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In exploring this, it is worth noting that there is an alternate to step Namely he used 7. It is possible that he never considered the problem of esa of atomic Hamiltonians, settling for a presumption that using the Friedrichs extension suffices as Kato suggests in [ 1 ] but I think that unlikely. It is possible that he thought about the problem but dismissed it as too difficult and never thought hard about it. Perhaps the most likely explanation involves Step 3: once you understand it, it is trivial, but until you conceive that it might be true, it might elude you.

In exploring extensions of Theorem 7. If it is not limit circle, we say it is limit point. It is a theorem that whether one is limit point or limit circle is independent of z. One has the basic. The ideas behind much of the theorem go back to Weyl [ , , ] in and predate the notion of self-adjointness. It was Stone [ ] who first realized the implications for self-adjointness and proved Theorem 7. Titchmarsh [ ] reworked the theory so much that it is sometimes called Weyl—Titchmarsh theory.

For additional literature, see [ 94 , , ]. It follows by general principles [ , Section 7. The quantum analog is the loss of esa. This is associated with the uncertainty principle. This result is essentially due to Rellich [ ] in He proved it using spherical symmetry and applying the Weyl limit—limit circle theory Theorem 7. This theory for systems was established by Kodaira [ ] in see also Weidmann [ ] so Theorem 7.

Interestingly enough, Kato seems to have been unaware of this result when he wrote his book second edition was Let V obey 7. Among the group were H. Cycon, H. Kalf, U. Schmincke, R. To understand this, it is useful to first consider one dimension. If it is finite, the particle gets to infinity in finite time and the motion is incomplete.

Ikebe—Kato had the important realization that one needs no global hypothesis on a , i. While they had too strong a local hypothesis on local behavior of a see Sect. For one of them, Kato made an important contribution. The first approach is due to Chernoff [ 90 , 92 ] as modified by Kato [ ] and the second approach is due to Faris—Lavine [ ]. Interesting enough, each utilizes a self-adjointness criterion of Ed Nelson but two different criteria that he developed in different contexts.

Chernoff turned this argument around! This is an expression of the fact that for the Dirac equation, no boundary condition is needed at infinity—intuitively, this is because the particle cannot get to infinity in finite time because speeds are bounded by the speed of light! Several years after his initial paper, Chernoff [ 92 ] used results on solutions of singular hyperbolic equations and proved the following version of the fact that Dirac equations have no boundary condition at infinity:.

Other results on esa for Dirac operators which are finite sums of Coulomb potentials include [ , , , , , , ]. This proof of the result appeared in Chernoff [ 90 ], but the result itself appeared earlier in Povzner [ ] and Wienholtz [ ]. This completes our discussion of the Chernoff approach. The underlying self-adjointness criterion of Nelson needed for the Faris—Lavine approach is. Let A be the operator closure of Open image in new window. By Theorem 9. The same method that proved 3. This section will discuss a self-adjointness method that appeared in Kato [ ] based on a remarkable distributional inequality.

Its consequences is a subject to which Kato returned often with at least seven additional papers [ 73 , , , , , , ]. It is also his work that most intersected my own—I motivated his initial paper and it, in turn, motivated several of my later papers. To explain the background, recall that in Sect. Of course this covers pretty wild growth at infinity but Theorem 9.

Within a few weeks of my sending out a preprint with Theorem 9. I will begin the discussion here by sketching a semigroup proof of Theorem 9. After the smoke cleared, it was apparent that my failure to get the full Theorem 9. As a warmup to the semigroup proof of Theorem 9. There is an analog of Theorem 9. Here is a sketch of a proof of Theorem 9. Step 3. Step 4. Step 5. He proved. We should pause to emphasize what a surprise this was.

Kato was a long established master of operator theory. He was 55 years old. Seemingly from left field, he pulled a distributional inequality out of his hat. Truly a remarkable discovery. The equivalence of a and b for M a finite set so A is a matrix is due to Beurling—Deny [ 54 ]. For a proof of the full theorem which is not hard , see Simon [ ] or [ , Theorem 7.

Simon [ ] 9. Indeed, our proof of 9. As with Theorem 9. For a proof, see the original papers or [ , Theorem 7. As one might expect, the ideas in Kato [ ] have generated an enormous literature. Some papers on these ideas include Devinatz [ ], Eastham et al. There is a review of Kato [ ]. For applications to higher order elliptic operators, see Davies—Hinz [ ], Deng et al. Kato himself applied these ideas to complex valued potentials in three papers [ 73 , , ]. For more on this theme, see [ 7 , ]. In the s and s, there was work in which the quadratic form point of view was implicit but it was only in the s that forms became explicitly discussed objects and Kato was a major player in this development.

Theorem We say that a quadratic form, q , is closable if and only if q has a closed extension. What makes quadratic forms so powerful is that, in a sense, Example Here are two versions of this result:. Let q be a closed quadratic form. In his book [ ], Kato calls Theorem He puts Theorem I put Theorem One can show it is immediate from Theorem It is called the form sum.

