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数理解析研究所講究録
第2074巻 2018年 133-147

TOSIO KATO’S WORK ON NON‐RELATIVISTICQUANTUM MECHANICS: AN OUTLINEBARRY SIMONIBM Professor of Mathematics and Theoretical Physics, Emeritus
California Institute of Technology, Pasadena, CA 91125\mathrm{E}‐mail: bsimon@caltech.edu

1. INTRODUCTION

In 2017, we are celebrating the 100^{th} anniversary of the birth of Tosio

Kato (August 25, 1917‐October 2, 1999), the founding father of thetheory of Schrödinger operators. There was a centennial held in Tokyo

in his memory and honor in September. I decided to write a review

article on his work in nonrelativistic quantum mechanics (NRQM),

which, as well see was only part of his opus. I originally guessed it
would be about 80 pages but it turned out to be more than 210! It willappear in Bull. Math. Sci. [42] with a shorter version also available[43]. I gave two lectures on Kato’s work at the conference in his honor
and this is an outline of what I said in those lectures.

The rest of this introduction will say a little about Kato’s life while
the next will summarize some major themes in his work. I will thendescribe in some depth (but less detail than in my Bull. Math. Sci.
article) five topics that were among the most important of Kato’s con‐tributions to NRQM.

Kato’s most significant paper was on self‐adjointness of atomic
Hamiltonians published in 1951 in Trans. A.M.S. (see Section 3). I

note that he was 34 when it was published (it was submitted a few
years earlier, as we’ll discuss in Section 3). Before it, his most impor‐tant work was his thesis, awarded in 1951 and published in 1949‐51.
One might be surprised at his age when this work was published but
not if one understands the impact of the war. Kato got his BS fromthe University of Tokyo in 1941 but during the war, he was evacuatedto the countryside. We were at a conference together one evening and
Kato described rather harrowing experiences in the camp he was as‐
signed to, especially an evacuation of the camp down a steep wet hill.
He contracted TB in the camp. In his acceptance for the Wiener Prize,
Date: October 18, 2017.

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B. SIMON

Kato says that his work on essential selfadjointness and on perturbation
theory were essentially complete by the end of the war.
In 1946, Kato returned to the University of Tokyo as an Assistant(a position common for students progressing towards their degrees) inphysics, was appointed Assistant Professor of Physics in 1951 and full
professor in 1958. Beginning in 1954, Kato started visiting the UnitedStates. This bland statement masks some drama. In 1954, Kato wasinvited to visit Berkeley for a year, I presume arranged by $\Gamma$ . Wolf. Of
course, Kato needed a visa and it is likely it would have been denied

due to his history of TB. Fortunately, just at the time (and only fora period of about a year), the scientific attaché at the US embassyin Tokyo was Otto Laporte (1902‐1971) on leave from a professorshipin Physics at the University of Michigan. Charles Dolph (1919‐1994),

a mathematician at Michigan, learned of the problem and contacted
Laporte who intervened to get Kato a visa. Dolph once told me thathe thought his most important contribution to American mathematics
was his helping to allow Kato to come to the US.
During the mid 1950\mathrm{s} , Kato spent close to three years visiting US
institutions, mainly Berkeley, but also the Courant Institute, American
University and Caltech. In 1962, he accepted a professorship in Mathe‐
matics from Berkeley where he spent the rest of his career and remained
after his retirement. One should not underestimate the courage it takes
for a 45 year old to move to a very different culture because of a scien‐tific opportunity. The reader can consult the Mathematics Genealogy

Project for a list of Kato’s students (24 listed there, 3 from Tokyo and21 from Berkeley; the best known are Ikebe and Kuroda from Tokyo

and Balslev and Howland from Berkeley) and [6] for a memorial article

with lots of reminisces of Kato.

Kato received the Wiener prize in 1980 and passed away in 1999.
One gets some sense of the impact of Kato’s work by looking at the
number of things named after him which include:(1) Kato’s Theorem (on self‐adjointness of atomic Hamiltonian)(2) Kato‐Rellich Theorem (stability of self adjointness under rela‐
tively bounded perturbations)(3) Kato‐Rosenblum Theorem and Kato‐Birman theory (traceclass scattering)(4) Kato dynamics (adiabatic theorem)(5) Kato cusp condition(6) Trotter‐Kato Theorem (3 different results!!)(7) Putnam‐Kato theorem (positive commutator \Rightarrow pure ac spec‐
trum)

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(8) Kato smoothness theory(9) Kato class
There are several things called Kato’s inequality(1) His distributional inequality on \triangle|u| (see Section 5)(2) His inequality \displaystyle \int|x|^{-1}| $\varphi$(x)|^{2}d^{3}x\leq\frac{ $\pi$}{2}\int|k| \hat{ $\varphi$}(k)|^{2}d^{3}k(3) A result on hyponormal operators that comes from Katosmoothness theory(4) A variant of the Heinz‐Loewner inequality on monotonicity ofthe square root that holds for maximal accretive operators.
2. OVERVIEW

One can roughly divide Kato’s extensive research opus into four dis‐tinct areas:(1) NRQM(2) Nonlinear equations of mathematical physics including Euler,
Navier‐Stokes, \mathrm{K}\mathrm{d}\mathrm{V} and non‐linear Schrödinger(3) Theory of linear semigroups on Banach spaces(4) Misc. Functional Analysis

