ, κ] and the proof follows. The multiplicity function of T can be determined, once m is known, using the results of [1] or [7] where the multiplicity function for a multiplication operator is computed. In our situation the multiplier is sign(·)|m|. Since |m(0)| = κ and |m(ξ)| → 0 as ξ → +∞, we see that sign(·)|m| is injective whenever |m| is strictly decreasing on [0, ∞[. In this case T has multiplicity 1.
Let us compute m in the case of the Laplace transform, that is, when K(t) = e−t . In this case t t H(t) = e−e e 2 and
m(ξ) =
Z
R
e
−et
t 2
e e
−2πiξt
dt =
Z
∞
e
−s − 21 −2πiξ log s
s
e
ds = Γ
0
Then, using the complement formula, 1 1 2 |m(ξ)| = Γ − 2πiξ Γ + 2πiξ = 2 2 sin π
1 − 2πiξ . 2
π 2π = 2π2 ξ , e + e−2π2 ξ − 2πiξ
1 2
ur
and |m(ξ)| is decreasing from [0, ∞[ onto ]0, π]. The preceding discussion therefore provides a different proof of the main result in [2], namely 4.4. The Laplace transform has absolutely continuous spectrum √Proposition √ [− π, π] and multiplicity 1.
Jo
The results above can be easily generalized to weighted spaces. Let Z ∞ (4.4) Tα f (λ) := K(λt)f (t)tα dt 2
0
α
where f ∈ L (I; t dt) and α is any real number. Using the isometry α
J : L2 (I) → L2 (I; tα dt), Jf 7→ f t− 2
Journal Pre-proof
SPECTRAL REPRESENTATION OF THE WEIGHTED LAPLACE TRANSFORM
we find that Tα is unitarily equivalent to Z ∞ Z α α J −1 Tα Jf (λ) = λ 2 K(λt)f (t)t− 2 tα dt = 0
∞
9
Kα (λt)f (t) dt
0
α
in the unweighted L2 (I), with Kα (s) = s 2 K(s). Theorem 4.1 and Proposition 4.3 can be applied with Kα instead of K. References 1. M. B. Abrahamse, T. L. Kriete, The spectral multiplicity of a multiplication operator, Indiana U. Math. Jour. 22 (1973), 845-857. 2. D. S. Gilliam, J. R. Schulenberger, J. R. Lund, Spectral representation of the Laplace and Stieltjes transforms, Mat. Apl. Comput. 7 (1988), no. 2, 101-107.
na lP repr oo f
3. G. R. Goldstein, Semigroups of Linear Operators and Applications, Oxford U. Press, New York and Oxford, 1985. 4. G. R. Goldstein, J. A. Goldstein, G. Metafune, L. Negro, The weighted Laplace transform, Functional Analysis and Geometry, ed. by P. Kuchment and E. Semenov, CONM Series of the American Mathematical Society, 733 (2019), 175-185. 5. T. Ikebe, Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory, Arch. Rat. Mech. Anal., 5 (1960), 1-34. 6. K. Yosida, Lectures on Differential and Integral Equations, Interscience, New York, 1960. 7. T. L. Kriete An Elementary approach to the multiplicity theory of multiplication operators, Rocky Mountain J. Math., 16 (1986), 23-32.
Department of Mathematical Sciences, 373 Dunn Hall, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected] Department of Mathematical Sciences, 373 Dunn Hall, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected] ` del Salento, C.P.193, Dipartimento di Matematica “Ennio De Giorgi”, Universita 73100, Lecce, Italy E-mail address: [email protected]
Jo
ur
` del Salento, C.P.193, Dipartimento di Matematica “Ennio De Giorgi”, Universita 73100, Lecce, Italy E-mail address: [email protected]