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The proof is really simple. This allows a definition of a self-adjoint sum of any two positive self-adjoint operators. There is a similar result for n arbitrary closed forms. The simplest proof is to use the Davies—Kato characterization below that closedness is equivalent to lower semicontinuity. There are closed symmetric operators which are not self-adjoint but every closed quadratic form is the form of a self-adjoint operator. Every symmetric operator has a smallest closed extension but there exist quadratic forms with no closed extensions. A follow-up paper of Freudenthal [ ] did Friedrichs extension in something close to form language.

We will need the following result of Simon [ ] see also [ , Theorem 7. The two monotone convergence theorems for positive quadratic forms are. For proofs, see [ , Theorem 7. This shows that in the decreasing case, the limit need not be closed or even closable. This material from the first edition was unchanged from the second edition.

In , Robinson [ ] proved Theorem In , Davies [ ] also proved this theorem. His proof relied on lower semicontinuity being equivalent to q being closed see below. When Kato published his second edition, he was clearly unaware of their work. The lower semicontinuity fits in nicely with even then well known work on variational problems that used the weak lower semicontinuity of Banach space norms so it was not surprising. Indeed Davies mentions it in passing in his paper without proof.

At the time I wrote the preprint, I was unaware of the relevant work of Davies and Robinson although I knew each of them personally. He stated a lovely result. For a proof, see [ , Theorem 7. I note that in precisely this context, Theorem Kato told me that he had no plans to publish his remark and approved my writing [ ] that explores consequences of Theorem In particular, he had the full Theorem In the Supplemental Notes, he quotes [ ] and [ ] but neither of the papers of Davies and Robinson, despite the fact that in response to their writing to me after the preprint, I added a Note Added in Proof to [ ] referencing their work.

Kato called C the pseudo-Friedrichs extension. Faris [ ] has a presentation that uses sesquilinear forms and makes this closer to the KLMN theorem. Herbst notes that this operator commutes with scaling, so after applying the Mellin transform, it commutes with translations and so, it is a convolution operator in Mellin transform space. The function it is convolution with is positive function so the norm is related to the computable integral of this explicit function. Typical is the following result of Nenciu [ ] which is a variant of Schmincke [ ] :.

Neither of them knew at the time that Loewner [ ] had already proven a much more general result in ! Despite the 17 year priority, the monotonicity of the square root is called variably, the Heinz inequality, the Heinz—Loewner inequality or even, sometimes, the Heinz—Kato inequality.

Heinz and Kato found equivalent results to the monotonicity of the square root one paper with lots of additional equivalent forms is [ ]. In particular, the following equivalent form is almost universally known as the Heinz—Kato inequality. I should warn the reader that I use two conventions that are universal among physicists but often the opposite of many mathematicians.

Nagumo [ ] used 2. Step 1. Finite Dimensional Theory. Analytic Resolvents and Spectral Projections. We can thus use 2. Because of 2. If a bound like 2. There exist unbounded B for which the relative bound is 0. Example 2. There has been considerable physical literature on this example. The first substantial rigorous mathematical work on the subject is a five part series of papers by Rellich [ , , , , ] published from to In particular, the second order term in this theory is called the Fermi golden rule, discussed, for example, in Landau—Lifshitz [ , pp.

Simon [ ] has a compact way to write this second order term. Put differently, it becomes a finite lifetime state that decays. He claimed to compute the lifetime but his calculation was wrong. Kato proves this by direct calculation rather than the more general strong convergence method in his book which we discuss below.

With these examples in mind, we turn to the general theory of asymptotic series. Recall [ , Section Theorem 3. Proof a follows from a simple use of the second resolvent formula; see [ , Theorem 7. There are two main ways that one can prove stability in cases where it is true. While there is some truth to this, Simon [ ] found the surprising fact that even in situations where perturbation theory diverges, one can have norm convergence of resolvents in a sector.

Remarks 1. The proof is easy. The ordinary approximates for a power series are by the polynomials obtained by truncating the power series. The other method is called Borel summability , introduced by Borel [ 65 ]. The result is due to Stieltjes [ , ] who discussed solutions of the moment problem 3. Starting around , Kato [ ] and Titchmarsh [ , , , , , ] considered what the perturbation series might mean for a problem like the Stark problem where a discrete eigenvalue is swallowed by continuous spectrum as soon as the perturbation is turned on.

Theorem 4. Riddell also has a converse. We begin with the two body case. First Proof of Theorem 5.

If we use 5. So we get another proof of Theorem 5. We can also say something about non-self-adjoint projections on Hilbert spaces and also about the general Banach space case. The spectral theory of general compact operators, A , is more subtle than the self-adjoint case [ , Section 3.