There is a kind of phase transition in his work around 1980. Be‐fore then, he mainly worked on NRQM with occasional papers in theother three. Afterwards, mainly in Nonlinear Equations. Its almost as
if when the theory of Schrödinger operators became overcrowded, heswitched to another area in which he could be a pioneer.
Kato’s non‐linear work is substantial. He has over 13,500 Math‐

SciNet citations (so far as I know the most citations of any author isjust under 22,000). The top three are some edition or printing of his
famous book Perturbation theory for linear operator\mathcal{S} [22] . The next
9 are nonlinear (if you include his paper, #11, with Brézis on apply‐
ing Kato’s inequality to Schrödinger operators with complex potentials
which I think is there because they have some lemmas widely used incurrent work on non‐linear equations).
Interestingly enough, the most quoted in NRQM (at least under

MathSciNet’s data which only counts references in math papers andsince about 2000) might surprise some of you. It’s his joint paper with
Arne Jensen [12] on time‐decay. By the way, I suspect many of you
will be even more surprised when I tell you later what is Kato’s mostquoted paper on Google Scholar!
Of course, one can’t compartmentalize all his work. I include his
famous paper on the Trotter product formula for general self‐adjoint
contraction semigroups in his NRQM work although it fits even betterin the linear semi‐group area and he has a non‐linear version with

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Masuda. Anyhow, talking about his NRQM work will keep us busy
enough so I won’t say anything about the other work.
In my article, I divide Kato’s work on NRQM into five subareas(1) Eigenvalue Perturbation Theory(2) Self‐Adjointness(3) Eigenvalue Results (only 2)(4) Spectral and Scattering Theory(5) Three Other Gems
Kato has only two papers on (3) and five for (5). For the others,

there are many, many papers.

Below, we will focus on five sets ofideas, two in are (2) (Sections 3 and 5), two are in (4) (Sections 6 and
7) and one in (5) (Section 4).
Interestingly enough, we will not discuss the NRQM paper with themost MathSciNet references (see above) and we’ll barely mention his
paper [15] with the most Google Scholar references. But I believe we’ll
discuss his most significant works.

3. FOUNDATIONS OF ATOMIC PHYSICSEver since the work of von Neumann about 1930, it has been clearthat self‐adjointness of quantum Hamiltonians is crucial. In this re‐gard, Kato has been \mathrm{a} , indeed, the key figure. His contributions include(1) Self‐adjointness of atomic Hamiltonians [14](2) His work with Ikebe [10] on V(x) > −cx2—d, an area whereWeinholtz was the pioneer(3) Kato’s inequality, including his theorem that if V \in L_{loc}^{2}(\mathbb{R}^{ $\nu$})
and V\geq 0 , then -\triangle+V is essentially self‐adjoint on C_{0}^{\infty}(\mathbb{R}^{ $\nu$})(see Section 5)(4) He was a pioneer on the use of quadratic forms including his
work on perturbations (KLMN theorem) and monotone conver‐gence for forms.

In 1951 Kato published a theorem [14] that he proved by 1944. Nbody quantum Hamiltonians on L^{2}(\mathbb{R}^{3N}) with two body potentials inL^{2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3}) are essentially self‐adjoint on C_{0}^{\infty}(\mathbb{R}^{3N}) and self‐

adjoint on D(- $\Delta$) . Since that class of potentials includes the Coulomb
potential, this includes atomic and molecular Hamiltonians.

This is one of the most significant theorem in mathematical physics.
It makes Kato the father of the theory of Schrödinger operators. As
Kato remarks in his Wiener prize acceptance, the proof is rather simple.
It relies on three steps. First the Kato‐Rellich theorem that if A is self‐
adjoint and B is A‐bounded with relatively bound a<1 , then A+B
is self‐adjoint for A and any core for A is a core for B.

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Secondly, by either a Sobolev estimate (or for Coulomb V , a Hardy
inequality), any L^{2}(\mathbb{R}^{3}) function, as an operator on L^{2}(\mathbb{R}^{3}) , is -\triangle-

bounded with relative bound zero. Finally, by integrating out the extra
variables, relative boundedness on \mathbb{R}^{3} implies relative boundedness ofthe pair potentials as operators on \mathbb{R}^{3N}
It remains surprising that this theorem wasn’t found earlier by Rel‐lich or Friedrichs. One factor is that von Neumann thought the prob‐
lem was impossibly hard which may have discouraged people. Rellichcertainly knew the Kato‐Rellich theorem by about 1940 and he used
Hardy’s inequality in one of his papers at that time. All I can think ofis that he missed the “trivial”’ idea of integrating out the extra variables(which isn’t trivial until you realize it!)For our discussion of Kato’s inequality below, we’ll need one no‐tion from later work extending Kato’s Theorem. Namely for generaldimension v , the replacement of Kato’s L^{2} for \mathbb{R}^{3} is Ư wherep=\left{bginary}{l 2,&\mathr{i}\mathr{f}v\leq3 >2,&\mathr{i}\mathr{f}l\tex{ノ}=4\ v/2&\mathr{i}\mathr{f}$\nu geq5.\nd{ary}\ight.

