CASEOLOGY

1. It is appropriate that we examine the history of Ken Case and the early years of transport theory in this setting because, quoting from the conference website: “La Fonda on the Plaza has it all. This landmark has been delighting travelers since the early 1920s when the original La Fonda was built on the oldest hotel corner in America. Records suggest that there has been an inn, or “fonda,” on this site for over 400 years, since the founding of Santa Fe in 1610.” CASEOLOGY

2. Paul F. Zweifel,1 Norman J. McCormick,2,* and Laurie H. Case3 1Virginia Tech, Blacksburg, VA 24062; zweifel@alumni.duke.edu 2University of Washington, Mechanical Engineering, Seattle, WA 98195-2600; mccor@uw.edu 3San Francisco Center for Psychoanalysis, San Francisco, CA 94103; lauriecasephd@gmail.com A BIOGRAPHICAL MEMOIR OF KEN CASE

3. Early Years Harvard and Los Alamos Institute for Advanced Study and the University of Rochester University of Michigan Rockefeller University and the Institute of Advanced Study Jason Retirement and UCSD Institute of Nonlinear Studies Outline of our ttsp submission

4. Harvard and Los Alamos Institute for Advanced Study and the University of Rochester University of Michigan Rockefeller University and the Institute of Advanced Study Outline of my presentation

5. Family name was changed from Cassoff to Case Taught lower level physics classes at Harvard Late 1943 went to Los Alamos with fellow student Fred de Hoffmann There met J Robert Oppenheimer Sometimes ate lunch with Klaus Fuchs His contributing to national security made lasting impression Harvard and Los Alamos

6. At Los Alamos shared office with Roy Glauber and worked on neutron diffusion and slowing down theory Calculated first atomic bomb test yield as 15 equivalent kilotons TNT---it was 20 but “exact as far as I am concerned” Returned to Harvard in 1945 and received PhD in 1948---from dissertation published only a 1.5 page letter to the editor of Physical Review

7. Why not a full length PR article? Probably because Case was scooped. In 1949 he published details of his calculation in “On nucleon moments and the neutron-electron interaction” (Phys. Rev. 76: 1—13) and in a footnote he acknowledged he received a preprint of Luttinger’swork prior to publication.

8. In 1949 at an APS meeting he presented “Case’s theorem” that asserted that two different interactions (so-called pseudoscalar and pseudovector couplings) gave the same result (to lowest order in perturbation theory) However M. Slotnickfound finite results for the pseudoscalar case, but infinite results for the pseudovectorcase INSTITUTE FOR ADVANCED STUDY AND THE UNIVERSITY OF ROCHESTER

9. Oppenheimer was in the audience and challenged Slotnick’s results as “violating Case’s theorem” Feynman also was present and that night he repeated Slotnick’s calculations using his new methods and proved, in more generality, that Slotnick was correct and challenged Case the next day after his talk The last paragraph of Case’s first paper in 1949 (Phys. Rev. 76:14—17) thus reads “Thanks are due to Dr. R. P. Feynman for pointing out an error in the original manuscript.” More on “Case’s Theorem”

10. “The difficulties arising in quantum mechanics when the potential is highly singular are considered. It is found that the Hamiltonian needs further specification in such cases. This may be done conveniently by requiring a fixed phase for the wave functions at the origin. A proof that all the well-known singular examples are amenable to this treatment is given. … This paper was cited at least 412 times “Singular potentials” (Phys. Rev. 80:797(1950)

11. Case argued that there is no intrinsic reason that neutrinos must have zero mass, which ultimately was shown to be true, and he clearly recognized that with maximal parity violation then neutrinoless double beta decay would be disfavored but not entirely disallowed if the neutrino has mass. This paper was cited at least 340 times “Reformulation of the Majorana Theory of the Neutrino” (phys. Rev. 107, 307 (1957)