Remark d was proven by Kalton [ ] using different methods. Our final result from the supersymmetric approach returns to the self-adjoint case. Then there exists a unitary map, U , obeying 5. For the converse, suppose that 5. Since 5. Proposition 5. These four spaces are mutually orthogonal.

There are alternate ways that this theorem is often expressed. Sometimes, 5. Moreover the decomposition of Q is Open image in new window. Theorem 6. Lemma 6. Proof The spectral theorem see [ , Chapter V and Section 7. Remark While we use the Spectral Theorem as Kato did , all we need is a spectral mapping theorem, i. Corollary 6. Proof of Theorem 6. As in the proof of 6. More generally, one wants to consider as Kato did Hamiltonians with the center of mass removed.

We discuss the kinematics of such removal in Sect.

## A BIBLIOGRAPHY OF MATHEMATICS EDUCATION

We note that the self-adjointness results on the Hamiltonians of the form 7. Of especial interest is the Hamiltonian of the form 7. Theorem 7. Step 2. Example 7. Proposition 7. We turn now to the extensions of Theorem 7. There are now and even then, but not so widely known sharper inequalities than 7. These esa results for ODEs were studied in the late s using limit point—limit circle methods. Theorem 8. Further developments all later than the Ikebe—Kato paper discussed below are due to Hellwig [ , , ], Rohde [ , , ] and Walter [ , ]. Unlike Wienholtz, they could allow local singularities such as atoms in Stark fields.

The argument is simple. Corollary 8. In these counterexamples, though, V is negative. It was known since the late s see Sect. But when I started looking at these issues around , there was presumption that for local singularities, there was no difference between the positive and negative parts. Theorem 9. I found Theorem 9. Arguments like those that proved 3. This step proves 9. Next, we provide our first proof of Theorem 9. By the just proven Theorem 9. The proof is not hard. Once we have 9. In his original paper, Kato [ ] proved more than 9.

In [ ], Kato followed his arguments to get Theorem 9. But there was a more important consequence of 9. In [ ], I noted that 9. There is one final aspect of [ ] which should be mentioned. Example We recall that the spectral theorem [ , Chapters 5 and Section 7. B might not be self-adjoint. The same argument that we used to prove 7. We end our discussion of the general theory by noting some distinctions between forms and symmetric operators.

If A and B are self-adjoint operators and B is an extension of A i. Having completed our discussion of the general theory, we turn to a brief indication of its history. Next, we turn to a discussion of monotone convergence of quadratic forms. The final topic of this section concerns pseudo-Friedrichs extensions and form definitions of the Dirac Coulomb operator. Recall that in Sect. In his book, Kato [ ] noted that by combining his definition of the pseudo-Friedrichs extension and his inequality, one can define a natural self-adjoint extension of Agmon, S.

Scuola Norm. Pisa Cl. Aguilar, J. Aizenman, M. Pure Appl. Albeverio, S. Ammann, B. Amrein, W. Linear Algebra Appl. Ando, T. Tohoku Math. Arai, M. Aronszajn, N. Pures Appl. Atkinson, F. Avron, J. General interactions. Duke Math. Separation of center of mass in homogeneous magnetic fields. Atoms in homogeneous magnetic field. Strongly bound states of hydrogen in intense magnetic field. A 20 , — CrossRef Google Scholar. Babbitt, D. Bachmann, S. Preprint arXiv SP] Baker, G.

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SSSR Ser. Mat, 27 , — Eng. In: Problems of Mathematical Physics, No. Wave Process, pp. SSSR , — Eng. Trans: Soviet Math. Borel, E. Borg, G. Born, M. Brascamp, H. Bratteli, O. Equilibrium States. Models in Quantum-statistical Mechanics. Brownell, F. Bruneau, L.

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Rank one operators. Deng, Q. Indiana Univ. Springer, New York Devinatz, A.

## Tosio Kato’s work on non-relativistic quantum mechanics: part 1

Dinu, V. Dixmier, J. Revue Sci. Dollard, J. Dolph, C. Donoghue, W. Dou, Y. Dunford, N. Convergence to projections. Dyatlov, S. Dyson, F. Eastham, M. In: Everitt, W. Ordinary and Partial Differential Equations, Dundee Lecture Notes in Mathematics, vol. Springer, Berlin Google Scholar. Research Notes in Mathematics. Evans, W. Eckmann, J. Effros, E. Efimov, V. B 33 , — CrossRef Google Scholar. Ehrenfest, P. Elgart, A. Enss, V. Short-range potentials. In: Blanchard, Ph, Streit, L. Epstein, P. Esteban, M.

Estienne, C. Faris, W. Feshbach, H. Figiel, T. Fock, V. To keep this review within bounds, we will not discuss this work. Before it, his most important work was his thesis, awarded in and published in — 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 from the University of Tokyo in , a year in which he published two not mathematical papers in theoretical physics. But during the war, he was evacuated to the countryside. We were at a conference together one evening and Kato described rather harrowing experiences in the camp he was assigned to, especially an evacuation of the camp down a steep wet hill.