For V ’s of general sign, this is optimal since limit point/limit circle

methods show if $\nu$\geq 5 , then for $\lambda$ large, - $\Delta$- $\lambda$|x|^{-2} is not self‐adjoint

on D(-\triangle) . Note that |x|^{-2}\in L^{p}+L^{\infty} for all p<v/2.

The paper [14] mentions that he can prove that eigenfunctions ofatomic Hamiltonians are bounded. He proved the details and much

more in his paper [15]. Included was the Kato cusp condition which

has been widely used in the atomic physics and quantum chemistrycommunities (which is why this has the most Google Scholar citations).
4. THE ADIABATIC THEOREM

In 1950, Kato published in J. Phys. Soc. Japan a paper [13] on the

adiabatic theorem that has been central to the vast literature on thesubject since then. This subject concerns a family of time dependent

Hamiltonians, H(s) , 0\leq s\leq 1 (which we’ll assume is a set of bounded
operators). One wants to run the dynamics very slowly. What thatmeans is that one fixes T large and looks at H(s/T) , 0 \leq s \leq T and
one wants to solve\displaystyle \frac{d}{ds}\tilde{U}_{T}(\mathcal{S})=-iH(s/T)\tilde{U}_{T}(s)Letting U_{T}(s)solves=,0\leq s\leq T ;\tilde{U}_{T}(0)=1\tilde{U}_{T}(sT) , 0 \leq s \leq 1 , we see that U_{T}(\mathcal{S}) , 0 \leq s \leq 1

138
B. SIMON\displaystyle \frac{d}{ds}U_{T}(\mathcal{S})=-iTH(s)U_{T}(s),U_{T}(0)=1

0\leq s\leq 1 ;Kato supposed that H(s) has an eigenvalue, $\lambda$(s) , which is continuous
in s and an isolated point of the spectrum of H(s) . While Kato was not
explicit about technical hypotheses, it is easy to see that he needed toassume that H(s) is C^{2} Let P(\mathcal{S}) be the projection onto the eigenspace\mathrm{k}\mathrm{e}\mathrm{r}(H(s)- $\lambda$(\mathcal{S}) . Then Kato proved the Adiabatic Theorem\displaystyle \lim_{T\rightarrow\infty}(1-P(s))U_{T}(s)P(0)=0
In other words, if $\psi$\in \mathrm{r}\mathrm{a}\mathrm{n}P(0) , then as T\rightarrow\infty, U_{T}(s) $\psi$ gets closer

to ranP (\mathcal{S}) .

In 1928, Born and Fock [4] proved the above theorem when H(s) has
purely discrete spectrum and $\lambda$(s) is a simple eigenvalue. The point of
Kato’s version is not merely the greater generality but that the proof
gives more information.
Recall that the theorem says that \displaystyle \lim_{T\rightarrow\infty}(1-P(s))U_{T}(s)P(0)=0.
Kato discovered a dynamics in which this formula is exact. He called
it the adiabatic dynamics and I’ll call it the Kato dynamics. Namely
if W(s) solves\displaystyle \frac{d}{ds}W(s)=iA(s)W(s), 0\leq s\leq 1 ;W(0)=1iA(\mathcal{S})\equiv[P'(\mathcal{S}), P(\mathcal{S})]Then W(s) obeys

W(s)P(0)W(\mathcal{S})^{-1}=P(s)\Rightarrow(1-P(s))W(s)P(0)=(1-P(s))P(\mathcal{S})W(s)=0
He proved the theorem by also proving that

\Vert e^{-iT\int_{0}^{s} $\lambda$(s)ds}U_{T}(\mathcal{S})^{*}W(\mathcal{S})P(0)-P(0)\Vert=\mathrm{O}(1/T)
The point is that Kato not only proved that U_{T}(\mathcal{S}) $\psi$ stayed inside
ranP (s) , he told you the exact vector it was. And in this regard Katoleft something on the table that took 30 years to be found and appre‐
ciated.

The dynamics, U_{T} , has two parts: the phase e^{-iT\int_{0}^{s} $\lambda$(s)ds} and theKato dynamics, W(s) . In fancy language, the set of k dimensional

subspaces of our Hilbert space (or a nice subset) is a manifold. For
each point in the manifold, we have the corresponding subspace. This

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defines a natural complex vector bundle (and, in case k=1 , which isinteresting, a complex line bundle). The map W(\mathcal{S}) defines a notion ofparallel transport, i.e. a connection.
The connection has a holonomy—in the line bundle case, there is a
finite phase associated to going in a closed path. And this holonomy isa line integral which can be written, using Stokes’ theorem, as a surface
integral the curl of the connection, called the curvature. The phase wasfound by Michael Berry [3] in 1983 and is called Berry’s phase and I
interpreted it [41] as holonomy in the underlying connection. As notedby Avron‐Seiler‐Simon [2] one can use Kato’s work to compute theconnection in the underlying bundle. This bundle and connection is

also behind the work of Thouless et al (TKNN) [45] used in their workon the quantum Hall effect for which Thouless recently got a Nobel
prize. I want to emphasize that Kato’s work is very much underlying
the mathematics behind Berry phase.
5. KATO ’ \mathrm{S} INEQUALITY