12. Why did Ken leave “fundamental physics” for more applied problems? Perhaps the fundamental problems became too difficult for the old-fashioned Schwinger-Tomonagaapproach used by most physicists except Ken, who did not use the Feynman-Dyson (Feynman diagram) approach At the Institute of ADVANCED STUDY, continued

13. Beginning in 1951 Case produced some important fundamental work carried out in support of the experimental physicists at Michigan who were attempting an accurate measurement of the gyromagnetic ratio of the free electron In two papers he and a student provided an analytical basis of the experiment for the calculations they did UNIVERSITY OF MICHIGAN

14. Another paper (Phys. Rev.101:874—876 1956), on fundamental physics possibly may have been the spark that ignited the introduction of strangeness as a quantum number in particle physics, and eventually led to the classification of elementary particles by the group SU(3) UNIVERSITY OF MICHIGAN, CONTINUED

15. PHYSICAL REVIEW: 101, NO. 2, JANUAR Y 15, 1956 Strange Particles and the Conservation of Isotopic Spin* K. M. CASE, ROBERT KARPLUS, AND C. N. YANG Radiation Laboratory, University of California, Berkeley, California Abstract The question is considered as to whether complete rotational symmetry in isotopic spin space is necessary. In particular, the classification of elementary particles on the basis of the representations of a finite group is attempted. It is found that for the particles whose reactions are known, the law of conservation of charge results in a scheme essentially equivalent with ones previously proposed. However, some additional freedom is found which would accommodate particles with rather unusual properties if such are ever observed.

16. Case’s new modus operandi came in 1957, with the publication of Rev. Mod. Phys.29:651 (1957): “Transfer Problems and the Reciprocity Principle” In the Introduction to this elegant piece of work he states (with his Eq. (1) the linear transport equation): “TRANSFER PROBLEMS AND THE RECIPROCITY PRINCIPLE”

17. “While (1) is, without simplifying assumptions, exceedingly complex, a few general statements can be made. These concern the reciprocity principle and questions of uniqueness. The first of these is most important. Besides enabling us to compare different experimental situations and simplifying much of the mathematics, it shows, it will be seen, how apparently difficult problems can be solved by relating them to simpler ones. Unfortunately, even the most elegant proofs have been rather complex.” At this point there is a reference to Chandrasekhar’s book. “TRANSFER PROBLEMS AND THE RECIPROCITY PRINCIPLE”

18. Around 1957 Case also became interested in plasma waves, probably because the linearized Vlasov equation is almost identical in form to the linear neutron transport equation His paper on plasma oscillations in Ann. Phys. (N.Y.) 7:349—364 (1959) contains the paragraph: UNIVERSITY OF MICHIGAN, CONTINUED

19. “The initial value problem for an electronic plasma has been solved by twostrikingly different methods. Landau has given a solution using a Laplacetransform technique. Van Kampenhas solved the problem by means of anormal mode expansion. Since both approaches have some puzzling features, it is interesting to see the complete identity of the solutions. This is shown below using an orthogonality property which is proved. The results … indicate that many of the classical completeness and orthogonality theorems hold for quite pathological operators.” Case, K. M., Plasma oscillations. Ann. Phys. (N.Y.)7:349—364 (1959)

20. By 1957 Case’s interests largely turned to what might be called “transport theory” involving the Boltzmann equation. Case’s 1960 paper, “Elementary solutions of the transport equation and their applications,” Ann. Phys. (N.Y.)9:1—23. Case overcame the difficulties of the Weiner-Hopf approach by transferring the disjoint eigenvalues to the much more manageable interval [-1,1] plus isolated values outside that interval and easily solved the infinite-medium problem. UNIVERSITY OF MICHIGAN, CONTINUED

21. ANNALS OF PHYSICS 9, 1-23 (1960) Elementary Solutions of the Transport Equation and Their Applications K. M. CASE Department of Physics, The University of Michigan, Ann Arbor, Michigan A new method of treating problems involving the transport equation is discussed. Starting from Van Kampen’s observation that it is sufficient that “solutions” be distributions, the elementary solutions of the homogeneous equation are considered. These are found to have completeness and, in some cases, orthogonality properties which lead to the solution of more interesting problems by a conventional eigenfunction expansion. While the method is illustrated here with the simplest examples of neutron diffusion, it seems to be generally applicable.