He contracted TB in the camp. In , Kato returned to the University of Tokyo as an Assistant a position common for students progressing towards their degrees in physics, was appointed Assistant Professor of Physics in and full professor in Kodaira got a BS in physics after his BA in mathematics and was given a joint appointment in , so there was clearly some sympathy towards pure mathematics in the physics department. Beginning in , Kato started visiting the United States. This bland statement masks some drama.

In , Kato was invited to visit Berkeley for a year, I presume arranged by F. Of course, Kato needed a visa and it is likely it would have been denied due to his history of TB. Charles Dolph — , a mathematician at Michigan, learned of the problem and contacted Laporte who intervened to get Kato a visa. Dolph once told me that he thought his most important contribution to American mathematics was his helping to allow Kato to come to the US.

In , he accepted a professorship in Mathematics from Berkeley where he spent the rest of his career and remained after his retirement. In [ ], Reed and I mentioned that Kato had this result independently. Although Kato never published anything on the subject, in recent times, it has come to be called the Kato—Seiler—Simon inequality.

One of its virtues is its comprehensive nature. We will not discuss every piece of work that Kato did in NRQM—for example, he wrote several papers on variational bounds on scattering phase shifts whose lasting impact was limited. Roughly speaking, this article is in five parts. There is a section on situations where either an eigenvalue is initially embedded in continuous spectrum or where as soon the perturbation is turned on the location of the spectrum is swamped by continuous spectrum i.

There are a pair of sections on two issues that Kato studied in connection with eigenvalue perturbation theory: pairs of projections and on the Temple—Kato inequalities. Next come four sections on self-adjointness. Finally his work on quadratic forms is discussed including his work on monotone convergence for forms. That will end Part 1. Part 2 begins with two pioneering works on aspects of bound states—his result on non-existence of positive energy bound states in certain two body systems and his paper on the infinity of bound states for Helium, at least for infinite nuclear mass.

Next four sections on scattering and spectral theory which discuss the Kato—Birman theory trace class scattering , Kato smoothness, Kato—Kuroda eigenfunction expansions and the Jensen—Kato paper on threshold behavior. Last is a set of three miscellaneous gems: his work on the adiabatic theorem, on the Trotter product formula and his pioneering look at eigenfunction regularity. Her will gave control of the pictures to H. Fujita, M. Ishiguro and S. I thank them for permission to use the pictures and H. Okamoto for providing digital versions. This is the first of five sections on eigenvalue perturbation theory; this section deals with the analytic case.

As a preliminary, we want to recall the theory of spectral projections for general bounded operators, A , on a Banach space, X. In —, this functional calculus was further developed in the United States by Dunford [ , ], Lorch [ ] and Taylor [ ]. In his book, Kato calls 2. With this formalism out of the way, we can turn to sketch the theory of regular perturbations.

For details see the book presentations of Kato [ , Chaps. The set of early significant results include two theorems of Rellich [ , , , , , Part I]. Step 5 Regular Families of Closed Operators. Kato [ , Section VII. Step 6 Criteria for Regular Families. He discussed this in his thesis and, in his book [ , Section VII. This completes our discussion of the theory of eigenvalue perturbation theory so we turn to some remarks on its history. Eigenvalue perturbation theory goes back to fundamental work of Lord Rayleigh on sound waves in [ , pp.

The first paper had a Hungarian language version [ ]. He defined type A perturbations via 2. His main advance is to exploit the definition of spectral projections via 2. As a student of F. Riesz, this is not surprising. Wolf [ ] also extended the Nagy approach to the Banach space case is Perhaps the most significant aspect of this work is that it served eventually to introduce Kato to Wolf for Wolf was a Professor at Berkeley who was essential to recruiting Kato to come to Berkeley both in and Wolf had spent time in Cambridge and did some significant work on trigonometric series under the influence of Littlewood.

When the Germans invaded Czechoslovakia in March , he was able to get an invitation to Mittag—Leffler. He was then able to get an instructorship at Macalester College in Minnesota. He made what turned out to be a fateful decision in terms of later developments. Because travel across the Atlantic was difficult, he took the trans-Siberian railroad across the Soviet Union and then through Japan and across the Pacific to the US.

This was mid before the US entered the war and made travel across the Pacific difficult. Wolf stopped in Berkeley to talk with G. Evans known for his work on potential theory who was then department chair. After the year he promised to Macalester, Wolf returned to Berkeley and worked his way up the ranks. At about the same time Nagy himself did similar work and in so did Kato. His thesis was published in a university journal in full [ ] in with parts published a year early in broader journals in both English [ , ] and Japanese [ ].

Two final early papers [ , ] dealt with the Banach space case and with further results on asymptotic perturbation theory discussed further in Sect. His main contribution beyond theirs concerns the use of reduced resolvents. In this section and the next, we discuss situations where the Kato—Nagy—Rellich theory of regular perturbations does not apply. Lest the reader think this is a strange pathology, we begin with six!