This section will discuss a self‐adjointness method that appearedin Kato [20] based on a remarkable distributional inequality. Its con‐sequences is a subject to which Kato returned to often with at least

seven additional papers [21, 23, 24, 5, 25, 26, 27]. It is also his workthat most intersected my own—I motivated his initial paper and it, inturn, motivated several of my later papers.
I want to set the stage by reminding you about Kato’s 1951 paper
and its aftermath. For this, two‐body suffices, so I consider -\triangle+V(x)on L^{2}(\mathbb{R}^{ $\nu$}) . We had the notion of p being v ‐canonical which I remindyou, for v\geq 5 means v/2 . By the late 1950' \mathrm{s} , it was known for \mathrm{e}\mathrm{s}\mathrm{a}-v(i.e. essential self‐adjointness on C_{0}^{\infty}(\mathbb{R}^{ $\nu$}) ) one needed some kind of

uniformly local U condition with p being v‐canonical. The example
V(x)=- $\lambda$|x|^{-2} for which \mathrm{e}\mathrm{s}\mathrm{a}-v can fail shows that p=v/2 is optimal.
When I entered the subject about 1970, there was an implicit belief
that this condition applied to both the positive and negative parts. In
hindsight, this should not have been the case! It was known there wasa global asymmetry: one can’t have behavior worse than \mathrm{O}(-|x|^{2}) at
spatial infinity without losing esa but if V\geq 0 and say, locally bounded,
one has esa no matter what the growth at infinity.
But it was assumed that for local singularities, there was no differ‐ence. In fact, the simplest example shows this to be wrong! Consider
the case V(x)= |x|^{- $\beta$} , where to have C_{0}^{\infty}\subset D(V) , one needs $\beta$<v/2.
Limit point/limit circle methods prove \mathrm{e}\mathrm{s}\mathrm{a}- $\nu$ for all such $\beta$ (but for-|x|^{- $\beta$}, \mathrm{e}\mathrm{s}\mathrm{a}-v fails if $\beta$>2 ). So there was no good reason for the belief

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B. SIMONbut, in fact, there was great surprise when I discovered the result I now
turn to.

One of the advantages of working in multiple areas is that there canbe technical cross‐pollination. In the fall of 1971, I was working on
two‐dimensional constructive quantum field theory. One of the toolsthat had been developed to get self‐adjointness of cut‐off Hamiltoni‐
ans concerned what are called hypercontractive semigroups. In 1970,\mathrm{H}\emptyset\mathrm{e}\mathrm{g}\mathrm{h}‐Krohn and I [44] had abstracted results of Segal [36], invented
the name hypercontractive and found an extension for positive poten‐

tials. Using, in addition, ideas of Nelson [33] and Konrady [31], I wasable to prove that [37] (d $\mu$(x)=e^{-|x|^{2}/2}dx)

V\in L^{2}(\mathbb{R}^{ $\nu$}, d $\mu$) & V\geq 0\Rightarrow-\triangle+V is \mathrm{e}\mathrm{s}\mathrm{a}-v
So the positive part can have L^{2} singularities. While L^{2}(\mathbb{R}^{ $\nu$}, d $\mu$)allows pretty wild growth at infinity, it was natural to conjecture, as I
did, that if V\geq 0 one only needs V\in L_{loc}^{2}(\mathbb{R}^{ $\nu$}) .
I wrote this up and sent out a preprint in early 1972 and within amonth got a letter back from Kato, then almost 55, with a proof [20] of
my conjecture! He used totally new ideas depending on a distributional

inequality that if u, \triangle u\in L_{loc}^{1}(\mathbb{R}^{ $\nu$}) (allowed to be complex valued) and\mathrm{s}\mathrm{g}\mathrm{n}(u)(x) is defined to 0 if u(x)=0 and otherwise |u(x)|/u(x) , then\triangle|u| \geq{\rm Re}[\mathrm{s}\mathrm{g}\mathrm{n}(u)\triangle u]
The proof isn’t hard; one proves it for u smooth by taking derivativesu^{*}u+$\epsilon$^{2} and taking $\epsilon$ \downar ow 0 . Then one usesof the definition |u|_{ $\epsilon$}^{2}mollifiers to get it for general distributions. If now V \in L_{loc}^{2} andV \geq 0 , one lets H-\triangle+V \mathrm{} C_{0^{\infty} \geq 0 and considers u \in D(H^{*})
H^{*}u=-u
with. This equation is a distributional equality so by Kato’s
inequality==\triangle|u| \geq(\mathrm{s}\mathrm{g}\mathrm{n}(u))(V+1)u=|u|(V+1)\geq|u|Thus (-\triangle+1)|u| \leq 0 so since (-\triangle+1)^{-1} has a positive integral
kernel, one sees that as a distribution |u| \leq 0 . So, we have proventhat \mathrm{k}\mathrm{e}\mathrm{r}(H^{*}+1)\{0\} , so by a theorem of von Neumann, H is esa.
This really was remarkable. The master of operator techniques had
suddenly pulled a distributional rabbit out of his hat.
Kato’s inequality is quite striking, so it is interesting to see what=

it is really saying. In this regard, I proved [38] the following result
for arbitrary positive self‐adjoint operators, A , on L^{2}(M, d $\mu$) .
following are equivalent