22. He proved the “half-range completeness theorem” that asserted that any function defined on the “half range” (0 < μ < 1) could be expanded in the positive half the eigenmodes. He was able to prove what he called the “half-range completeness theorem.” He learned how to solve the resulting equations involving principal value integrals from Muskhelishvili’sSingular Integral Equations book (translated from Russian in 1953) “Elementary solutions of the transport equation and their applications”, continued

23. The half-range completeness theorem is far from intuitive: the analogue in the case of Fourier series would be that a function defined for positive x, say, could be expanded in terms of the positive Fourier components alone, which is not the case. Half-range orthogonality relations then were derived in 1964 by Ivan Kuščer et al. to expedite the solution of half-space problems. “Elementary solutions of the transport equation and their applications”, continued

24. By 1973 at least 150 papers directly utilized the eigenmode approach to solve problems, including many not tackled with the Chandrasekhar and Wiener-Hopf approaches. The 1960 paper has been cited at least 485 times. Ed Larsen has estimated the number of Ph.D. theses it has engendered as being in the hundreds. It is interesting that Case’s 1960 paper, only one of his 100+ publications, impacted linear transport theory so much out of proportion to the time he devoted to it. “Elementary solutions of the transport equation and their applications”, continued

25. Publication of this monograph greatly expanded the number of “transport theory practitioners.” The book has been cited at least 2,133 times. (By comparison, Boris Davison’s 1957 book on Neutron Transport Theory has been cited at least 1,074 times.) K M Case and P F Zweifel, linear transport theory, Addison-Wesley, 1967

26. One forward problem is the determination of the angular flux given the properties of a spatially homogeneous medium and the boundary and initial conditions A corresponding inverse problem is the determination of the properties of a spatially homogeneous medium given the incident and emerging angular fluxes and the initial conditions Examples of “Forward” and “inverse” problems

27. Ken extended Appendix H of Case and Zweifel’s book thatevaluates even-order integral moments of the densities everywhere in an infinite medium that arise from localized point- and plane-sources By directly integrating the transport equation he inferred the properties of the medium He introduced the concept of “inverse problem” to our transport theorycommunity “Inverse Problem in transport theory,” Phys. Fluids 16, 1607--1611 (1973)

28. His analytical technique is an “explicit” method, in contrast to a traditional “implicit” or iterative method involving, e.g., the method of steepest descent or conjugate gradient methods. Research by Larsen, Siewert, Sanchez, and others led to the solution of more realistic (e.g., half-space and slab) problems that require only measurements exterior to the medium. “Inverse Problem in transport theory,” continued

29. Ken’s paper, completed with only a small effort, is another example of his opening up the study of new transport problems. Subsequent research on analytical algorithms to solve various time-independent and time-dependent problems led to the publication of more than 50 papers. “Inverse Problem in transport theory,” continued

30. In 1969 he left Michigan for Rockefeller where he remained until he retired in 1988 Ken’s later publications usually were not in transport theory except for the following: Rockefeller University AND THE IAS

31. “For a simple model of a linearized plasma the normal modes are discussed for the case when an n fold degenerate eigenvalue is embedded in the continuum. It is found that for each such eigenvalue there are 2n generalized eigenfunctions and 2nadjoint functions. The set consisting of the continuum modes, and discrete modes of both complex and real eigenvalues are complete. Expansions are readily found using biorthogonality relations which are obtained.” “Plasma oscillations,” Phys. Fluids 21, 249 (1978)