For obvious reasons, these are called autoionizing states. For our purposes, it is useful to look at states with angular momentum 2 and azimuthal angular momentum 2 which are simple. There are actually 15 subspaces with definite symmetry. In one, there is a doubly degenerate embedded eigenvalue, in 3 an isolated eigenvalue and in 11 a simple embedded eigenvalue. What does the perturbation series have to do with the resonance energy? Can one mathematically justify the Fermi golden rule? What are the higher terms? Is there a convergent series?

Many of the same questions occur as for Example 3. What is the meaning of the divergent perturbation series? What is the difference between 3. In his first example, he takes B to be multiplication by x. A good example is given [ , Table after So it is interesting and important to know that a series is asymptotic but if one knows the series and wants to know f , it is disappointing not to know more. He used what are now called Temple—Kato inequalities to obtain asymptotic series to all orders in [ , ]. We discuss this approach in Sect. His work relied heavily on ODE techniques.

There is some overlap of this work from his book and work of Huet [ ], Kramer [ , ], Krieger [ ] and Simon [ ]. Here is a theorem, going back to Rellich [ , , , , , Part 2] describing some results critical for asymptotic perturbation theory:. For b , one first proves 3. The basic theorem is then due to Trotter [ ] in his thesis written under the direction of Feller, whose interest in semigroups was motivated by Markov processes.

This theorem has also been called the Trotter—Kato—Neveu or Trotter—Kato—Neveu—Kurtz—Sova theorem after related contributions by these authors [ , , , ]. There is another related result of this genre sometimes called the Trotter—Kato theorem. Returning to perturbation theory, Kato introduced and developed the key notion of stability. To state results on asymptotic series, we focus on getting series for all orders. Kato [ ] is interested mainly in first and second order, so he needs much weaker hypotheses.

This is not an issue that Kato seems to have written about but it is an important part of the picture, so we will say a little about it. This leads to a notion of strong asymptotic condition and an associated result of there being at most one function obeying that condition and so a strong asymptotic series determines E —see Simon [ , ] or Reed—Simon [ , Section XII. Algorithms for recovering a function from a possibly divergent series are called summability methods.

Hardy [ ] has a famous book on the subject. Many methods, such as Abel summability i. The series that arise in eigenvalue perturbation theory are usually badly divergent but, fortunately, there are some methods that work even in that case. It follows from results of Loeffel et al. The key convergence result for Borel sums is a theorem of Watson [ ]; see Hardy [ ] for a proof:. Avron—Herbst—Simon [ 25 , 26 , 27 , 28 , Part III] proved that for the Zeeman effect in arbitrary atoms, the perturbation series of the discrete eigenvalues is Borel summable.

Clearly, when it can be proven, Borel summability is an important improvement over the mere asymptotic series that concerned Kato. Kato discussed things in terms of what he called pseudo-eigenvalues and pseudo-eigenvectors. He later realized that these notions imply a concentration of spectrum like that used by Titchmarsh. In his book [ ], he emphasized what he formally defined as spectral concentration and linked the two approaches.

By Theorem 3. Taking into account that we may want to also have T shrink in cases like Example 3. These ideas were used by Friedrichs and Rejto [ ] to prove spectral concentration in Example 3. Riddell [ ] proved spectral concentration to all orders for the Stark effect for Hydrogen using pseudo-eigenvectors and Rejto [ , ] proved the analog for Helium see below for more on spectral concentration for the Stark effect. This theme was developed by James Howland, a student of Kato, in 5 papers [ , , , , ].

Howland discussed two situations. Thus the resonance energy could be interpreted as the analytic continuation of a perturbed eigenvalue. Perhaps the most successful approach to the study of resonances, one that handles problems in atomic physics like Examples 3. The idea appeared initially in a technical appendix of a never published note by J. Combes and collaborators knew that the formalism, which they used to prove the absence of singular continuous spectrum, provided a possible definition of a resonance.

It was Simon [ ] who realized that the formalism was ideal for studying eigenvalues embedded in the continuous spectrum like autoionizing states. Example 3. This first implies there is a convergent perturbation series i. In spite of this accepted wisdom, a quantum chemist, Bill Reinhardt, did calculations for the Stark problem using complex scaling [ ] and got sensible results. Motivated by this, Herbst [ ] was able to define complex scaling for a class of two body Hamiltonians including the Hydrogen Stark problem.

It is a theorem that elements in Banach algebras and, in particular, bounded operators on any Banach space, have non-empty spectrum but that is only for bounded operators. The physical value is then determined by analytic continuation. Graffi—Grecchi [ ] had proven Borel summability slightly earlier using very different methods. Graffi—Grecchi [ ] and Herbst—Simon [ ] also proved Borel summability for discrete eigenvalues of general atoms. For Hydrogen, Herbst—Simon conjectured 3. Shortly thereafter, Harrell—Simon [ ] proved the Oppenheimer formula for the complex scaled defined Stark resonance and so also 3.

