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• 1
Electronic Resource
Springer
Communications in mathematical physics 29 (1973), S. 105-111
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract Two examples are presented: The first shows that a potentialV(x) can be in the limit circle case at ∞ even if the classical travel time to ∞ is infinite. The second shows thatV(x) can be in the limit point case at ∞ even though the classical travel time to infinity is finite. The first example illustrates the reflection of quantum waves at sharp steps. The second example illustrates the tunnel effect.
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We study the real, Euclidean, classical field equation $$(\mu ^2 + \Delta )\varphi + \lambda F(\varphi ) = f,\mu ^2 〉 0$$ where φ: ℝ d →ℝ is suitably small at infinity. We study existence and regularity assuming that λ≧0,F∈C ∞(ℝ), andaF(a)≧0∀a∈∝. These hypotheses allow strongly nonlinearF and nonunique solutions forf≠0. WhenF′≧0, we prove uniqueness, various contractivity properties, analytic dependence on the coupling constant λ, and differentiability in the external sourcef. For applications in the loop expansion in quantum field theory, it is useful to know that φ is in the Schwartz classL wheneverf is, and we provide a proof of this fact. The technical innovations of the problem lie in treating the noncompactness of R d , the strong nonlinearity ofF, and the polynomial weights in the seminorms definingL.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
ISSN: 1432-0673
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract The qualitative behavior of solutions of the mixed problem utt = Δu-a(x)ut in IR x Ω, u=0 on IR x ∂Ω, is studied in the case when a〉0 and Ω⊂IRn is bounded. Roughly speaking, if a≧amin〉0, then solutions decay at least as fast as exp t(ɛ −1/2amin), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If amax is sufficiently small, depending on Ω, then no overdamping occurs.
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• 4
Electronic Resource
Springer
Communications in mathematical physics 61 (1978), S. 149-168
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract The main theorem asserts that ifH=Δ+gV is a Schrödinger Hamiltonian with short rangeV, φεL compact 2 (IR3), andR〉0, then ‖exp(iHt)Π S φ‖ L 2 (|x|〈R)=O(t −1/2), ast→∞ where Π S is projection onto the orthogonal complement of the real eigenvectors ofH. For all but a discrete set ofg,O(t −1/2) may be replaced byO(t −3/2).
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• 5
Electronic Resource
Springer
Communications in mathematical physics 106 (1986), S. 481-484
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We show that for most non-scalar systems of conservation laws in dimension greater than one, one does not have BV estimates of the form $$\begin{gathered} \parallel \overline V u(\overline t )\parallel _{T.V.} \leqq F(\parallel \overline V u(0)\parallel _{T.V.} ), \hfill \\ F \in C(\mathbb{R}),F(0) = 0,F Lipshitzean at 0, \hfill \\ \end{gathered}$$ even for smooth solutions close to constants. Analogous estimates forL p norms $$\parallel u(\overline t ) - \overline u \parallel _{L^p } \leqq F(\parallel u(0) - \overline u \parallel _{L^p } ),p \ne 2$$ withF as above are also false. In one dimension such estimates are the backbone of the existing theory.
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• 6
Electronic Resource
Springer
Communications in mathematical physics 81 (1981), S. 203-227
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract The creation and propagation of jump discontinuities in the solutions of semilinear strictly hyperbolic systems is studied in the case where the initial data has a discrete set, {x i } i =1n , of jump discontinuities. LetS be the smallest closed set which satisfies: (i) S is a union of forward characteristics. (ii) S contains all the forward characteristics from the points {x i } i =1n . (iii) if two forward characteristics inS intersect, then all forward characteristics from the point of intersection lie inS. We prove that the singular support of the solution lies inS. We derive a sum law which gives a lower bound on the smoothness of the solution across forward characteristics from an intersection point. We prove a sufficient condition which guarantees that in many cases the lower bound is also an upper bound.
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• 7
Electronic Resource
Springer
Numerische Mathematik 51 (1987), S. 459-467
ISSN: 0945-3245
Keywords: AMS(MOS): 35L20 ; CR: G1.8
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary An estimate is derived for the error committed by the introduction of artificial boundaries and corresponding artificial boundary conditions when solving wave equations on unbounded domains. The estimate has two terms. One is proportional to the largest reflection coefficient for the artificial boundary condition, the maximum taken only on those rays which appear in the computation. The second term is proportional to 1/k wherek is a measure of the average frequency present in the solution.
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