The

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(1) e^{-tA} is positivity preserving:\forall u\in L^{2}, u\geq 0, t\geq 0\Rightarrow e^{-tA}u\geq 0(2) (Beurling‐Deny criterion) u\in Q(A)\Rightarrow|u| \in Q(A) and

q_{A}(|u|) \leq q_{A}(u)(3) (Abstract Kato Inequality) u\in D(A)\Rightarrow |u| \in Q(A) and for all$\varphi$\in Q(A) with $\varphi$\geq 0 , one has that\langle A^{1/2} $\varphi$, A^{1/2}|u|\rangle\geq{\rm Re}\langle $\varphi$, \mathrm{s}\mathrm{g}\mathrm{n}(u)Au\rangle
This shows that the Kato inequality methods are not unrelated to
my original hypercontractive result and led to a direct semigroup proofof the L_{loc}^{2} result [39].

Kato also proved a version of his inequality when there is a C^{1} mag‐netic

\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}1\rightar ow_{a}$\Delta$|u| \geq{\rm Re}[\mathrm{s}\mathrm{g}\mathrm{n}(u)(\#-i\vec{a})^{2}u]This eventually led me (with a suggestion of Nelson) to what is
known as diamagnetic inequalities [38, 39]. If H(a, V)=-(\nabla-ia)^{2}+V,

then|e^{-tH(a,V)} $\varphi$| \leq e^{-tH(a=0,V)}| $\varphi$|pointwise. There is a direct line from Kato’s paper to this result.
As with the ordinary Kato inequality, there is an abstract semigroupversion of this result found by Simon [40] and by Hess‐Schrader‐
Unhlenbrock [9].
6. KATO‐ROSENBLUM AND KATO‐BiRMAN

There was an explosion of papers on time‐dependent scattering the‐ory in 1957‐58 with work by Cook, Jauch, Kato and Rosenblum. So
far as I can tell, these were independent but several were clearly mo‐tivated by Friedrichs [8]. In particular, Kato extended the notion ofwave operator. If A and B are arbitrary self‐adjoint operators and
P_{ac}(X) the projection onto the absolutely continuous subspace for X,

Kato [16] considered$\Omega$^{\pm}(A, B)=\displaystyle \mathrm{s}-\lim_{t\rightar ow\mp\infty}e^{itA}e^{-itB}P_{ac}(B)(the crazy \pm vs. \mp convention is from the physics literature and notwhat Kato used). If these limits exist, one says the wave operators
exist and if ran $\Omega$^{\pm}(A, B)=\mathrm{r}\mathrm{a}\mathrm{n}P_{ac}(A) , Kato called the wave operatorscomplete.

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B. SIMON

Kato proved a basic fact: $\Omega$^{\pm}(A, B) are complete \Leftrightarrow $\Omega$^{\pm}(B, A)
exist. The proof is almost trivial but this result is nevertheless very
important since it reduces completeness to an existence property. For

example, motivated by this theorem, Deift and I (in 1977) showed [7]that completeness of general N‐body quantum scattering was equiv‐
alent to the existence of certain geometrically defined “inverse” wave
operators. So far as I know, every proof of N‐body completeness uses
the Deift‐Simon wave operators.
One consequence of this theorem is that existence under a symmetriccondition on A and B implies both existence and completeness. In
the same paper where the above fact appears, Kato proved that wave

operators exist (and are complete) if A-B is finite rank.
In short order after the finite rank results (in terms of paper publi‐
cation dates), Rosenblum [35] proved that if A and B both have purely
a.c. spectrum and A-B is trace class, then the wave operators provide

a unitary equivalence. After that Kato [17] proved what is often called

the Kato‐Rosenblum theorem: if A-B is trace class, then the Katowave operators exist and are complete.
In terms of the sequence of papers, \mathrm{I}' \mathrm{d} always assumed that Rosen‐
blum’s paper was a rapid reaction to Kato’s finite rank paper which, inturn, motivated Kato’s trace class paper. But I recently learned thatis wrong. Rosenblum was a graduate student of Wolf at Berkeley who
submitted his thesis in March 1955. It contained his trace class resultwith some additional technical hypotheses; a Dec. 1955 Berkeley tech‐nical report had his full result. Rosenblum submitted a paper to theAmerican Journal of Mathematics which took a long time refereeing it
before rejecting it. In Apri11956, Rosenblum submitted a revised pa‐

per to the Pacific Journal in which it eventually appeared (this version
dropped the technical condition; I’ve no idea what the original jour‐

nal submission had). So Rosenblum’s work was independent of and(presumably) earlier than Kato’s.
Kato’s finite rank paper was submitted to J. Math. Soc. Japan on

March 15, 1957 and was published in the issue dated April, 1957 (!) .