They used similar arguments to prove the Bender—Wu formula for the anharmonic oscillator. While the free Stark problem has scaled Hamiltonians with empty spectrum when there is one positive charge and N particles of equal mass and equal negative charge, there are charges and masses, where the spectrum is not empty. We end this discussion by noting that I have reason to believe that, at least at one time, Kato had severe doubts about the physical relevance of the complex scaling approach to resonances.

I had some of the report quoted to me. The referee said that the complex scaling definition of resonance was arbitrary and physically unmotivated with limited significance. There is at least one missing point in a reply to this criticism: however it is defined, a resonance must correspond to a pole of the scattering amplitude. While this is surely true for resonances defined via complex scaling, as of this day, it has not been proven for the models of greatest interest.

So far, resonance poles of scattering amplitudes in quantum systems have only been proven for two and three cluster scattering with potentials decaying faster often much faster than Coulomb and not for Stark scattering; see Babbitt—Balslev [ 33 ], Balslev [ 39 , 40 , 41 ], Hagedorn [ ], Jensen [ ] and Sigal [ , ]. E for these unphysical values of the parameters is an eigenvalue of these corresponding H. Thus resonances can be viewed as analytic continuations of actual eigenvalues from unphysical to physical values of the parameters.

One called distortion analyticity works sometimes for potentials which are the sum of a dilation analytic potential and a potential with exponential decay but not necessarily any x -space analyticity. The basic papers include Jensen [ ], Sigal [ ], Cycon [ ], and Nakamura [ , ].

There is an enormous literature on the theory of resonances from many points of view. A review of the occurrence of resonances in NR Quantum Electrodynamics and of the smooth Feshbach—Schur map is Sigal [ ] and a book on techniques relevant to some approaches to resonances is Martinez [ ]. Recall [ , Section 2. There is a one-one correspondence between such direct sum decompositions and bounded projections.

We saw in Sect. Theorem 5. His formula for U looks more involved than 5. He did this using the same formalism he had developed for his treatment of the adiabatic theorem Kato [ ] and Sect. In [ ], Kato noted that his expression was equal to the object found by Sz-Nagy [ ] but in the Banach space case, one could get better estimates from his formula for the object. There are two approaches. Here is a typical use of this method:. The results a — c and the proof we give of d is new in the present paper. So, in this case, the theorem says that U obeying 5.

This special case is in [ 31 ]. The general case of this theorem is due to Wang, Du and Dou [ ] whose proof used the Halmos representation discussed below. Our proof here is from Simon [ ]. Two recent papers [ 70 , ] classify all solutions of 5. Operators obeying 5. Our final big topic in this section concerns the Halmos representation. As a first step, we note that. The proof of the second statement is similar. The two spaces in the first statement are orthonormal by b and the mutual orthogonality of eigenspaces.

The second relation has a similar proof. Immediate from the orthogonality of different eigenspaces of a self-adjoint operator. Immediate from c. The Halmos two projection theorem says. This result is due to Halmos [ ]. There were earlier related results by Krein et. The proof we give here is due to Amrein—Sinha [ 13 ]. We mention also Lenard [ ] who computes the joint numerical range i. This range is a union of certain ellipses. Kato has this as a Lemma in a technical appendix to [ ], but it is now regarded as a significant enough result that Szyld [ ] wrote an article to advertise it and explain myriad proofs [ 69 ] also discusses proofs.

Del Pasqua [ ] and Ljance [ ] found proofs slightly before Kato but the methods are different and independent; indeed, for many years, no user of the result seemed to know of more than one of these three papers. Del Pasqua [ ] noted that 5. While strictly speaking the central material in this section is not so much about perturbation theory as variational methods, the subjects are related as Kato mentioned in several places, so we put it here.

Kato also had several other papers about variational methods for scattering phase shifts [ , , ] and for an aspect of Thomas—Fermi theory [ ] not the energy variational principle central to TF theory but one concerning a technical issue connected to the density at the nucleus. But none of these other papers had the impact of the work we discuss in this review, so we will not discuss them further. Here is his theorem:. The spectral theorem see [ , Chapter V and Section 7. While we use the Spectral Theorem as Kato did , all we need is a spectral mapping theorem, i.

The spectral mapping theorem for polynomials holds for elements of any Banach algebra and the proof in [ , Theorem 2. That this lemma follows from considerations of resolvents only was noted by Temple [ ]. Taking contrapositives in 6. This is Lemma 6. Kato exploited what are now called the Temple—Kato inequalities in his thesis to prove results on asymptotic perturbation theory.