32. Ken’s work on plasma oscillations is part of the “Van Kampen-Case-Siewert” method referred to in the Phys. Fluids B2, 1154 (1990) paper that uses of the continuous and discrete parts of the eigenvalue spectrum. The Uba and Subbaraopaper entitled "Intrinsic chaos in plasma waves" shows that "a single plasma wave in a hot plasma can give rise to fake heating effects among other quasilinear effects.“ “Plasma oscillations,” Phys. Fluids 21, 249 (1978)

33. One of his more highly cited papers (at least 49 times) was “Sum rules for zeros of polynomials. 1.” in J. Math. Phys. 21:702—708 (1980) Rockefeller University, Continued

34. Abstract It is shown that for polynomials satisfying differential equations of a particular form it is easy to generate sum rules for the powers of the zeros. All of the classical orthogonal polynomials are of this form. Examples are given for the Hermite, Laguerre, Tchebycheff, and Jacobi polynomials. In particular an explicit formula is given for the sums of all powers of Tchebycheff zeros. This same formula gives the sums for general Jacobi polynomials in the limit of large N. “Sum rules for zeros of polynomials. 1”

35. Beginning in 1960, a hush-hush organization of scientists (mainly theoretical physicists) was formed to meet for six weeks every summer in various locations to give the U.S. government advice on scientific aspects of defense matters Case was a member from 1961 until the early 2000s Membership in the group also was an intensely social experience for their families jason

36. A huge success of the Jasons was their development of so-called “adaptive optics” to compensate for turbulent atmospheric distortion. It improved the capability of detecting Soviet spy satellites by telescopes and is now used by astronomers to studyblack holes in the early universe Case's involvement with Jason was deeply important to him, both professionally and personally, due in part to his experience with the Manhattan Project Jason

37. Members of Jason reveled in their intellectual freedom to choose projects that interested them and to express their opinions to the Defense Department without censorship. Case felt that in this way Jason performed an invaluable service as a check on defense policy. jason

38. Case was involved in developing the so-called electronic barrier that was designed to halt the movement from north to south along the Ho Chi Minh trail A prototype of the electronic barrier was ultimately declared a failure but Case maintained it failed because it had not been implemented on a large-enough scale jason

39. He was interviewed by Science in which he proudly claimed that “The detector could hear a soldier peeing.” Later Case became one of about ten members of Jason with a top security Navy clearanceand began his work on understanding the action of soliton waves that fail to dissipate. jason

40. The work remains classified, but was part of a project aimed at understanding how to detect the presence of a nuclear submarine or, conversely, prevent such a submarine from being detected. Many of Case’s later papers were most likely inspired by work he carried out for Jason; a good example is “The N-soliton solution of the Benjamin-Ono equation,” Proc. Nat. Acad. Sci. USA 75:3562—3563 (1978). jason

41. Another example of work he carried out for Jason is “The N-soliton solution of the Benjamin-Ono equation,” Proc. Nat. Acad. Sci. USA75:3562—3563 (1978) Many of Ken and Pat’s closest friends were with Jason and for that reason they retired to La Jolla, CA, to be near them after Ken retired in 1988 jason

42. In retirement he returned to Rockefeller University for two months every year until 2002 He also served as an adjunct professor at the UCSD Institute of Nonlinear Studies He spent many happy hours gardening before his death on February 1, 2006 RETIREMENT

43. We thank E.G.D. Cohen, Mitchell Feigenbaum, and Nicola Khuri, colleagues of Ken’s at Rockefeller, as well as Ray Aronson, Noel Corngold, Norman Francis, Roy Glauber, Edward Larsen, the late EugenMerzbacher, and John Wilkerson for the inputs acknowledgments

44. I want the world to know what a good friend Ken was, and how kindly he treated me through the years, like the big brother I never had. Our families were often together, in Ann Arbor, Princeton, New York, Blacksburg, and La Jolla. Ken did everything to promote my professional career; without his support I would have had much less success. My acknowledgment to ken