The full trace class result was submitted to Proc. Japan Acad. on May
15, 1957. Kato’s first paper quotes an abstract of a talk Rosenblum
gave to an A.M.S. meeting but I don’t think that abstract contained
many details. This finite rank paper has a note added in proof thank‐
ing Rosenblum for sending the technical report to Kato, quoting its
main result and saying that Kato had found the full trace class results(“Details will be published elsewhere ). That second paper used sometechnical ideas from Rosenblum’s paper.

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KATO’S WORK

I’ve heard that Rosenblum always felt that he’d not received suffi‐
cient credit for his trace class paper. There is some justice to this. The
realization that trace class is the natural class is significant. In thisregard, it is important that Weyl [47] and von Neumann [46] provedthat for any self‐adjoint, A , there is a Hilbert‐Schmidt operator, C , sothat A+C has only point spectrum (so there is no chance that wave
operators exist if A has any a.c. spectrum). And Kato’s student, S.
T. Kuroda, extended this [32] to allow C in any trace ideal other than

trace class.

Kato was at Berkeley in 1954 when Rosenblum was a student (albeit
some time before his thesis was completed) and Kato was in contact
with Wolf. However, there is no indication that Kato knew anything
about Rosenblum’s work until shortly before he wrote up his finite rank
paper when he became aware of Rosenblum’s abstract. My surmise isthat both, motivated by Friedrichs, independently became interested
in scattering.

I end by noting some of the important later developments (for refer‐ences, see [42]):
The original proofs were time‐independent. In his book Kato
found a lovely time‐dependent approach.
\bulletInvariance principle and Two‐Hilbert space scattering— bothof them where Kato made important contributions\bullet\bullet\bullet\bulletExtensions by Kuroda and especially Birman (so much so it iscalled Kato‐Birman)
Pearson’s formulation which encompasses most extensions
Krein spectral shift and the Birman‐Krein formula
7. KATO SMOOTHNESSFor the 25 years starting in 1955, spectral and scattering theory were
one of Kato’s major focuses. For example, when he was invited to give
one of the plenary talks at the 1970 ICM, he chose Scattering theory

and perturbation of continuous spectra [28] as his subject.

There are two approaches to proving results for scattering theory:time‐dependent and time‐independent based, respectively, on wave
operators and resolvent boundary values. For many years, time inde‐
pendent was regarded as the more powerful but the Enss work changed
that attitude to some extent.

In 1966 and 1968, Kato published two remarkable papers [18, 19] (tome, the 1951 self‐adjointness paper is Kato’s most significant work,
Kato’s inequality his deepest and the subject \mathrm{I}' \mathrm{m} about to discuss his

most beautiful). One of the things that is so beautiful is that there

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B. SIMON

isn’t just a relation between the two approach‐ there is an equivalence
that depends on the observation that ( R(z)=(H-z)^{-1} for z\in \mathbb{C}\backslash \mathbb{R} )\displaystyle \int_{0}^{\infty}e^{- $\epsilon$ t}e^{i $\lambda$ t}e^{-iHt} $\varphi$ dt=-iR( $\lambda$+i $\epsilon$) $\varphi$

this plus the Planceherel formula proves that\displaystyle \int_{-\infty}^{\infty}(\Vert AR( $\lambda$+i $\epsilon$) $\varphi$\Vert^{2}+\Vert AR( $\lambda$-i $\epsilon$) $\varphi$\Vert^{2}) d $\lambda$=
2 $\pi$\displaystyle \int_{-\infty}^{\infty}e^{-2 $\epsilon$|t}\Vert Ae^{-itH} $\varphi$\Vert^{2}dt

We say that A is H‐smooth if there is C<\infty so that for all $\epsilon$>0,
the left side of this is bounded by (2 $\pi$ C)^{2}\Vert $\varphi$\Vert^{2}

Thus we have the first two of a remarkable set of equivalences [18]:(1) A is H‐smooth with constant C(2) \displaystyle \sup_{\Vert $\varphi$\Vert=1}\int_{-\infty}^{\infty}\Vert Ae^{-itH} $\varphi$\Vert^{2}dt\leq C^{2}(3) \displaystyle \sup_{\Vert $\varphi$\Vert=1,-\infty<a<b<\infty}\frac{\Vert AP_{(a,b)}(H) $\varphi$\Vert^{2} {b-a}\leq C^{2}(4)
R( $\mu$)A^{*} $\varphi$\rangle| \leq C^{2}\displaystyle \sup_{ $\mu$\not\in \mathb {R}, $\varphi$\in D(A^{*}) \Vert $\varphi$|=1,(2 $\pi$)^{-1}|2{\rm Im}\langle A^{*} $\varphi$,(5) \displaystyle \sup_{\Vert $\varphi$\Vert=1, $\mu$\not\in \mathbb{R}, $\varphi$\in D(A^{*})}( $\pi$)^{-1}\Vert R( $\mu$)A^{*} $\varphi$\Vert^{2}|{\rm Im} $\mu$| \leq C^{2}
From (4) and Stone’s formula, we immediately see that ran (A^{*}) \subset\mathcal{H}_{ac}(H) so the existence of H‐smooth operators has spectral conse‐
quence for H.

The uniformity in the time‐dependent condition immediately implies
that if H=H_{0}+A^{*}B and if B is H_{0} ‐smooth and A is H‐smooth, thenby noting the derivative is integrable, we see that $\Omega$(H, H_{0}) exists.
H-B^{*}A , we also have the inverse wave operators exist
Since H_{0}=and so completeness.