Below are two typical results whose proofs are very much in the spirit of this work of Kato—see Sect. But in [ ], he put in a Note Added in Proof announcing he had solved the problem! The solution appeared in [ ]. Not surprisingly, in addition to estimates of Temple—Kato type, the proofs use a variant of quadratic form methods. I note that Kato did not put any of these results in his book where his discussion of asymptotic series applies to general Banach space settings and not just positive operators and the ideas are closer to what we put in Sect.

Besides the original short paper on Temple—Kato inequalities, Kato returned to the subject several times. Variants of the Temple—Kato inequality for operators of this form are the subject of two papers of Kato [ , ]. Kato et al. George Frederick James Temple — was a mathematician with a keen interest in physics—he wrote two early books on quantum mechanics in and At age 82, he became a benedictine monk and spent the last years of his life in a monastery on the Isle of Wright.

Turner [ ] and Harrell [ ] have extensions to the case where A is normal rather than self-adjoint and Kuroda [ ] to n commuting self-adjoint operators and so including the normal case. Cape et al. Golub—van der Vost [ ] have a long review on eigenvalue values bounds mentioning that by the time of their review in , Temple—Kato inequalities had become a standard part of linear algebra.

This is the first of four sections on self-adjointness issues. We assume the reader knows the basic notions, including what an operator closure and an operator core are and the meaning of essential self-adjointness.

A reference for these things is [ , Section 7. This section concerns the Kato—Rellich theorem and its application to prove the essential self-adjointness of atomic and molecular Hamiltonians. The quantum mechanical Hamiltonians typically treated by this method are bounded from below.

Then the Hamiltonian of 7. The same results holds with the terms in 7. This relies on a simple Sobolev estimate. Kato states in the paper that he had found the results by Kato originally submitted the paper to Physical Review. Physical Review transferred the manuscript to the Transactions of the AMS where it eventually appeared. They had trouble finding a referee and in the process the manuscript was lost a serious problem in pre-Xerox days! Eventually, von Neumann got involved and helped get the paper accepted. The receipt date of October 15, on the version published in the Transactions shows a long lag compared to the other papers in the same issue of the Transactions which have receipt dates of Dec.

There are plans to publish an edited version [ ]. It is a puzzle why it took so long for this theorem to be found. Bargmann told me of a conversation several young mathematicians had with von Neumann around in which von Neumann told them that self-adjointness for atomic Hamiltonians was an impossibly hard problem and that even for the Hydrogen atom, the problem was difficult and open. This is a little strange since, using spherical symmetry, Hydrogen can be reduced to a direct sum of one dimensional problems.

### THE NATURE OF MATHEMATICS

For such ODEs, there is a powerful limit point—limit circle method named after Weyl and Titchmarsh although it was Stone, in his book, who first made it explicit. Still it is surprising that neither Friedrichs nor Rellich found this result. In exploring this, it is worth noting that there is an alternate to step Namely he used 7. It is possible that he never considered the problem of esa of atomic Hamiltonians, settling for a presumption that using the Friedrichs extension suffices as Kato suggests in [ 1 ] but I think that unlikely.

It is possible that he thought about the problem but dismissed it as too difficult and never thought hard about it. Perhaps the most likely explanation involves Step 3: once you understand it, it is trivial, but until you conceive that it might be true, it might elude you. In exploring extensions of Theorem 7. If it is not limit circle, we say it is limit point. It is a theorem that whether one is limit point or limit circle is independent of z. One has the basic.

The ideas behind much of the theorem go back to Weyl [ , , ] in and predate the notion of self-adjointness. It was Stone [ ] who first realized the implications for self-adjointness and proved Theorem 7. Titchmarsh [ ] reworked the theory so much that it is sometimes called Weyl—Titchmarsh theory.

For additional literature, see [ 94 , , ]. It follows by general principles [ , Section 7. The quantum analog is the loss of esa. This is associated with the uncertainty principle. This result is essentially due to Rellich [ ] in He proved it using spherical symmetry and applying the Weyl limit—limit circle theory Theorem 7. This theory for systems was established by Kodaira [ ] in see also Weidmann [ ] so Theorem 7. Interestingly enough, Kato seems to have been unaware of this result when he wrote his book second edition was Let V obey 7.

Among the group were H. Cycon, H. Kalf, U. Schmincke, R. To understand this, it is useful to first consider one dimension. If it is finite, the particle gets to infinity in finite time and the motion is incomplete. Ikebe—Kato had the important realization that one needs no global hypothesis on a , i. While they had too strong a local hypothesis on local behavior of a see Sect. For one of them, Kato made an important contribution.

The first approach is due to Chernoff [ 90 , 92 ] as modified by Kato [ ] and the second approach is due to Faris—Lavine [ ]. Interesting enough, each utilizes a self-adjointness criterion of Ed Nelson but two different criteria that he developed in different contexts. Chernoff turned this argument around! This is an expression of the fact that for the Dirac equation, no boundary condition is needed at infinity—intuitively, this is because the particle cannot get to infinity in finite time because speeds are bounded by the speed of light!

