$p$-Adic Pseudodifferential Operators: Methods and Applications

Various methods of the theory of $p$-adic pseudodifferential operators are discussed, such as constructing a basis of $p$-adic wavelets and constructing a large class of $p$-adic pseudodifferential operators that are diagonal in the basis of $p$-adic wavelets but cannot be diagonalized by the Fourier transform. The application of the methods of the theory of $p$-adic pseudodifferential operators to describing replica symmetry breaking in the replica method of the spin-glass theory is demonstrated.