Kato‐Yajima [29] (1989) say that A is H‐supersmooth if\displaystyle \sup (2 $\pi$)^{-1}|\langle A^{*} $\varphi$, R( $\mu$)A^{*} $\varphi$\rangle| <\infty\Vert $\varphi$\Vert=1,$\mu$\not\in \mathbb{R}, $\varphi$\in D(A^{*})The notion (which appeared already in Kato [18]) remains useful but,alas, the name didn’t stick. In any event, the above implies that theimaginary part is bounded so supersmoothness \Rightarrow smoothness. More

importantly, Kato [18] noted (as follows from a geometric series), that

if the above \displaystyle \sup is less than 1 and if \Vert C\Vert \leq 1 , then A is also H_{1} ‐smooth
if H_{1}=H+A^{*}CA so the wave operators exist and are unitary.

Moreover, it is known (Kato‐Yajima [29], implicit in Kenig‐Ruiz‐
Sogge [30]) that if V\in L^{ $\nu$/2}(\mathbb{R}^{ $\nu$}) , l\text{ノ\geq }3 , then |V|^{1/2} is H‐supersmooth.
So for such V' \mathrm{s} , one has a small coupling result (Kato’s original paper

145
KATO’S WORK

had a slightly weaker result). Iorio‐O’Carroll [11] (Iorio was Kato’s
student) have an analogous N‐body weak coupling result.
In 1968, Kato [19] found that smooth perturbations are an ideal tool
for proving a theorem found the year before by Putnam [34] and now

called the Kato‐Putnam theorem: if A and B are bounded self‐adjoint
operators so that i[A, B] is strictly positive (i.e. for all $\varphi$\neq 0 , we havethat \langle $\varphi$, i[A, B] $\varphi$\rangle >0 ), then A and B both have purely a.c. spectrum.

For let C be the square root of i[A, B] . Then \displaystyle \frac{d}{dt}\langle e^{-itA} $\varphi$, Be^{-itA} $\varphi$\rangle^{2}=\Vert Ce^{-itA} $\varphi$\Vert^{2} so the integral of \Vert Ce^{-itA} $\varphi$\Vert^{2} from s to t is bounded by
2\Vert B\Vert\Vert $\varphi$\Vert^{2} Thus C is A‐smooth and A has only a.c. spectrum on theclosure of ran (C) which is all of \mathcal{H}.
Further developments, discussed in [42], include Lavine’s local

smoothness theory and its application to repulsive potentials and awonderful and not well‐known application by Vakulenko.
REFERENCES[1] 1980 Wiener and Steele Prizes Awarded, Notices A.M.S. 27 (1980), 528‐533.
[2] J. Avron, R. Seiler, and B. Simon, Homotopy and Quantization in Condensed
Matter Physics Phys. Rev. Lett. 51 (1983), 51‐53.
[3] M. V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes, Proc.
Roy. Soc. A392, (1984) 45‐57.
[4] M. Born and V. A. Fock, Beweis des Adiabatensatzes, Zeit. für Physik A51(1928), 165‐180.
[5] H. Brézis, and T. Kato, Remarks on the Schrödinger operator with singularcomplex potentials, J. Math. Pures Appl. 58 (1979), 137‐151.
[6] H. Cordes, A. Jensen, S. T. Kuroda, G. Ponce, B. Simon, and M. Taylor,
Tosio Kato (1917−1999), Notices A.M.S. 47 (2000), 650‐657.
[7] P. Deift and B. Simon, A time‐dependent approach to the completeness ofmultiparticle quantum systems, Commun. Pure Appl. Math. 30 (1977), 573‐
583.

[8] K. O. Friedrichs, On the perturbation of continuous spectra, Comm. on Appl.
Math. 1 (1948), 361‐406.
[9] H. Hess, R. Schrader, and D. A. Uhlenbrock, Domination of semigroups andgeneralization of Kato’s inequality, Duke Math. J. 44 (1977), 893‐904.
[10] T. Ikebe, and T. Kato, Uniqueness of the self‐adjoint extensions of singularelliptic differential operators, Arch. Rational Mech. Anal. 9 (1962), 77‐92.
[11] R. J. Iorio and M. O’Carroll, Asymptotic completeness for multi‐particle
Schroedinger Hamiltonians with weak potentials, Comm. Math. Phys. 27(1972), 137‐145.
[12] A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time‐
decay of the wave functions, Duke Math. J. 46 (1979), 583‐611.
[13] T. Kato, On the adiabatic theorem for quantum mechanics, J. Phys. Soc.
Japan 5 (1950), 435‐439.
[14] T. Kato, Fundamental properties of Hamiltonian operators of Schrödinger
type, Trans. A.M.S. 70 (1951), 195‐211.