Several years after his initial paper, Chernoff [ 92 ] used results on solutions of singular hyperbolic equations and proved the following version of the fact that Dirac equations have no boundary condition at infinity:. Other results on esa for Dirac operators which are finite sums of Coulomb potentials include [ , , , , , , ]. This proof of the result appeared in Chernoff [ 90 ], but the result itself appeared earlier in Povzner [ ] and Wienholtz [ ]. This completes our discussion of the Chernoff approach. The underlying self-adjointness criterion of Nelson needed for the Faris—Lavine approach is. Let A be the operator closure of Open image in new window.

By Theorem 9. The same method that proved 3. This section will discuss a self-adjointness method that appeared in Kato [ ] based on a remarkable distributional inequality. Its consequences is a subject to which Kato returned often with at least seven additional papers [ 73 , , , , , , ]. It is also his work that most intersected my own—I motivated his initial paper and it, in turn, motivated several of my later papers. To explain the background, recall that in Sect.

Of course this covers pretty wild growth at infinity but Theorem 9. Within a few weeks of my sending out a preprint with Theorem 9. I will begin the discussion here by sketching a semigroup proof of Theorem 9. After the smoke cleared, it was apparent that my failure to get the full Theorem 9. As a warmup to the semigroup proof of Theorem 9. There is an analog of Theorem 9. Here is a sketch of a proof of Theorem 9. Step 3. Step 4. Step 5. He proved. We should pause to emphasize what a surprise this was.

Kato was a long established master of operator theory. He was 55 years old. Seemingly from left field, he pulled a distributional inequality out of his hat. Truly a remarkable discovery. The equivalence of a and b for M a finite set so A is a matrix is due to Beurling—Deny [ 54 ]. For a proof of the full theorem which is not hard , see Simon [ ] or [ , Theorem 7. Simon [ ] 9. Indeed, our proof of 9. As with Theorem 9. For a proof, see the original papers or [ , Theorem 7.

As one might expect, the ideas in Kato [ ] have generated an enormous literature. Some papers on these ideas include Devinatz [ ], Eastham et al. There is a review of Kato [ ]. For applications to higher order elliptic operators, see Davies—Hinz [ ], Deng et al. Kato himself applied these ideas to complex valued potentials in three papers [ 73 , , ].

For more on this theme, see [ 7 , ]. In the s and s, there was work in which the quadratic form point of view was implicit but it was only in the s that forms became explicitly discussed objects and Kato was a major player in this development. Theorem We say that a quadratic form, q , is closable if and only if q has a closed extension.

What makes quadratic forms so powerful is that, in a sense, Example Here are two versions of this result:. Let q be a closed quadratic form. In his book [ ], Kato calls Theorem He puts Theorem I put Theorem One can show it is immediate from Theorem It is called the form sum. The proof is really simple. This allows a definition of a self-adjoint sum of any two positive self-adjoint operators.

There is a similar result for n arbitrary closed forms. The simplest proof is to use the Davies—Kato characterization below that closedness is equivalent to lower semicontinuity. There are closed symmetric operators which are not self-adjoint but every closed quadratic form is the form of a self-adjoint operator. Every symmetric operator has a smallest closed extension but there exist quadratic forms with no closed extensions.

A follow-up paper of Freudenthal [ ] did Friedrichs extension in something close to form language. We will need the following result of Simon [ ] see also [ , Theorem 7. The two monotone convergence theorems for positive quadratic forms are. For proofs, see [ , Theorem 7. This shows that in the decreasing case, the limit need not be closed or even closable. This material from the first edition was unchanged from the second edition. In , Robinson [ ] proved Theorem In , Davies [ ] also proved this theorem.

His proof relied on lower semicontinuity being equivalent to q being closed see below. When Kato published his second edition, he was clearly unaware of their work. The lower semicontinuity fits in nicely with even then well known work on variational problems that used the weak lower semicontinuity of Banach space norms so it was not surprising. Indeed Davies mentions it in passing in his paper without proof. At the time I wrote the preprint, I was unaware of the relevant work of Davies and Robinson although I knew each of them personally.

He stated a lovely result. For a proof, see [ , Theorem 7. I note that in precisely this context, Theorem Kato told me that he had no plans to publish his remark and approved my writing [ ] that explores consequences of Theorem In particular, he had the full Theorem In the Supplemental Notes, he quotes [ ] and [ ] but neither of the papers of Davies and Robinson, despite the fact that in response to their writing to me after the preprint, I added a Note Added in Proof to [ ] referencing their work. Kato called C the pseudo-Friedrichs extension.

Faris [ ] has a presentation that uses sesquilinear forms and makes this closer to the KLMN theorem. Herbst notes that this operator commutes with scaling, so after applying the Mellin transform, it commutes with translations and so, it is a convolution operator in Mellin transform space. The function it is convolution with is positive function so the norm is related to the computable integral of this explicit function.