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Focus on Mathematics

太鼓の形を聴く-幾何学的逆問題- 勝田篤

  Two techniques can be considered for investigating a certain object: if given that object, its properties can be studied one after another, and conversely, if a set of properties is presented beforehand, a search can be made for the object that possesses those properties.The former technique corresponds to a direct problem in the sense that output in the form of properties is obtained when given input, and the latter technique corresponds to an inverse problem in the sense that input is determined from output. For example, in the field of geometry that studies the shapes of objects, studying the curvature of individual objects is a direct problem while finding what objects satisfy the condition of having curvatureis an inverse problem. Here, we take up inverse problems and introduce, in particular, the problem given by Mark Kac, which can be described by the following slogan-likeexpression:

Can one hear the shape of a drum?

  This expression asks the question: “Given that the sound of a drum is generated by the vibration of its surface membrane and that membrane vibration can be broken down into characteristic vibrations, can the shape of that surface membrane be understood by understanding those vibrations (by listening to that sound)?” First, before we consider this question with respect to membrane vibration (two dimensional), we consider the simpler case of string vibration (one dimensional). As studied in high-school physics, the characteristic vibrations of a string fixed at both ends can be expressed as sine curves in which half of the period is an integer fraction of the length of the string.

sine curve 1

sine curve 2

sine curve 3

This is equivalent to the problem of searching for a non-zero solution u = u(x) to the following equation, which is called an eigenvalue problem. In this equation, string length is denoted as L.

Here, λ is called the eigenvalue of operator and u its eigenfunction. Kac’s problem can therefore be phrased as follows: “If the eigenvalue (or eigenvalue and eigenfunction) is known, can the shape of the string (meaning string length here, since it is length that determines the string) be understood? In this case, given that the length of the string is half of the longest period of the string’s characteristic vibrations, and since the period is 2π times the reciprocal of the square root of the eigenvalue, the shape of the string can be understood if the eigenvalue is known. In this sense, we can say that listening to the sound of a guitar would enable us to understand where the player’s fingers are pressing on the neck of the guitar. The above equation can be generalized to an eigenvalue problem in two dimensions in the following way, where Ω denotes a planar domain.

  Here, Δ is an operator (called the Laplacian) defined as follows:

The value of Δu at a certain point can be taken to bethe approximate difference between the average value of u at nearby points and the value of u at that point. As one interpretation of Kac’s problem, we first consider whether Ω can be determined from only eigenvalues. The information that can be understood from eigenvalues includes whether Ω is a circular disc, the area of Ω, and the circumference of Ω. However, as to whether Ω can be determined in general from only eigenvalues, the answer is no. For example, Chapman [1] gave the following example of two different domains having the same eigenvalues.

Chapman’s example

A number of similar examples are known, some of which include major Japanese contributions. At present, however, researchers are far from finding a solution to the general problem of finding a test as to whether a domain different from given domain Ω but having the same eigenvalues exists. The question therefore arises as to whether we can consider not just eigenvaluesas in Kac’s problem but eigenfunction information too, and as it turns out, it is relatively easy to determine domain Ω in such a case. Accordingly, Kac’s problem normally refers to the case of being given only eigenvalue information. Here, we introduce the Gelfand problem as a non-trivial problem on whether a domain can be determined on the basis of eigenvalue and Laplacian information. First, we consider the following eigenvalue problem in a form similar to the equations given above:

Here, denotes the differential in the direction orthogonal to the Ω boundary. Boundary conditions in the previous problem are called Dirichlet conditions while in this problem they are called Neumann conditions. The former corresponds to membrane vibration when boundary conditions are fixed and the latter to that when boundary conditions are free. It is known that the results obtained under Neumann conditions from only eigenvalue information are nearly the same as those under Dirichlet conditions. The Gelfand problem can be expressed as follows:

  Under Neumann conditions, can a domain be determined from eigenvalue and eigenfunction values at the boundary?

This problem is related to those that investigate the inner conditions of the earth, the human body, industrial products, etc. from surface information. In fact, in the 1990s, the above question was answered in the affirmative not only for domains but also for a Riemann manifold with boundary (curved space). However, this does not mean that the problem ends here. For example, only information that includes errors can be obtained from the surface of an actual substance, but to what extent can domain-related information still be obtained from such initial information? This is a stability problem, which is also important. It is also known that qualitative results can be obtained for this problem in the following form: “If initial information is sufficiently close to the real thing, the domain information so obtained is likewise close to the actual domain.”Here, though, researchers have yet to determine exactly what “sufficiently close” means. Since an inverse problem usually exists given a direct problem, the former is not restricted to certain fields. Nevertheless, much research on inverse problems is taking place out of practical necessity as described above.

  [1] S.J. Chapman,Drums that sound the same,Amer. Math. Monthly, 102(1995) 124-138.