146
B. SIMON[15] T. Kato, On the eigenfunctions of many particle systems in quantum mechan‐ics, Comm. Pure Appl. Math. 10 (1957), 151‐177.
[16] T. Kato, On finite‐dimensional perturbations of self‐adjoint operators, J.
Math. Soc. Japan 9 (1957), 239‐249.
[17] T. Kato, Perturbation of continuous spectra by trace class operators, Proc.
Japan. Acad. 33 (1957), 260‐264.
[18] T. Kato, Wave operators and similarity for some non‐selfadjoint operators,
Math. Ann. 162 (1966), 258‐279.
[19] T. Kato, Smooth operators and commutators, Studia Mathematica 31 (1968),
535‐546.

[20] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13(1972), 135‐148.
[21] T. Kato, A second look at the essential self‐adjointness of the Schrödinger
operators, in Physical reality and mathematical description, ed. C. Enz and
J. Mehra, pp. 193‐201, Reidel, Dordrecht, 1974.

[22] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehrender Mathematischen Wissenschaften, Band 132, Springer, Berlin‐New York,

1976, (1st ed. 1966; there is also a revised and corrected 1980 printing[23] T. Kato, Remarks on Schrödinger operators with vector potentials, Integral
Equations Operator Theory 1 (1978), 103‐113.
[24] T. Kato, On some Schrödinger operators with a singular complex potential,
Ann. Scuola Norm. Super. Pisa, Cl. Sci., IV. 5 (1978), 105‐114.
[25] T. Kato, Remarks on the selfadjointness and related problems for differential
operators, in Spectral Theory of Differential Operators (Proc. Conf., Birm‐
ingham, USA 1981) ed. I. Knowles and R. Lewis, pp. 253‐266, North‐Holland(1981).
[26] T. Kato, Nonselfadjoint Schrödinger operators with singular first‐order coef‐
ficients, Proc. Roy. Soc. Edinb., Sect. A 96 (1984), 323‐329.
[27] T. Kato, L^{p} ‐theory of Schrödinger operators with a singular potential, in As‐
pects of Positivity in Functional Analysis, ed. R. Nagel, U. Schlotterbeck and

M. Wolff., pp 63‐78, North‐Holland (1986).
[28] T. Kato, Scattering theory and perturbation of continuous spectra, Actes duCongrès International des Mathématiciens 1 (1971), 135‐140.
[29] T. Kato and K. Yajima, Some examples of smooth operators and the associ‐ated smoothing effect, Rev. Math. Phys. 1 (1989), 481‐496.
[30] C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalitie\mathcal{S} andunique continuation for second order constant coefficient differential opera‐

tors, Duke Math. J. 55 (1987), 329‐347.
[31] J. Konrady, Almost p_{0\mathcal{S}}itive perturbations of positive selfadjoint Operators,
Comm.Math. Phys. 22 (1971), 295‐299.
[32] S. T. Kuroda, On a theorem of Weyl‐von Neumann, Proc. Japan Acad. 34(1958), 11‐15.
[33] E. Nelson, A quartic interaction in two dimensions, in Mathematical Theory
of Elementary Particles, ed. R. Goodman and I. Segal, pp. 69‐73, M.I.T.
Press, Cambridge, MA, 1966.

[34] C. R. Putnam, Commutation properties of Hilbert space operators and related
topics, Springer‐Verlag, New York, 1967.

147
KATO’S WORK[35] M. Rosenblum, Perturbation of the continuous spectrum and unitary equiva‐
lence, Pacific J. Math. 7 (1957), 997‐1010.
[36] I. E. Segal, Construction of nonlinear local quantum processes, I, II, Ann.
Math. 91 (1970), 462−481; Invent. Math. 14 (1971), 211‐241.
[37] B. Simon, Essential self‐adjointness of Schrödinger operators with positive
potentials, Math. Ann. 201 (1973), 211‐220.
[38] B. Simon, An abstract Kato’s inequality for generators of positivity preserving
semigroups, Indiana Univ. Math. J. 26 (1977), 1067‐1073.
[39] B. Simon, Maximal and minimal Schrödinger forms, J. Operator Theory 1(1979), 37‐47.
[40] B. Simon, Kato’s inequality and the comparison of semigroups, J. Func. Anal.
32 (1979), 97‐101.
[41] B. Simon, Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase,
Phys. Rev. Lett. 51 (1983), 2167‐2170.
[42] B. Simon, Tosio Kato’s Work on Non‐Relativistic Quantum Mechanics, Bull.
Math. Sci., to appear.

[43] B. Simon, Tosio Kato’s Work on Non‐Relativistic Quantum Mechanics: A
Brief Report, Bull IAMP, to appear; also to appear in Analysis and Operator
Theory, ed. T. Rassias and V. Zagrebnov, Springer[44] B. Simon and R. \mathrm{H}\emptyset\mathrm{e}\mathrm{g}\mathrm{h}‐Krohn, Hypercontractive semi‐groups and two di‐mensional self‐coupled Bose fields, J. Func. Anal. 9 (1972), 121‐180.
[45] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, QuantizedHall Conductance in a Two‐Dimensional Periodic Potential, Phys. Rev. Lett.
49, (1982) 405‐408.
[46] J. von Neumann, Charakterisierung des Spektrums eines Integraloperators,
Actualités Sci. Industr. 229 (1935), 3‐20.

[47] H. Weyl, Über beschränkte quadratische Formen, deren Differenz vollstetig
ist, Palermo Rend. 27 (1909), 373‐392, 402.